PhD Thesis:- On the Singularity of the World: an Actual Counterpart Theory of Modality

On the Singularity of the World:-
An Actual-Counterpart Theory of Modality










Richard Murray Craven














A dissertation submitted to the University of Bristol in accordance with the requirements of the degree of Doctor of Philosophy in the Faculty of Arts



Wordcount:- 71, 520 words


Abstract

This dissertation consists in the motivation, formal development, and partial defence of a novel theory of alethic modality, Actual Counterpart Theory (act). act is counterpart-theoretical, analysing an individual a’s possible/necessary F-ness in terms of some/all of a’s counterparts being F. However, unlike the familiar Classical Counterpart Theory (cct) of David Lewis, act locates a’s counterparts, not in other possible worlds (pw’s), but rather in spatiotemporal regions of the actual world α.

The formal theory of act is provisionally based on the formalization of cct in [Lewis 1968], but with the following differences. α functions as the space of possibility in act, replacing the set W of pw’s, which occupies this role in cct. The quantifiers in act range over α-regions r instead of worlds w. Lastly, a privileged α-region, intuitively the spatiotemporal vicinity of utterance, which I call @, usurps α’s cct role as the locus of evaluation of claims.

The motivations for act accrue from its being:-
i. model-theoretical; ii. counterpart-theoretical; iii. not pw-theoretical;
iv. permissive towards actual counterparts.
These facts imply that act is e.g. extensional, compositional, immune to problems of identity across worlds, in possession of both a plausible epistemology and the safest and sanest ontology, and able to assert very many de re contingencies.

act is defended against two objections:-
a. act does not deliver a complete account of modal truth, because there are insufficient α-individuals to serve as counterparts.
b. The Actuality Objection [Hazen, M.Fara & Williamson]:- There is no correct translation into counterpart theory of formalized modal actuality claims.
Objection a is addressed by appealing to theoretical apparatuses and a plausible error-theory; Meyer’s [2012] solution to Objection b calls for an SQML logic with fixed quantifiers. This explains the provisional nature of the formalization of act.


Dedication & Acknowledgements

I dedicate this dissertation to my parents, Sir John Craven and Gillie Morris, to my children Sophie and Joe, to my brothers and sisters Nicola Phillimore, James Craven, Will Morris, Ben Craven, Caroline van Kuffeler and Tom Craven, to my step-father Peter Morris, to my step-mother Lady Jane Craven, and to the memory of my late step-mother Lady Ning-Ning Zhang Craven.

My gratitude to James Ladyman, who has most kindly acted as my supervisor/advisor ever since I began my MA all those years ago; and to Anthony Everett and Øystein Linnebo who have also generously undertaken an advisory role over the course of my PhD. All three have been unstinting with their time, insights and encouragement. I cannot thank them enough. My thanks also go to John Divers and Richard Pettigrew, who have kindly agreed to act as my PhD examiners. Both are recognized as experts of international stature in their respective fields. I am flattered and humbled that they have agreed to examine my case. 

Warm salutations to Tim Crane, who was my tutor at U.C.L nearly twenty years ago, and to Dominic Gregory, my undergraduate contemporary there, and since renowned as the author of outstanding work on modal epistemology and the a priori. My appreciation also to David Liggins, for his unfailingly warm and constructive approach to our discussions of the Autism Objection. I trust I have reciprocated in kind.

I acknowledge the friendship and encouragement of past and present fellow postgraduates and postdoctoral researchers at Bristol University, in particular:- Edit Talpsepp, Simone Duca, Ellen Clark, Zoe Drayson, Zoe Gilbert, Giulia Terzian Roly Perera, Justin Alam, Hannah Edwards, Alex Malpas, Chris Gifford, Cédric Paternotte, Matt Farr, Milena Ivanova, Tom Richardson, Haris Shekeris, Stavros Ioannides, Elena Pechlivanidis, Steve Horvath and Chris Clarke. 

Profound thanks to Susan Frost, and latterly Deborah Hughes, for their invaluable and invariably good-humoured help in all my dealings with University and general administrative issues. Thanks also to the many dozens of undergraduates who have participated in the seminars I have taught during my time here, for their enthusiasm, commitment and naive faith in me as a source of philosophical insight.

My especial thanks to:- Many Sok and Kim Thea, and their friends, who welcomed me into their circle during my sojourn in Paris, and whose friendship I will always value; to Henri Gallinon, Marion Vorms, Isabelle Drouet, Johannes Martens, and the other denizens of neighbouring offices at the Institut Jean Nicod.


author’s declaration









I declare that the work in this dissertation was carried out in accordance with the requirements of
the University's Regulations and Code of Practice for Research Degree Programmes and that it
has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate's own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of
the author.


SIGNED: ......................................... DATE:.......................... 


Table of Contents

Introduction 1
§ The Theory and Motivations for act 1
§ Dissertation Structure 4

ch1. Modalism and Possible Worlds 9
§1. Modalism 9
1.1. Modalism & QML 10
1.2. Alleged Theoretical Defects of Modalism 14
1.3. Alleged Expressive Deficit of Modalism 25
§2. PW Theory 32
2.1. Models and Worlds 33
2.2. Theory and Expressivity Revisited 38
Conclusion 49

ch2. Modal Realism 50
§1. Actualism 51
§2. Domain-Inclusion Actualism 52
2.1. DI-Actualism and the Barcan Formula 53
2.2. DI-Actualism With Fixed Domains 55
2.3. Against Proxy Actualism:- Metaphysical Queerness and
Converse Meinongianism 57
§3. Non-Domain-Inclusion Actualism 59
3.1. Representation and Formalism 60
3.2. Lewis vs. Ersatzism 61
3.3. Lewis on Ersatzism and Primitive Modality 61
3.4. Lewis on Ersatzism and Descriptive Power 64
§4. Possibilism 65
4.1. Absolute or Relative Actuality? 66
4.2. Transworld Identity? 68
4.3. The Best Theory Argument 70
§5. Against Modal Realism 72
5.1. Melia’s Refutation of the Best Theory Argument 72
5.2. The Actuality Objection 74
5.3. The Cardinality Objection(s) 75
5.4. The Epistemology Objection 83
Conclusion 90

ch3. Modal Anti-Realism 92
§1. PW-Theoretical Atheism 93
1.1. Atheism’s Modal Collapse 93
1.2. Atheism and Determinism 94
1.3. Atheism, Assertibility Gaps, and Subsidiary Norms 95
§2. Agnosticism 97
2.1. Moderate Agnosticism and Necessity/Impossibility 99
2.2. Actual Counterparts and Contingency 101
2.3. The Residual Assertibility Gap and Rational Dispensation 102
§3. Agnosticism Collapses Into ‘Weak Possibilism’ 106
§4. Actual Counterparts and the Demotivation of Agnosticism 107
§5. Actual Counterparts and the Redundancy of Worlds 110
Conclusion 113

ch4. Actual Counterpart Theory 114
§1. act 115
1.1. The Theory 117
1.2. The Translation 124
1.3 Necessity De Dicto and De Re 127
§2. act and Modalism 127
2.1. Necessity and Essence 128
2.2. Counterfactuals 129
2.3. Intensional Failure 130
2.4. Iteration 130
2.5. Modal Inference and Validity 132
§3. act and Subscripted-Operator Modalism 133
3.1. SO-Modalism:- a Recapitulation 134
3.2. Is SO-Modalism a Notational Variant of PW-Theory? 135
3.3. Is SO-Modalism a Notational Variant of Region-
Theory? 136
§4. act and Cardinality 136
4.1. act and FAL 137
4.2. act and Nolan 139
§5. act and Epistemology 139
Conclusion 141

ch5. Does act Deliver a Complete Account of Modal Truth? 143
§1. The Completeness Objection 144
§2. From Counterparts of Properties to Deeply Alien Properties 146
2.1. Implementing the C-Method 147
2.2. Why Property Counterparts Are Not Ad Hoc 148
2.3. The Residual Problem of Deeply Alien Properties 148
§3. From Inflationary Cosmology to Merely Metaphysical Possibilities 149
3.1. KOV’s Argument 150
3.2. But is Inflationary Cosmology True? 151
3.3. Liberal Inflationary Cosmology:- a Codicil 152
§4. Deeply Alien Possibilities, Merely Metaphysical Possibilities, and
Error Theories 152
4.1. Error Theory 154
§5. A Posteriori Necessity 155
5.1 Mathematical Truth and A Posteriori Necessity 155
5.2. Identity and A Posteriori Necessity 155
5.3. Material Constitution and A Posteriori Necessity 156
5.4. Dispositional Properties and A Posteriori Necessity 158
§6. Conceivability and Possibility 159
§7. A Posteriori Necessity and Two-Dimensionalism 162
7.1. Chalmers’s Taxonomy of Conceivability 165
7.2. Primary Possibility 166
7.3. Does Ideal Primary Positive Conceivability Really Entail
Primary Possibility? 166
7.4. Is X’s Primary Possibility Really a Metaphysical Possibility
of X? 166
Conclusion 167


ch6. Soundness:- An Actuality Objection 169
§1. Setting Up the Actuality Problem 170
1.1. Actuality, PW Theory and Counterpart Theory 171
1.2. Counterpart Theory, Non-Standard Scenarios, and the
Actuality Objection 172
1.3. An Overview of Five Translation Schemes 173
1.4. The Logic of Actuality 174
§2. Hazen vs. Lewis 176
§3. MFW vs. Lewis 177
3.1. Lewis and the Case of No Actual Counterparts 177
3.2. Lewis and the Case of Multiple Actual Counterparts 180
§4. MFW vs. Forbes 184
4.1. How tsf Works 184
4.2. tsf and the Case of Relational Predicates 185
4.3. A Pertinent Digression: act and the Necessity of Identity 186
4.4. A Less Pertinent Digression: Strong Necessity and the
Falsehood Convention 187
4.5. tsf and the Case of Monadic Predicates 188
§5. MFW vs. Ramachandran 188
5.1. tsr Translates Fa in Terms of & 189
5.2. tsr and the Case of Relational Predicates 189
5.3. tsr and the Case of Monadic Predicates 190
5.4. Three Actuality Problems for tsr 191
§6. The Symmetry Argument 193
§7. DGF’s Eternal Recurrence Objection 194
7.1. The Qualitative Problem and its Cheap Haecceitist Solution 195 as an AC Actuality Objection 197
§8. First Response:- Multiple Counterpart Relations 201
8.1. Lewis’s  Current Problem with His Own Body 202
8.2. Lewis’s Solution 204
8.3. Two Methods of Implementing Lewis’s Solution 204
8.4. DGF’s Criticisms 205
§9. Second Response:- Can A be Eliminated? 207
9.1. Meyer’s Antecedent Elimination Strategy – the Generalities 208
9.2. The Eliminability of A in Propositional Modal Logic 209
9.3. The Eliminability of A in Quantified Modal Logic 210
§10. Actuality, Meyer and act 215
10.1. Actuality and act 215
10.2 . Meyer and act 217
10.3. The Barcan Formula and Necessary Existence 218
Conclusion 223

Bibliography 236

List of Tables
Ch1 Tab. 1. Melia’s Model Q 36
Tab. 2. Modelling Collapsing Iteration 45
Tab. 3. BF Countermodel 48
Ch4 Tab. 1. cct vs act Interpretations of ms Elements 115
Ch6 Tab. 1. Hazen vs. tsl:- MC 177
Tab. 2 Disputed Sentences of QMLA 202


Introduction
Modality is the study of necessity, possibility, contingency and related notions. Modal philosophy addresses various questions about the metaphysics, ontology, semantics and epistemology of modality:- what things are necessary or possible or contingent, what their necessity, possibility or contingency consists in and what are the truth conditions of modal sentences, and the whether and how of the justification of modal belief. A modality M is described as alethic when M-necessity implies M-truth, and M-truth implies M-possibility. For instance, the metaphysical modality is, whereas the ethical modality is not, alethic.

The main body of this dissertation consists in the motivation and development of a novel theory of alethic modality, viz. Actual Counterpart Theory (act), and in the defence of act against some of the objections which are likely to be raised against it. This Introduction divides into two sections. §I.1 consists in a description of the theory and motivations of act,  and §I.2 explicates the structure of the remainder of this dissertation.


(I.1) The Theory and Motivations of act

As its name suggests, act is a counterpart theory of modality. That is to say, given an individual a and property F, act analyses the de re modal claim “a is possibly/necessarily F” in terms of some/all of a’s counterparts being F. However, act is not a traditional counterpart theory as exemplified by David Lewis’s Classical Counterpart Theory (cct) [Lewis 1968, 1986]. Traditional theories like cct are possible-world (pw)-theoretical, locating the counterparts of a in other worlds w, whereas act is region-theoretical, locating a’s counterparts in spatiotemporal regions r of the actual world α.

The factors motivating act fall under the following headings:- act is model-theoretical, act is counterpart-theoretical, act is not modal realist i.e. realist about other worlds, act is not pw-theoretical, act permits actual counterparts.

act is model-theoretical. Truth at the worlds of cct models is replaced by truth at the α-regions of act models. Hence, act confers many of the benefits associated with pw theory, despite not being itself pw-theoretical. For instance, just like a typical pw theory it delivers:-
* Extensionality, and thereby a compositional account of the truth conditions of modal sentences, which does not render mysterious the ability of ordinary speakers to understand new modal sentences.
* An unified account of counterfactuals and modality. This is a matter of adapting the Stalnaker-Lewis pw-account of counterfactuals, i.e. φ⎕→ψ is analysed in terms of the most similar φ α-regions being ψ α-regions.
* Interpretations of the meanings of iterated modal strings, given various constraints on accessibility between the α-regions of the model.
* Explanations of the validity or otherwise of various modal inferences, e.g. the validity of ⎕→◊φ is explained in terms of φ’s holding at all regions implying that φ holds at some regions.

act is counterpart-theoretical. Counterpart theory is motivated by its avoidance of the problems of transworld identity and “accidental intrinsics”, raised by Lewis [1986] against traditional non-counterpart-theoretical pw theories associated with Kripke. Consider Senator Humphrey, who intuitively is intrinsically five-fingered but possibly six-fingered. Kripkean pw-theory analyses this possibility in terms of Humphrey existing at a world w and being six-fingered there. This raises two problems. Firstly, if w has a concrete interpretation in the theory, being interpreted as a concrete world distinct from α, then it is hard to see how Humphrey can exist in distinct worlds. Secondly, the intrinsic nature of his five-fingeredness becomes accidental. In cct, worlds are conceived of as bounded spacetimes, which are mutually spatiotemporally unconnected and therefore non-overlapping. Given this architecture, cct solves both problems by the simple expedient of Humphrey not existing at more than one world. In act, regions are conceived of as bounded, but mutually spatiotemporally connected and therefore in some cases overlapping – Humphrey may exist at more than one region, but in any such region he – Humphrey! The Senator himself! – is five-fingered. His possible six-fingeredness is accounted for by the six-fingeredness of counterparts of his in regions of α. Thus, we have both that his intrinsic five-fingeredness is essential and that he is possibly six-fingered.

act is not modal realist. As a modal realist theory, besides running directly counter to the widespread intuition that other worlds do not exist, cct is vulnerable to a pressing objection from modal epistemology. act is of course in accordance with the sceptical intuition, and avoids the epistemological objection:-

The Objection from Epistemology charges modal realism with implying that modal belief is unjustifiable. In realist theories like cct, the objects of our modal beliefs occupy different bounded spacetimes from us, and we cannot justify beliefs about objects to which we are not related spatiotemporally. However, many of our modal beliefs most certainly are justified. Ergo modal realism is false.

act avoids the Epistemology Objection, because it makes the objects of our modal beliefs occupy the same bounded spacetime, viz. α. act also answers to the phenomenology of the acquisition of modal belief in a much more satisfying way than what is on offer from cct. Consider my modal belief that e.g. I could have been a dry-stone waller. As a matter of fact, I acquired this belief because the Stevenson brothers of my direct personal acquaintance are dry-stone wallers, and the job they do looks to me like something I could do. However, according to cct, my belief is not justified by anything to do with the Stevensons; instead, it’s justified by facts about my counterparts working as dry-stone wallers in other spacetimes, which are spatiotemporally unconnected with our spacetime α. On the act alternative, my belief that I could have been a dry-stone waller is justified, it is reasonable to suppose, much by the means of its acquisition, viz. observing the Stevensons doing something which appears within my capacities.

act permits actual counterparts. There now follows an argument against pw-semantics, which I call the P5 Argument, and which I believe to be novel. The cct postulate P5 [Lewis 1968] prohibits an individual having worldmate counterparts distinct from itself. I argue (ch3) that P5 is arbitrary, inconsistent with the purely qualitative nature of the counterpart relation as Lewis would have it, and takes its place amongst cct’s other postulates only as an ad hoc measure to preserve the modal relevance of possible worlds. Rescinding it opens up the prospect of our being able to express a very great deal of what we want to say regarding the modal, entirely without recourse to pw-theory with its undesirable ontology and implausible epistemology, having recourse only to the safest and sanest apparatus of all, sc. counterparts resident in regions of α. Hence, counterpart theorists are best advised to rescind P5 and abandon pw-semantics.

(I.2) Dissertation Structure
Overall, this work has a tripartite structure. Ch’s 1-3 are expository, ch4 presents the formal theory of act, and ch’s 5-6 defend act against a pair of important objections.

Ch1 focuses on the dialectic between non-pw-theoretical accounts of modality as exemplified by the modalism of Graeme Forbes, and pw-theory as the interpreted model theory of Kripke. In particular, I show how modalism, which aspires to deliver an account of modality with recourse only to the apparatus of quantified modal logic (QML), faces various theoretical worries as well as serious concerns about its expressive adequacy, in particular over modal actuality. In the second part of ch1, I show how pw theory avoids these worries.

Ch2 focuses on modal realists – pw-theorists who accept the existence of other worlds. These are distinguished into actualists, who believe that everything in the space of possibility actually exists, and possibilists, who believe that some things exist in the space of possibility but do not actually exist – the so-called ‘mere’ possibilia.
Actualism is shown to lead either to an ontologically over-laden theory called ‘proxy actualism’ [K.Bennett 2004, 2006], or to a representation-functional theory, familiar to us as linguistic ersatzism – famously criticized by Lewis [1986] on account of its unreduced modality and expressive inadequacy. 
Numbered among the possibilist theories is cct, but also Leibnizian possibilism, which claims that actuality is absolute rather than relative, as well as Kripkean possibilism which is based on transworld identity; continuing with the previous example, Senator Humphrey is possibly six-fingered in virtue of really being in w and being six-fingered there. cct claims to the contrary that actuality is indexical, and that there is no transworld identity; Humphrey’s counterpart rather than Humphrey himself is six-fingered at the other world. I argue that possibilists should adopt the cct approach. However, cct is then exposed to various objections, the most serious of which is the Epistemology Objection related above.

Ch3 concerns the anti-realist strand of the pw project. Modal anti-realism is scepticism about other worlds; either strong scepticism/‘atheism’ which accepts the non-existence of other worlds, or weak scepticism/’agnosticism’ which does not accept the existence of other worlds, but does not accept their non-existence either.
Atheism implies ‘modal collapse’ aka the denial of contingency. The proof: assuming atheism, if a is F simpliciter, then a is F in every world. But “a is F in every world” is the pw translation of “necessarily a is F”. So a being F simpliciter implies a being F necessarily, QED.
Agnosticism is of fairly recent provenance, having been developed in two papers by John Divers [2004, 2006]. Divers recognizes the existence of an “assertibility gap”: some modal claims asserted by the folk cannot be asserted by the agnostic. Divers appeals to various apparatuses in order to shrink the gap or mitigate its significance. Amongst these measures, crucially, is the rescinding of P5, and the exploitation of actual counterparts, in order to enable the assertion of a range of de re contingencies. I take this as the cue to instigate the P5 argument, and undertake the same actual counterpart policy on behalf of atheism, paving the way for the introduction of the formal theory of act in ch4.

The provisional formalization of act in ch4 is based on Lewis’s [1968] formalization of cct. That is to say, quantifier domains are variable as with cct, α itself takes the role played by the space of possibility in cct, spatiotemporal regions r of α take the role occupied by worlds w in cct, and a privileged region of α, which is intuitively the spatiotemporal vicinity of utterance and which I call @, takes α’s place as the locus of evaluation of claims. To restate an earlier point more explicitly, act differs from Kripkean pw theories and allies itself with cct insofar as it is a counterpart theory; and differs from cct and allies itself with Kripkean theories in permitting overlap. In the course of stating the formal theory, I investigate the benefits of introducing a second order relation of counterpart-hood between properties. The purpose of adopting this measure is to address the completeness problem in relation to ch5, alluded to in the paragraph following. This can be brought about, either by augmenting the list of primitive predicates of the theory, or by modifying the counterpart relation as traditionally conceived. Also included in ch4 are some specifications of act’s translation of counterfactuals, and brief prognoses concerning whether act translates certain modal principles as theorems.  The latter part of ch4 is for the most part concerned with demonstrating the successes of the formal theory in addressing the concerns raised against modalism and pw theory, in particular cct.

The concern of ch5 is with what is perhaps the obvious complaint against act, sc. that it fails to deliver a complete account of modal truth. In dispensing with the apparatus of worlds, act generates cases in which we intuit that an individual a is possibly F, however there are not enough actual individuals with the right sort of properties to serve as a’s counterparts.

I suggest two responses. Firstly, appealing to property counterparts as mentioned above results in only what I call deeply alien possibilities escaping capture. Secondly, pursuing an insight due to Knobe, Olum & Vilenkin [2006], I argue that the widely accepted theory of inflationary cosmology implies that only what I call merely metaphysical possibilities escape capture. These results considerably increase the plausibility of the error theory which I propose in order to account for the residue of deeply alien/merely metaphysical possibilities.
In the latter part of ch5, I argue that error theory is particularly appropriate as a means of disposal of beliefs corresponding to denials of a posteriori necessity (APN). I acknowledge that this treatment of APN’s involves refuting the corresponding conceivability-possibility arguments (CP’s) terminating in APN denials, and that it is not easy to see how considerations of APN block CP’s in non-APN cases. However, I argue that most non-APN cases can be disposed by renewing the earlier appeals to property counterparts or to inflationary cosmology. Finally, I defend error-theory about APN-denials  against a notorious objection of David Chalmers, which is based on two-dimensional semantics.

Ch6 addresses an objection against counterpart theory, which concerns modal actuality, what might have actually been the case. The Objection has early and later versions.
The early version [Hazen 1979] contends that there is no correct way of extending Lewis’s QML-counterpart theory translation scheme tsl, so as to translate modal actuality.
The later version [M.Fara & Williamson 2005] insists that there is no correct scheme at all for translating modal actuality from QML into counterpart theory. All schemes translate at least some inconsistent QML sentences into satisfiable sentences of counterpart theory.
Crucially, the problem arises specifically when an individual a at a world w has either no actual counterparts or more than one. Let A denote an actuality operator, such that the truth conditions of “AFa” are defined in terms of actual counterparts of a being F. Suppose a has both an F and a ¬F counterpart. Then depending on whether AFa is characterised in terms of some or of all of a’s counterparts being F, the result is that a comes out either both Aly F and Aly ¬F, or neither Aly F nor Aly ¬F. In short, the QML sentences to be generated from this are plainly inconsistent, and the pw-hypothesi plainly satisfied.
I canvass two possible solutions to the Objection:-
Firstly, Lewis has postulated multiple counterpart relations [1979, 1986 p230], which holds out the prospect of some unique individual always being the sole actual counterpart of a relative to any given context. However, as Delia Graff Fara has pointed out [2009] this requires A not to e.g. distribute over &, which is highly implausible.
Secondly, in a forthcoming paper, Ulrich Meyer proposes that A is eliminable in a Simplest Quantified Modal Logic (SQML) with fixed quantifiers, and that SQML regimentations of natural language avoid the Actuality Objection – inconsistent SQML sentences about modal actuality never translated to satisfiable sentences of counterpart theory. I show how the Hazen/Fara/Williamson problematic presses on act much as it presses on counterpart theory in general. Next, I demonstrate the applicability of Meyer’s solution in relation to act. Finally, I advert to a potential drawback of Meyer’s solution in respect of SQML’s fixed quantifiers, which lead to SQML validating the Barcan Formula and the thesis of Necessary Existence; I argue that act palliates the consequences of these validations more convincingly than do certain rival SQML proposals e.g. those of Plantinga, Linsky & Zalta and Williamson.



Chapter 1. Modalism and Possible Worlds


In this chapter, I survey firstly the dialectic between modalist and pw theories of modality, and secondly the intra-pw debate between the Kripkean and Lewisian (i.e. counterpart-theoretical) tendencies which between eachother have bestrode modal philosophy since the 1960’s.

In brief, like the Actual Counterpart Theory (act) which forms the central subject-matter of the present work, modalism purports to deliver a viable account of modality not involving reference to possible worlds or possibilia. The act theorist’s intention is to achieve a viable account of modality by exploiting the rescinding of Lewis’s 5th postulate prohibition of same-world counterparts [Lewis 1968], but doing so whilst retaining the extensional, model-theoretical aspect of pw-theory, as well as the counterpart-theoretical aspect of Lewis’s theory. In contrast, the modalist’s intention is to achieve a viable account of modality merely by exploiting the resources of a minimally enriched quantified modal logic (QML).

Modalism is widely acknowledged to be beset by several worries over its theoretical and expressive adequacy (q.v. §1). Its travails serve to motivate a pw account of modality (q.v. §2), since pw accounts are for the most part shown not to face these concerns.



(1.1) Modalism

Modalism agrees with act in claiming that an adequate account of modality can be achieved without reference to or quantification over possible worlds. However, modalism differs from act as to the means of attaining this Arcadia.  We will presently see, in ch4, how act applies Lewis’s counterpart apparatus to actual individuals, in order to deliver the de re component crucial to any viable account of modality. In contrast, modalism relies largely on the familiar ⎕ and ◊ of QML.

It is, as Melia remarks, fairly natural to think of modalism as
“closer to our normal conception of the modal than the philosopher’s apparently artificial apparatus of possible worlds and possibilia. From the very beginning it may have seemed that, although we had no difficulties in accepting that Joe could have been taller than Bruno ... it just seems wrong to treat [such] truths as involving other entities [Melia’s italics] ... The QML sentence ◊Rab is just right [ditto] about the logical form of the English sentence “Joe could have been taller than Bruno”. [2003 p82]
Nonetheless, as Melia goes on to reveal, modalism faces a compendium of theoretical and expressive concerns. These concerns do not generally appear devastating to modalism’s status as a viable account of modality. Nevertheless, a significant element in the overall motivation for the uptake of a model-theoretical or pw system is the fact that such systems are for the most part demonstrably free of the concerns besetting modalism. In §1.1 now following, I explicate QML and its uptake by modalism. In §§1.2-3, I address a selection of the theoretical and expressive issues faced by modalism.




(1.1.1) Modalism and QML
The QML with which I now treat is at bottom the familiar first order predicate calculus with identity esteemed by Quine, supplemented with the ⎕ and ◊ operators (very much less to Quine’s liking), the meanings of which are commonly apprehended as, respectively, “it is necessarily the case that ...” and “it is possibly the case that ...”. I now briefly recapitulate the construction of the first order predicate calculus.

The first order predicate calculus contains a basic vocabulary and some formation rules, determining which formulae of the language can be considered well-formed.

The basic vocabulary consists in:-
(a) the familiar connectives ¬,&,v,→,↔, denoting respectively negation, conjunction, disjunction, material implication, and biconditional material implication.
(b) terms; these being divided into names a,b,c..., a1,a2...an,b,b1... ; and variables x,x1,x2...xn,y,y1....
(c) predicate letters F,G,H...
(d) the quantifiers ∀ and ∃, whose respective meanings are “every” and “some”.
(e) the 2-ary identity predicate =.
(f) the brackets ( and ).

If the items just listed are considered as the expressions of the language, then the formation rules determine which formulae, consisting of strings of such letters, count as well-formed formulae (wff’s). The rules are needed in order to distinguish formulae such as (Fa ↔ ∀xFx), which is intuitively sensible, from manifest gibberish such as (a∀ v ↔ xF). Before setting out a formal definition of a wff, one further notion is required, namely that of an atomic formula. An atomic formula is a formula consisting of, for a number n, an n-ary predicate concatenated with n terms. For example, given our basic vocabulary, if F is a 2-ary predicate, then Fax is an atomic formula.

With the notion of an atomic formula in place, we are now ready to set down a formal definition of a wff in the first order predicate calculus. This proceeds as follows:-
(i) Every atomic formula is a wff.
(ii) If ϕ and ψ are wff’s, then ¬ϕ, ϕ & ψ, ϕ v ψ, ϕ→ψ, and ϕ↔ψ are all wff’s.
(iii) If ϕ is a wff and y is a variable, then ∀yϕ and ∃yϕ are both wff’s.
(iv) The only wff’s are formulae established as wff’s on the basis of (i) to (iii) above.

In order to generate QML from the traditional first order predicate calculus, we start by adding to the basic vocabulary listed above the following items:-
(g) the operators ⎕ and ◊.
(h) the 1-ary predicate letter E, whose meaning is “... exists”.

Unsurprisingly, the addition of new vocabulary requires additional formation rules and/or modifications to the existing rules, in order to determine the well-formedness or otherwise of formulae featuring the new vocabulary. To illustrate the issue, ⎕Fa is intuitively well-formed, whereas Fa⎕ looks like gibberish. Hence, ⎕ and ◊ are treated as operating in a manner grammatically analogous to ¬; just as putting ¬ in front of a wff generates a new wff per (ii) above, so putting ⎕ or ◊ in front of a wff generates a new wff.

With this understanding, we are now ready to set down a formal definition of a wff in QML. This involves, firstly, retaining each of rules (i) to (iii) above; in effect, all first order predicate wff’s are QML wff’s. Next, the analogy between ¬ and the new operators gives us:-
(iii’) If ϕ is a wff, then ⎕ϕ and ◊ϕ are both wff’s.
Next, the introduction of the new predicate letter E requires, for any name a or variable x
(iii’’) Ea and Ex are both wff’s
Lastly, (iv) above is replaced with the requisite modification:-
(iv’) The only wff’s are formulae established as wff’s on the basis of (i) to (iii), (iii’), or (iii’’) above.

A naive modalist, perhaps recycling the sentiment expressed in the passage of Melia’s quoted near the beginning of §1, might at this stage reckon on having all the apparatus required in order to deliver a viable account of modality. However, consider the matter of modal actuality, as exemplified in the sentence
1. There could have been things which do not actually exist.
The naive modalist who relies solely on QML appears to lack the means to express sentences such as 1. The obvious QML candidate for expressing 1 is
2. ◊∃x¬Ex.
But 2 expresses the absurd Meinongian sentence
3. There may be things which do not exist.
There is, as Melia points out [ibid. p83] no a priori reason compelling the modalist to repose all his trust in the resources of QML alone. The natural solution to the modal actuality problem involves introducing some new expression, which captures what is conveyed in the context of 1 and suchlike sentences by the English word ‘actually’. The technical fix, urged originally by Hazen [1976, 1979], involves supplementing QML with an actuality operator, A. A functions as, so to speak, a rigidifying operator, rigidly picking out what is actual or, when translated by means of pw-terminology, the actual world α.

In pw-terminology, prefixing A to a sentence S generates a sentence which is true at a world iff S is actually true. In general, pw-theory has the following semantics for A:-
Aφpw. Aφ is true at a world w iff φ is true at α.
Of course, modalists are generally motivated by disdain for the notion of quantifying over possible worlds, so they are likely to reject Aφpw. It is quite natural to envisage modalists analysing A homophonically instead, as per
Ahom. Aφ is true iff φ is actual.

An argument for adopting the homophonic approach is that doing so makes the semantics for A continuous, both with the standard homophonic semantics for the Boolean connectives and the first-order quantifiers, and with modalism’s already implicitly homophonic semantics for modality. Respective examples of which:-
&hom. (φ&ψ) is true iff φ and ψ.
∃hom. ∃xFx is true iff some x is F.
⎕hom. ⎕φ is true iff φ is necessary.

Call QML enriched with A QMLA. QMLA appears to have the resources to express 1, namely by means of
4. ◊∃xA¬Ex.

The foregoing indicates that the modalist has good reason to adopt QMLA. In the subsection immediately following, I investigate several theory-based concerns with modalism, which feature in the compendium of issues raised by Melia [2003]. In the succeeding subsection, I investigate one of the problems raised by Melia against QMLA-based modalism on grounds of its expressivity; specifically, this is the allegation that QMLA is incapable of expressing comparative modal actuality claims, such as
5. There could have been more things than there actually are.

(1.1.2) Alleged Theoretical Defects of Modalism
Over the course of his [2003 ch2-4], Melia discusses a fairly comprehensive list of potential problems for modalism, taking these as sources of motivation for the pw theories which he proceeds to investigate later in the same work . Amongst these are the following five:-
i. Modalism cannot easily capture the distinction between necessity and essence. 
ii. Modalism does not provide an unified account account of modality and the intuitively related notion of counterfactuals.
iii. The intensional character of the QML on which modalism is based renders mysterious – at least, to modalists – the ability of ordinary speakers to understand and use new modal sentences. 
iv. Modalism inherits QML’s liberal stance towards the iterability of modal operators. Thereby, it is committed to treating even immensely long iterated modal strings as meaningful, However, modalism cannot give an account of the meanings of these strings.
v. In taking the ⎕ and ◊ of QML as primitives not requiring semantic analysis, modalism affords no opportunity of testing for soundness and completeness the many competing systems of modal logic (out of S4, B, S5 etc), as the necessary precursor for deciding which of these systems is the correct logic of modality.

My findings are as follows. Firstly, modalism seems to have the resources to effect a fairly punctilious refutation of (i). Secondly, the modalist can defuse (ii)-(iv) mainly by appealing to homophonic analyses of modality, counterfactuals etc. However, the homophonic strategy raises a secondary worry, about the status of homophonic analysis as analysis – my contention being that this secondary worry is what faute de mieux warrants investigating a model-theoretical or pw-theoretical approach. Thirdly, (v) exerts a pressure against modalism which is not decisive against it, but which again seems sufficient to motivate investigating alternative approaches.

(i) Necessity/Essence
To say that there is a distinction between necessity and essence is to say that de dicto and de re necessity are not equivalent; i.e. for some individual a and property F, it is not the case that
6. “a is F” is necessarily true iff a is essentially F.
It is highly plausible that necessity and essence should be distinguished. Intuitively, there are many a’s and F’s for which 6 does not hold right-left, even supposing that it holds left-right. Some a is F essentially without “a is F” being necessarily true. The obvious way for this to happen is when contingent existences have essential properties. For instance, we human beings exist contingently and are essentially human. Indeed, contingency seems to be built into the standard understanding of what it is for a to be essentially F: a ceases to exist on ceasing to be F.

Melia’s complaint against modalism is then that it does not capture this natural distinction. It wrongly conflates a’s having F essentially with “a is F” being necessarily true. That things go wrong in this way for modalism is attributed to the formation rules of QML. The modalist would perhaps prefer to express
7. a is essentially F
by means of
8. a⎕F.
but is manifestly precluded from doing so by the formation rules for QML. The only viable alternative then seems to be
9. ⎕Fa
But surely, given the standard interpretation of ⎕, 9 is to be reserved for the purpose of expressing
10. “a is F” is necessarily true.

Nevertheless, it is easy enough to fall out of sympathy with this particular objection of Melia’s. It will seem to most people that modalism does have a way of capturing essentialism, in a way that defines it distinctly from necessity:-
ess. a is essentially F =def ⎕(Ea → Fa)
Indeed, several commentators have investigated allegations that modalism has trouble expressing anti-essentialist claims. It is puzzling that Melia does not mention the possibility of the modalist adopting ess, as ess involves no technical complication, and is in fact in standard use pretty much throughout the modal-theorist community. It may be that Melia has in mind Kit Fine’s [1994] argument against ess. Fine’s objection is that ess’s right-left direction does not hold. Consider F as the property of being a member of the singleton set {Socrates} ({s}) which has Socrates (s) as its sole member. Clearly, s is the only individual who is F under this description. Moreover, ess’s right hand side is true; it is necessarily the case that, if s exists, then s is a member of {s}. However, Fine argues, ess’s left hand side is not true; being a member of {Socrates} is not an essential property of Socrates. However:- 
* Firstly, Melia nowhere evinces scruples of a Finean nature, either explicitly or implicitly.
* Secondly, Fine’s objection is posed as the objection that modality does not capture the idea of essence – essence is not a modal notion at all. So, to the extent that Melia accepts Fine’s argument, it is unfair of him to single out modalism for its failure to capture the necessity/essence distinction. Fine considers that any modal theory will fail to capture the distinction.
* Thirdly, Fine’s assertion that being a member of {s} is an essential property of s is itself controversial. The intuition on which it is founded is not very clear-cut, and is likely just to be denied by robust defenders of the idea of essence as a modal notion.



(ii) Counterfactuals
The second theoretical concern is that modalism does not, qua account of modality, automatically account for the intuitively related notion of counterfactuals. To this end, the modalist must – disappointingly – introduce a new primitive connective. This objection does not aspire to be a knock-down refutation of modalism. However, there is inescapably comparison to be made between modalism and pw-theory which, as we presently see [§1.2.2] does provide a unified account of modality and counterfactuals, viz. in the terminology of worlds. So the worry over counterfactuals threatens to motivate a preference for pw-theory over modalism.

To explicate a little. A counterfactual is a sentence of the form
11. If it was the case that p, then it would be the case that q
 with the antecedent ‘(it is the case that) p’ assumed false. In 11, the word ‘would’ certainly appears to encapsulate something modal, perhaps in some sense ‘midway’ between the ‘could’ of ◊ and the ‘must’ of ⎕. At any rate, something non-indicative is afoot. So it is quite natural to suppose that any purported analysis of modality ought also to function with a minimum of adaptation as an analysis of counterfactuals. The charge is then that the modalist’s QML-based analysis of modality cannot easily be made to function as an analysis of counterfactuals. And indeed, modalism certainly seems to flounder. Its only possible candidates for rendering 11 seem to be the following:-
12a. p → ⎕q
12b. ⎕(p → q)
12c. p → ◊q
12d. ◊(p → q)
Manifestly, none of these four alternatives captures 11. In 11, the connection between p and q is stronger than the mere possibility allowed for in 12c and 12d, but weaker than the necessity insisted on in 12a and 12b; it says that q would be the case, and not that it either might or must be the case. In order to accommodate counterfactuals, modalists are put to the expedient of introducing a new primitive ⎕→, translating 11 by means of
12e. p ⎕→ q
But it is not clear what this achieves. It looks as though 12e, rather than analysing 11, let alone introducing us to whatever logic governs counterfactuals, merely introduces additional jargon. The comparison with pw-theory is stark. In pw theory, ◊p is analysed in terms of p’s truth at some possible world, and p⎕→q in terms of the nearest p-worlds being q-worlds. Unity of analysis. Enough, we may suppose, to motivate a preference for pw-theory over modalism.

The modalist will most likely seek to dispel this worry by readverting to the homophonic nature of his semantics. The analytical unity desired by the modalist’s critic seems to be a matter of ontological commitment; as we presently see, it is represented as a virtue of the pw approach, that it commits users of modal and counterfactual discourse only to one range of entia, viz. the worlds and their inhabitants. The homophonic modalist will translate ⎕ or ◊ or A as ‘necessarily’ or ‘possibly’ or ‘actually’, and ⎕→ as ‘if it was the case that _, then it would be the case that _’. It is not clear that his semantics commits its users to anything except to the additional propositional connective ⎕→. Moreover, even with the additional connective conceded as a cost to the theory, surely this cost is offset by the continuity of methodology inherent in the modalist’s adoption of QML and homophonic semantics; continuity, it should be emphasized, with the treatment of the Boolean connectives, as well as with that of the modal operators and A.

The homophonic response convincingly refutes any suggestion that the worry over counterfactuals amounts to a conclusive objection against modalism. However, themodalist’s reliance on homophonic semantics is itself likely to motivate a preference for model/pw-theoretical semantics. Such a preference will be motivated in particular for anyone who thinks that a desirable feature of the analyses of notions such as the modal and the counterfactual is that they be in some way reductive or ampliative or illuminating, and notes that homophonic analyses do not have this feature. By contrast, heterophonic analyses typically do have this feature. In particular, the pw analysis of counterfactuals and modality quite patently exhibits it in spades! Moreover, as we presently see, analogous considerations carry over to the cases of (iii) and (iv) below.

(iii) Intensionality
The third worry raised by Melia [ibid. ch3] concerns the intensional character of modalist analyses of modal sentences. The intensionality argument turns on the notions of extension and intension. These notions are explained as follows.

It is linguistic entities – names, predicates and sentences – which are apprehended as having extensions and intensions.
Thus:-
* the extension of a name is the object to which the name refers;
* the extension of a predicate is the set of objects satisfying the predicate;
* the extension of a sentence is the sentence’s truth value.
Hence:-
* the extension of ‘Obama’ is Obama;
* the extension of ‘born in Hawaii’ is the set of native-born Hawaiians;
* the extension of ‘Obama was born in Hawaii’ is the true.
In this context, to describe a discourse as extensional is to say that substitution of co-extensional names, predicates and sentences preserves truth value. Moreover, if a discourse is not extensional, then it is intensional.

That the modalist’s QML-based language is intensional is easily shown. If it was extensional then, by the foregoing definition, for any co-extensional pair of sentences ϕ and ψ, ⎕ϕ would have the same truth value as ⎕ψ. Notoriously, however, this is often not the case. To see this, consider the pair of sentences:-
13. Quine wrote Word & Object
and
14. 2+2=4
Both sentences are true, and are therefore co-extensional. However, substituting one for the other in modal contexts does not preserve truth value. For
15. ⎕Quine wrote Word & Object
is false, since Quine might have eloped with a travelling circus rather than pursuing a career in philosophy. By contrast,
16. ⎕2+2=4
is true if anything is.

It can be difficult to understand the precise nature of the problem posed by modalism’s intensionality. It is sometimes parsed as the complaint that modalism’s intensionality threatens to commit modalism to the non-compositionality of modal sentences – a language being compositional when the meaning of a sentence of the language depends upon the meanings of its parts. The implications of the intensionality→non-compositionality argument are pretty stark, having both metaphysical and epistemic elements:-
(a) No systematic truth conditions for modal sentences; and
(b) The patent ability of ordinary speakers to understand and use new modal sentences viewed as uncanny and inexplicable. 

However, under this interpretation the complaint is unsound. As Peacocke [1978] shows, the modalist’s homophonic semantics is precisely an example of a semantics which is intensional yet compositional. Indeed, it might occur with the benefit of hindsight to ask how much substance there was in the first place underlying the intuition that, for some φ and ψ, just because ⎕φ and ⎕ψ do not share the same truth value whereas φ and ψ do share the same truth value, therefore the meaning of ⎕φ/ψ cannot be determined from the meanings of ⎕ and φ/ψ.

Not withstanding Peacocke’s rebuttal of the intensionality→non-compositionality version of the complaint, there is yet some residual savour to the complaint, relating to its epistemic element (b) above. Even if the meaning of ⎕φ/ψ can be determined from the meanings of ⎕ and φ/ψ, there is still some mystery about how ordinary users know how to use modal sentences correctly – are able to say quite so readily which ones are true etc – given that, for some φ and ψ, ⎕φ and ⎕ψ do not share the same truth value whereas φ and ψ do share the same truth value. Although this consideration does not provide the basis for anything amounting to a conclusive refutation of modalism, nevertheless it is taken later in the present chapter to motivate the investigation of pw-theory, this latter being extensional.


(iv) Iteration
The fourth theoretical worry [Melia ibid. pp27-30] concerns the account that modalists give of the meanings of iterated modal formulae – formulae featuring iterations of the modal operators ⎕ and ◊. The problem is that modalism is committed to the meaningfulness of iterated modal formulae, yet this commitment sits awkwardly with the intuition in some quarters that they are meaningless.

To describe an operator • as iterable is to say that, if a formula •ϕ is well-formed, then ••ϕ is well-formed. The formation rules for QML have as a consequence not only the iterability of so-to-speak pure formulae – formulae featuring only ⎕’s or only ◊’s, but also the iterability of mixed formulae - formulae featuring both ⎕’s and ◊’s. This is easily shown. Let ϕ be a wff. Then, by the rules, ⎕ϕ and ◊ϕ are both wff’s. Now let ϕ be ⎕ψ. This allows us to derive ⎕⎕ψ and ◊⎕ψ as wff’s. Analogous procedures give us ◊◊ζ and ⎕◊ζ, and any number and combination of further iterations. In short, QML treats as well-formed strings of any finite length.

And therein lies the problem. As Melia complains,
“even a single iteration of the modal operators produces a sentence that is baffling. While we may have fairly firm beliefs about whether or not P is necessary, many of us find ourselves at a complete loss when wondering whether or not P is necessarily necessary. Intuitively, it may not even be clear to us whether such a proposition makes sense.” [ibid. p28]
Moreover, if things are this bad for single iterations, so much the worse when it comes to “strange beasts” [p27] such as e.g. ◊◊⎕⎕⎕◊⎕⎕◊◊⎕◊◊⎕◊◊ϕ.

One response, investigated by Melia [pp28-29], is to adjust the rules for wff’s so as to reflect the intuition that iterated modal formulae are nonsensical. A natural way of effecting this adjustment is to say that ⎕ϕ and ◊ϕ are to count as wff’s only if ϕ itself contains no modal operators. However, as Melia points out, adopting this rule threatens modalism’s expressive adequacy. In particular, the modalist will no longer be able to express plausible theses about the modal properties of merely possible objects, since to formulate such theses requires modal operators to appear within the scope of other modal operators. I illustrate Melia’s point with an example from later in the present work. It seems plausible that Obama could have had another daughter who – this very woman! – might have been a florist. The natural formulation of this thesis, with D, o and F given their expected meanings, is:
17.  ◊∃x(Dxo & ◊Fx)
In 17, the fragment after the first  ◊ contains a modal operator – the second ◊. So, given the adjusted rule, 17 counts as ill-formed. Melia summarises current proceedings in the form of a dilemma:-
“If we allow the modal operators … syntactic freedom, then we generate sentences that are almost impossible to understand or assess, and that, we fear, may be nothing more than nonsense. Yet we must be careful not to restrict these operators too far, else we will find our language unable to express natural modal theses.” [ibid. p29]

The modalist will defend himself against this objection by adverting once more to Peacocke’s [1978], arguing that this provides a perfectly good understanding of iterated modal sentences, much as the standard homophonic semantics for first-order quantifiers provides a perfectly good understanding of iterated quantifier sentences.

The modalist’s opponent may yet motivate a pw-based approach on pragmatic grounds, even if the blow thereby inflicted on modalism is not fatal. Historically speaking, attempts to make sense of modal iteration have revolved around the axiomatisation of systems of logic (e.g. S4, B, S5, etc). Such systems have tended to make sense of modal iteration by reduction, i.e. by asserting equivalences between long strings and much shorter strings, typically either single iterations or single-operator formulae. As is argued in (v) immediately following, pw theory, with its apparatus of worlds and relation of accessibility between worlds, affords a good understanding of what must be the case for such reductive equivalences to obtain. For instance, ◊◊⎕⎕⎕◊⎕⎕◊◊⎕◊◊⎕◊◊ϕ is equivalent to ◊ϕ as per S5, only if all worlds are mutually accessible. There can be no doubt that the method of reduction makes understanding iteration just plain easier. In contrast to the pw-theorist, the modalist entirely lacks any basis for reducing modal iteration, being wedded to the cumbersome homophonic analysis of ⎕ and ◊.

(v) Modal Inference and Validity
The fifth, and final theoretical objection against modalism is that, unlike pw theory in particular, it leaves uninterpreted the ⎕ and ◊ of modal logic. As a result, we gain no purchase on the question of what are to be understood as the correct theorems, axioms, and rules governing inference and validity in modal logic. In contrast, pw theory does interpret ⎕ and ◊, in terms of truth at respectively all and some possible worlds, enabling at least some purchase on the question in question. For instance, ⎕φ→◊φ is readily explained, in terms of φ’s holding at all worlds entailing φ’s holding at some worlds

For modalists, with QML uninterpreted and taken as basic, the logic of modality presents formidable obstacles to comprehension and formalization. To some extent, intuitions may be relied upon; either of the weak systems K and T of modal logic alluded to below may lay some claim to be thought of as intuition-conserving logics. However, there remains a large number of candidate theorems, axioms and rules in respect of whether to include which our intuitions give out. These difficulties are reflected in the existence of a plethora of systems of logic (K, T, S4, B and S5 being perhaps the best known candidates), distinguished by their inclusions of different axioms and theorems, and each vying to be recognized as the correct logic of modality. In fairness, it cannot be pretended that any given modal theory, even the pw theories of Kripke and Lewis, comes close to solving these difficulties to the satisfaction of all concerned. The criticism of modalism is that, in leaving ⎕ and ◊ uninterpreted, it does not even provide the opportunity to understand the different systems, let alone to test them for soundness and completeness, as the necessary precursor to the eventual selection of one to be treated as the correct logic of modality.

Although modal logic presents the difficulties just described, this is not to deny that at least some modal axioms and theorems are reasonably uncontentious. Many of these latter are collected within the weak system K of modal logic. K includes all the tautologies of propositional logic, plus the necessitation rule and the distribution axiom which are, respectively:-
Nec.  ϕ is a theorem → ⎕ϕ is a theorem
Dist. ⎕(ϕ→ψ) → (⎕ϕ→⎕ψ)

However, K is almost universally acknowledged as being too weak to provide an adequate account of modality. Most obviously, an adequate account of alethic modality will validate the reflexivity axiom:-
Ref. ⎕ϕ → ϕ.
But Ref is not provable within K.

What has become known as the system T (or M; but T hereafter, for no particular reason) is the system K supplemented with Ref. However, T is also commonly acknowledged as being too weak. It has nothing to say about whether or not to include the Barcan formula and/or its converse:-
BF. ◊∃xϕx → ∃x◊ϕx
CBF. ∃x◊ϕx → ◊∃xϕx
Moreover, a nice result would be if the transitivity axiom
Trans. ⎕ϕ → ⎕⎕ϕ
was found to be correct, as this would greatly reduce the iteration difficulty raised earlier.

The modal logic S4, which is the system T supplemented with Trans, validates the equivalence of ⎕⎕ϕ with ⎕ϕ, and of ◊◊ϕ with ◊ϕ, which amounts to the claim that iteration of strings composed purely of ⎕’s or of ◊’s is superfluous.

The modal logic B is the system T supplemented with the symmetry axiom
Sym. ϕ → €◊ϕ
It would be a very good result if both Trans and Sym were found to be correct, because this would entirely solve the iteration difficulty. The strong modal logic S5, which is T supplemented with both Trans and Sym, amounts to the claim that an impure string, i.e. an iterated string composed of both €’s and ◊’s, is equivalent to the last operator in the string; for instance, ◊⎕◊◊⎕◊⎕◊⎕⎕⎕◊⎕⎕◊◊⎕◊◊⎕◊◊ϕ is equivalent to ◊ϕ. In other words, S5 entails that even mixed iteration is superfluous.

The trouble with T is that it includes neither Trans nor Sym. And once we move beyond T, as we are impelled to by our recognition of its inadequacies, our intuition loses its power to guide us in choosing the right modal logic – whether this be B, S4, S5 or some other less well-known system. This might not matter if we at least felt that there was some purchase on the various arguments for and against the problematic formulae of the previous paragraph. To anticipate somewhat, we tend to get such purchase from pw systems because, unlike modalism, they interpret the ⎕ and ◊ of modal logic. Although we may never find ourselves in the position to say for certain whether or not e.g. ◊ϕ entails ⎕◊ϕ, at least the worldly systems provide us with a formal semantics against which we can test the candidate systems (T, S4, B, S5 etc) for soundness and completeness; for example, Sym is validated in B and S5 but not in S4. In contrast, modalism does not interpret ⎕ and ◊, affording us no opportunity to test the candidate systems.

So concludes the enumeration of five theoretical objections against modalism. I turn now to another kind of objection against modalism, an allegation of expressive rather than theoretical inadequacy.

(1.1.3) Alleged Expressive Deficit of Modalism
Consider an analysand discourse δ featuring a number of sentences s1...sn, with a candidate analysing theory T. T is said to be expressively inadequate or deficient with respect to δ iff, for some si:1≤i≤n, T does not analyse si. It is objected that modalism is expressively inadequate, that there are some sentences of natural modal discourse for which modalism fails to produce an analysis. A precis of what presently follows:- Firstly, Hazen and Melia each raise a problem to do with modal actuality. Secondly, Forbes shows that even more complex modal actuality sentences, including the comparative sentences cited by Melia, can be accommodated within a QML enriched with a denumerable number of subscripted operators. Finally, Melia complains, fairly persuasively in my view, that QML thus enriched is a mere notational variant of the worldly systems, rather than a genuinely competing system.

(i) Hazen: Modal Actuality
Hazen’s [1976] consists in a comparison of the expressivity results of the following artificial languages:-
a. A ‘weak’ modal language, that is to say a modal language accommodating the expression of essentialist claims, that is to say a QML in which ⎕Fa is consistent with ◊¬∃x x=a;
b. A ‘strong’ modal language, that is to say a modal language not accommodating essentialism, that is to say a QML in which ⎕Fa entails ⎕∃x x=a;
c. A modal language enriched with A, in effect QMLA;
d. A first order language quantifying over worlds. For purposes of simplification, this pw language is Kripkean rather than Lewisian, in the sense that it allows for transworld identities and does not invoke a counterpart relation.

Briefly, Hazen finds the expressivity of strong QML superior to that of weak QML, and the expressivity of QMLA superior to that of strong QML, but all three modal languages expressively inferior to the pw language. In particular, A can be added in to facilitate the expression – within QMLA now – of simple modal actuality sentences. To recapitulate an earlier prognosis (§1.1),
1. There could have been things which do not actually exist
is adequately rendered by means of
4. ◊∃xA¬Ex.

Nevertheless, according to Hazen, other sentences resist expression within QMLA. He shows this by investigating certain sentences, the expression of which within a QML language requires the language’s enrichment with what he calls outer quantifiers, and then showing that some sentences lie beyond the expressive reach even of QMLA with outer quantifiers.  Outer quantifiers are quantifiers <∀x> and <∃x> which range over the domain D of individuals of W, the set of all possible worlds. In contrast, inner quantifiers, which Hazen takes to be the natural quantifiers of ordinary modal discourse, range only over the domain d(w) of individuals in the world w from the point of view of which (modal) claims are evaluated.  Hazen shows how outer quantifiers can be put to use in expressing a sentence about two worlds, each containing an individual which does not exist in the other world:-
18. <∃x><∃x’>(◊(Ex & ¬Ex’) & ◊(Ex’ & ¬Ex)).
However, QMLA even with outer quantifiers is incapable, according to Hazen, of rendering a sentence about there being some world such that every world contains an individual contained in that world, even though such a sentence is rendered quite straightforwardly in a worldly language:-
19. ∃w1∀w∃x(Ixw & Ixw1)

(ii) Melia: Modal Comparative Actuality
Melia grants that QMLA captures simple modal actuality sentences like 1 by means of 4; but complains that more complex modal actuality sentences, in particular those involving modally-embedded comparatives, continue to elude the modalist’s expressive grasp. Melia’s example of one such recalcitrant sentence is, as before,
5. There could have been more things than there actually are.

To begin with,
4. ◊∃xA¬Ex
 obviously fails to capture 5. 4 is satisfied by a world containing just one non-actual individual all alone, in which case there would be fewer things than there actually are. We need a QML sentence which captures the idea of there being a world w which contains both all the actual individuals and further individuals besides. To this purpose, Melia then runs through two further QMLA candidates. These are 5a and 5b:-

5a. ◊(∀x(AEx → Ex) & (∃y¬AEy))
Sentence 5a says that it might have been the case that all the actually existing things existed simpliciter, and there also existed some mere possibilia. But this will not do. Tellingly, perhaps, the easiest way to see why not is in terms of possible worlds. 5a is satisfied by a world w distinct from α, containing just one of α’s actual inhabitants in company with a merely possible individual, i.e. who inhabits w but not α. The problem is that, because it falls within the scope of ◊, the ∀ occurring in 5a ranges only over the inhabitants of the world introduced by ◊, sc. w. And this has the effect in turn of further rigidifying the embedded A, so that now only actual individuals who also inhabit w are taken into account. Grant that all the inhabitants of this world who also happen to inhabit α exist simpliciter; then, even if there is only one such dual-worlder, 5a is satisfied by the addition of just one merely possible individual. But this does not capture 5.

5b. ◊((⎕∀x(AEx → Ex)) & ∃y¬AEy)
Since the problem was with ∀ falling within the scope of a modal operator, some means of releasing ∀ from its bondage must be found, in order to allow it to range over all the actual individuals in the way required to capture 5. Sentence 5b attempts to do this by modifying 5a, inserting a ⎕ immediately to the left of the ∀. Thus, 5b says that it might have been that necessarily all the actually existing things existed simpliciter, and there also existed some mere possibilia. Where ∀ occurs at the leftmost end of a string, i.e. outside the scope of a modal operator, the effect of inserting ⎕ immediately to the left of ∀ is functionally equivalent to the effect of introducing the outer operators alluded to by Hazen; not only the individuals in d(w) come into consideration; all the merely possible individuals are taken into account as well. Where ∀ falls within the scope of ◊ as in the case of 5a, the effect of inserting ⎕ immediately to the left of ∀ is to restrict the domain of individuals within the scope of ⎕. No longer extending its scope over everything in D as previously, ⎕ now picks out only the possibilia in the worlds accessible from w. Thus 5b is satisfied by a world w2 such that, at all the worlds w accessible from w2 all the individuals in w2 who also happen to inhabit α exist simpliciter, and some merely possible individual also inhabits w. But this still says nothing about how many dual-worlders inhabit w. Indeed, why should it? Widening the domain of consideration, from the individuals in w to the individuals in the worlds accessible from w, still does nothing to ensure that all the individuals in α are also in w. Thus, again, no sense has been captured of a possible case of there being more things than actually exist.

(iii) Forbes:- Subscripted Operators?
Forbes, who along with Peacocke is fairly described as one of modalism’s principal apologists, has accepted the failures of 5a and 5b. He has responded to the expressive difficulties highlighted by Melia et al, by enriching QML with denumerably more operators [Forbes 1989]. Thus, in addition to the standard QMLA operators ⎕, ◊ and A, Forbes introduces, for each number n, the operators ⎕n, ◊n and An.

The standard actuality operator A and the subscripted actuality operator An can be contrasted using the ‘rigid/non-rigid’ terminology coined by Kripke during the course of his investigation of the referring behaviour of names. The standard actuality operator is so to speak rigid, in that in all contexts it picks out us inhabitants of α. In contrast, subscripted actuality operators are non-rigid. Each individual subscripted actuality operator picks out the inhabitants of the world introduced by the possibility operator bearing the corresponding subscript.

To anticipate a little, the introduction of these subscripted operators enables 5 to be rendered in the following way
5c. ◊1((⎕∀x(AEx → A1Ex)) & ∃y¬AEy)
5c says that it might have been the case that necessarily, all the actual things – all the things that really are in this world – actually existed, and there also existed some mere possibilia. Melia accepts that 5c captures the sense of 5 [ibid. p90], and is right to do so. This is because the subscripted operator An has the effect, returning to pw-terminology, of causing us to evaluate what is actually the case from the point of view of the world introduced by the correspondingly subscripted ⎕n or ◊n, no matter what other embedments the An in question is subjected to. Thus 5c is satisfied only by a world containing all the actual individuals contained in all the worlds; which is the required result.

Although all parties agree that 5c captures 5, Melia objects that the modalist ought to give up the pretence that subscripted-operator modalism constitutes a genuine account of modality in its own right. That is, he doubts that it counts as a genuine competitor against the worldly theories of Kripke and Lewis in particular, rather than a mere notational variant of (one of) them [ibid. pp92-97].

The source of Melia’s objection is the striking isomorphy, in terms of grammar and syntactic structure, between the QML sentences of subscripted-operator modalism and the corresponding sentences of pw theory. To revert to the recent example,
5. There could have been more things than there actually are
has the agreed subscripted-operator-modalism translation
5c. ◊1((⎕∀x(AEx → A1Ex)) & ∃y¬AEy)
and has the obvious pw-theoretical translation
5d. ∃w1((∀w∀x(Exα → Exw1)) & ∃y¬Eyα)
Sentences 5c and 5d are palpably isomorphic with regard to their respective grammars and syntactic structures. 5d can be obtained from 5c by straightforward substitution of terms, without any change in the connectives: working from left-to-right, ∃w1 replaces ◊1, ∀w replaces ⎕, ∀x replaces ∀x, Exα replaces AEx, and so on. As Melia remarks [ibid. p93], a linguist who came across a tribe who used boxes and diamonds in this way would be “strongly tempted to conclude” that the tribe in question were using a language which was a notational variant of the possible-world language, rather than a genuine ontological reduction thereof.

This is to say that we should expect ontological reduction by paraphrase to be revelatory of differences in structure and grammar, between subject sentence and intended paraphrase. Certainly, paraphrastic ontological reduction is revelatory of such differences in other cases. As Melia shows, two examples of this phenomenon are numerical quantification and “the average X” phrases:-

* In the case of numerical quantification, consider
20. Two is the number of the sheep on my waterbed.
Sentence 20 bears, it might be supposed, an undesirable commitment to Platonism about numbers. The obvious reducing paraphrase of 20 is
21. ∃x∃y∀z(x is a sheep on my waterbed & y is a sheep on my waterbed & ¬x=y & (z is a sheep on my waterbed → (z=x v z=y)))
Sentences 20 and 21 are palpably non-isomorphic with regard to their grammars and structures, so it seems safe to count 21 as a genuine ontological reduction of 20.

* Average X’s become ontologically problematic in a way that is very obvious when fractions are involved. Melia’s example:
22. The average woman has 2.4 children.
Going by its surface grammar, 22 bears undesirable commitments to average women and fractional children. The obvious way of ontologically reducing 22 is
23. The number of children divided by the number of women is 2.4.
Again, sentences 22 and 23 are palpably non-isomorphic, so it seems safe to count 23 as a genuine ontological reduction of 22.

To summarise, the two cases just presented indicate that grammatical and structural difference are the typical concomitants of ontological reduction by paraphrase iff there genuinely is ontological reduction; Melia’s ‘linguist’ thought experiment indicates that grammatical and structural isomorphy can be considered as grounds for inferring mere notational variance; and 5c and 5d are revelatory of isomorphy between a modalist language enriched with subscipted operators and (one of the) pw-theoretical languages. It is then natural to conclude that subscripted-operator modalism only avoids these actuality-related difficulties with expressivity, at the cost of its integrity as a theory distinct from the worldly theories it supposedly reduces. It is also worth noting note that all these new operators have to be taken as primitive by the modalist.  A sparse ontology is purchased at the price of an inflated 'ideology,' as Quine might well put it.

This concludes the investigation of modalism. Modalism has been found vulnerable to a range of theoretical and expressive objections. It has not been my purpose to argue that any of these objections is singly conclusive against modalism. Nevertheless, viewed collectively, they seem formidable. Indeed, I take it that a substantial factor in the motivation of the pw-theoretical accounts to which I now turn, is the immunity of such accounts to the troubles besetting modalism.

(1.2) PW Theory

The pw method of modal theorising is not motivated only by dissatisfaction with modalism. Historically speaking, a positive motivating factor has been the formal analogy which obtains between the logics of the ‘traditional’ modal operators ⎕/◊ and the quantifiers ∀/∃, naturally inviting the supposition that the intensional ⎕/◊ can be analysed in terms of the extensional ∀/∃. The analogy is manifested in two ways. Firstly, both pairs are definitional duals which make directly analogous use of the apparatus of negation:-
⎕ϕ =def ¬◊¬ϕ ∀xϕ =def ¬∃x¬ϕ
◊ϕ =def ¬⎕¬ϕ ∃xϕ =def ¬∀x¬ϕ
Secondly, ⎕ϕ is understood to entail ◊ϕ much as ∀xϕ, at least in a free logic, is understood to entail ∃xϕ.

The following remarks of Hazen’s attest to the motivating force of the analogy:-
“So strong is this analogy that it has seemed that the only reasonable course is to consider modal language as involving a peculiar sort of quantification over some more or less arcane sort of entity. One prominent tradition has taken these entities to be ‘possible worlds’ or ‘cases’ – things each of which, whatever its other properties, is able to serve as a model for the semantic interpretation of the non-modal portion of the language under investigation.” [ibid. pp25-26]
Leibniz is widely (although controversially) credited with having introduced the early modern era to the notion of accounting for necessity and possibility in terms of quantification over possible worlds. During the twentieth century, and mostly building on developments in formal logic and modal logic initiated by, respectively, Frege and C.I.Lewis, a number of attempts were made to place pw systems on a formal footing, eventually culminating in the modern model theory widely attributed to Kripke [1963], which forms the subject-matter of the remainder of the current section.

In the following subsections, I introduce the pw system as a model-theoretical semantics interpreting ⎕ and ◊, and show how it withstands the objections levelled against modalism in §1.

(1.2.1) Models and Worlds
Model theory as devised by Kripke is intended as a semantics for so-called ‘normal’ modal logics. These are systems closed under modus ponens and necessitation, and containing all the rules and theorems of the modal logic T. In effect, T itself, S4, B and S5 are brought under consideration, but non-normal systems such as S2 and S3 are not.

The modal semantics which Kripke aims to provide is model-theoretical, in that modal truth is made a matter of truth in models. In the original Kripkean notation, a model structure ms is an ordered triple <G,K,R>, such that K is a set, G is a member of K, and R is a reflexive relation on K. Intuitively, K is understood as the set W of all possible worlds, and G as our world α; G has this privileged place in the Kripkean system as the locus of evaluation for modal claims, much as α is the locus of evaluation for the modal claims we make as α’s inhabitants. In recent years, model-theoretical semantics as it is practiced, acting perhaps in accordance with Kripke’s intuition, has often replaced the G and K of Kripke’s [1963] with, respectively, the w*/α and W of the intuitive interpretation, and I will accordingly regard ms for present purposes as an ordered triple <α,W,R>.

Now, R is understood as an accessibility relation; for any two arbitrarily selected worlds w1 and w2 in W, Rw1w2 is understood as meaning ‘w1 is possible relative to w2’ i.e. every sentence true in w1 is possible in w2. In the context of R, reflexivity, transitivity and symmetry have the following understandings:-
I. To describe R as reflexive is to say that, for all worlds w, Rww, answering the intuition that worlds are possible relative to themselves. If R is reflexive but neither transitive nor symmetric then ms is a T structure.
II. To describe R as transitive is to say that for any three arbitrarily selected worlds w1, w2, w3, ((Rw1w2 & Rw2w3)→Rw1w3). If R is reflexive and transitive but not symmetrical, then ms is an S4 structure.
III.To describe R as symmetric is to say that for any two arbitrarily selected worlds w1, w2, (Rw1w2 → Rw2w1). If R is reflexive and symmetrical but not transitive, then ms is a B structure.
IV. If R is reflexive, transitive and symmetric, then ms is an S5 structure.

A model M on ms is a binary function val(ϕ,w) assigning a truth value t or f to each atomic formula ϕ in each world w∈W. Assignment of truth values to non-atomic formulae proceeds in the familiar manner by induction, as functions of the truth values of the constituent atomic formulae and the meanings of the connectives. That is to say, for n-adic formulae ϕ and ψ and world w, we obtain the following rules in respect of negation and conjunction, from which those for disjunction, material implication and the biconditional are easily derived:-
PW¬. val(¬ϕ,w) = t iff val(ϕ,w) = f
PW&. val(ϕ&ψ,w) = t iff val(ϕ,w) = val(ψ,w) = t
Furthermore, we now have enough structure to define the assignments to truth values for modal formulae. It can be seen that these straightforwardly implement the original Leibnizian intuition envisaging necessity/possibility as a matter of truth at all/some worlds:-
PW⎕. val(⎕ϕ,w1) = t iff val(ϕ,w) = t for all w∈W such that Rww1
PW◊. val(◊ϕ,w1) = t iff val(ϕ,w) = t for some w∈W such that Rww1

Additional structure is needed in order to define assignments to truth values for quantified formulae. Accordingly, we define the notion of a quantificational model structure qms. A qms is a n-tuple of the form <W,α,R,D,d,val,w>. The additional four elements have the following explanations:-
*D is intuitively the set of all possible objects;
*d is a function associating subsets of D with elements w of W. On a typical understanding, d(w) denotes the set whose members are the inhabitants of w, the individuals existing at w; thus, persisting with the typical understanding, d(w) may fairly be described as the extension of the predicate Inhabitant-of-w. (However, note that we presently encounter other conceptions, according to which d(w) denotes the set of individuals which are the subjects of predications at w, and individuals do not need to exist in w in order to be the subjects of predications there. Instead, d(w) may be thought of as denoting the set of individuals represented as existing at w.)
*val is, as before, a binary function from ordered pairs <ϕ,w> to truth values t/f.
*w is an element of W like α, but is typically distinct from α. Intuitively, of course, the w’s are possible worlds.

An exemplifying model will render more perspicuous the form taken by the schemata for quantified formulae, which follow shortly. Any single one of any number of models will do this job perfectly well. For convenience, the present quantificational model Q is based on a model provided by Melia.

Q contains just three worlds w1, w2 and w3. Moreover, three individuals a, b and c constitute D(Q), of which a is in w1 and w2, b is in all three worlds, and c is in w3 only. Hence, it can be said that d(w1) = d(w2) = {a,b}, and d(w3) = {b,c}. The language of Q is impoverished, featuring just two predicates, F and G. a is F in w1 but not in w2; and is G in w2 but not in w1. b is F in w3 only, but is G in all three worlds, c is F but not G. Now suppose that modal claims are evaluated at w2, i.e. w2 is treated as α in Q. Q is represented in Table 1. below:-

Tab.1. Melia’s Model Q α
w1 w2 w3
a a in w? yes yes no
F val(Fa,w) t f ?
G val(Ga,w) f t ?

b b in w? yes yes yes
F val(Fb,w) f f t
G val(Gb,w) t t t
c c in w? no no yes
F val(Fc,w) ? ? t
G val(Gc,w) ? ? f

As Table 1. makes plain, qms contains the structure required for the quantificational analysis of modality. Two or three examples, selected at random, suffice to show this. Accordingly, given that b is G in every world in Q, we have both ∃x⎕Gx and ⎕∃xGx; moreover, since no world w in the model is such that everything is F at w, we have ¬◊∀xFx. And besides explicating the structural utility of model theory, Q also explicates the rationale underlying truth assignments to quantified formulae. Where (x)<a> signifies that an individual a is assigned to a variable x, the respective schemata for ∀ and ∃ are as follows:-
PW∀. val(∀xiϕ(x1...xn),w) = t iff val(ϕ,w) = t for every ai(a1...an)∈d(w) s.t. (x1...xn)<a1...an>
PW∃. val(∃xiϕ(x1...xn),w) = t iff val(ϕ,w) = t for some ai(a1...an)∈d(w) s.t. (x1...xn)<a1...an>

Two matters outstanding are the following:-
1. The formal representation in pw-theoretical analysis (as opposed to analysand) of truth at a world and truth simpliciter.
2. The Frege-Russell or ‘present King of France’ problem; i.e. what is the truth value of Fa when ¬∃x(x=a).

Truth in a world.
If an inhabitant of α makes the (false) claim that donkeys talk simpliciter, then we understand him to be saying that donkeys talk in his world, which is our world and which he and we all think of as the world. In this way, it is a fairly straightforward matter to reinterpret what we ordinarily think of as truth simpliciter, as a special case of truth in a world, viz. truth in α. But this is only straightforward because α is the world of evaluation. Let TDx be ‘x is a talking donkey’. We have already specified TDa’s translation: ‘TDa at α’. It seems that we want ‘TDa at w’ to translate a pw formula, but that such a formula is lacking.

This difficulty is susceptible of at least two fairly simple and plausible technical fixes:-
* Define satisfaction as a relation between predicates and worlds. Fx’s truth in w becomes w’s satisfying Fx. Truth in α is truth simpliciter. If w satisfies Fx, then that w satisfies Fx is true in α, and therefore true simpliciter. [Melia]
* For almost any n-ary predicate Fn, increase F’s n-adicity by one, to make Fn+1, the extra argument place being taken by the world which is in play. Truth in α is truth simpliciter. F(x1…xn,w) is true in α, and therefore true simpliciter. [Hazen ibid., Forbes 1982 et passim]
The advantages and disadvantages of the two fixes are the converses of eachother. The first forces on us a new relation of satisfaction between predicates and worlds, but not extra predicate n-adicity; the second forces on us extra predicate n-adicity, but no new relation. However, both expedients seem fairly well-attested; it would be very odd if predicates were not related to the worlds we understood them to be true at; and when we speak of truth in a world, we seem already to be increasing predicate n-adicity. Ultimately, it seems, either option is tenable. Henceforth, by-and-large and where relevant and for no particular reason, I choose the second of these.

Frege-Russell.
The Frege-Russell or ‘present King of France’ problem, a notorious and ongoing issue in the philosophy of language, concerns the assignment of a truth value to Faw when a is not assigned a value in d(w). To illustrate, consider:-
24. The present King of France lives in Bognor.
Grant the meaningfulness of 24. According to the Bivalence Thesis, 24 if meaningful must be either true or false. Now 24 is certainly not true. But is 24 false? To say so suggests that His Majesty exists but has upped-sticks to Bexhill. Again, we have two options, roughly corresponding to the stances occupied by the Frege/Strawson and Russell factions in the dispute.
* The Frege-Strawson option involves denying bivalence.  In particular,
Strawson holds that unsaturated formulae, i.e. formulae featuring empty designations e.g. 24, do not express propositions and so are not subject to the constraint of bivalence, but make statements which not subject to this constraint and which lack truth values.
*The Russell option involves attaching so-called Russellian truth conditions to propositions which, like 24, contain definite descriptions. Let F be the property of being a present King of France, and let B be the property of living in Bognor. Then the Russellian option recasts 24 as
24r. ∃x∀y(Fx & (Fy→y=x) & Bx).
24r is false, but false in a way that is unproblematic. Kripke acknowledges that either option is tenable, but chooses to proceed on the basis of the Russellian solution. For present purposes, I follow his example.

(2.2) Theory and Expressivity Revisited
Even bearing in mind the analogy between ⎕/◊ and ∀/∃, the pw system would be seriously under-motivated if it did not represent a significant improvement in terms of both theory and expressivity over modalism. §1 consisted mainly in setting out a (probably not exhaustive) list of the problems besetting modalism in respect of theory and expressivity. It is now time to gauge the responses available to pw-theorists in respect of these several problems.

(i) Necessity/Essence Revisited
Earlier, we saw Melia complaining, in my view incorrectly, that modalism does not account for the intuitive distinction between necessity and essence. It certainly seems right to most people that there should be such a distinction. It is not just possible for there to be states of affairs such that (a is essentially F & it is not necessarily the case that a is F): as a matter of fact, states of affairs of this form quite obviously abound. The existence of human beings attests to this, since we are essentially human and exist contingently. Nevertheless, complains Melia, modalism’s reliance on the formation rules of QML prevents it from capturing this distinction.

The pw system’s model-theoretic articulation enables it to capture the desired distinction. And we can expect some model to show this. Let Q1 be such a model. In Q1, the following overall state of affairs holds. There are just three worlds w1, w2, and w3; a single individual a inhabiting the first two but not the third of these worlds; and an impoverished language featuring just one predicate F; finally, let it be supposed that the following facts obtain:-
w2 is the locus of evaluation – the α world of the model.
(Faw1 & Faw2).
a is F in every world in which a exists, so ⎕(Ea → Fa) is true in Q1, so a is essentially F, on the standard QML understanding of what it is to be essentially F. However, given a’s absence from w3, we have ¬⎕(a is F). Ergo, pw theory captures the necessity/essence distinction, i.e. by means of a model such as Q1.

(ii) Counterfactuals Revisited
Earlier, we saw that modalism is unable to provide a unified account of modality and counterfactuals, as intuitively an account of modality should be able to. There are some indications that counterfactuals are (related to) modal notions. For example, if the consequent of a counterfactual is necessary, then the counterfactual is true. To cut to the chase, it seems that the best the modalist can hope for is to introduce a new primitive ⎕→. Moreover, it is highly doubtful that the introduction of ⎕→ resolves the modalist’s difficulty, since it does nothing to illuminate the relation between counterfactuals and other intuitively related modal notions.

In contrast, the Stalnaker-Lewis approach now to be explicated arguably succeeds in showing that a suitably modified pw system can enable an account of counterfactuals. The modification in question is the introduction into the pw system of a relation of overall similarity/closeness, which orders worlds and is defined by the extent to which the worlds being assessed match in terms of the facts and the laws holding at them. Although the introduction of new terms into a theory is naturally viewed with suspicion, in this case the initial suspicion can be allayed, since the notion of ordering things in terms of their relative similarity to eachother is a well-attested feature of discourse. Accordingly, as worlds are also individuals, there appears to be no particular prima facie reason why we should prohibit an overall similarity relation from holding between worlds, much as we would suppose it to hold between other kinds of individual.

Two conceptions of the idea underlying a world-similarity-based account of counterfactuals are as follows. For a pair of propositions p and q, with p assumed false at α, the counterfactual p⎕→q is non-vacuously true iff …
… the closest p-world (i.e. the closest world where p holds) is also a q world. This conception, associated with Stalnaker, requires that there be a unique closest world. 
… some (p&q)-world is nearer to α than any (p&¬q)-world is. This conception, associated with Lewis does not require there to be a unique closest world.

Jonathan Bennett [1974] and Kit Fine [1975] have both objected that the Stalnaker-Lewis approach yields the wrong truth values for some counterfactuals:-

* Bennett asks us to consider Oswald’s role in Kennedy’s death:
25. If Oswald had not killed him, Kennedy would not have been killed.
On the supposition endorsed by the Warren Commission that Oswald acted alone, 25 is presumably true. Unfortunately for the pw theorist, on the Stalnaker-Lewis approach 25 comes out false, since it is highly unlikely that some world in which nobody kills Kennedy more closely resembles α than any world in which someone other than Oswald kills him, as would have to be the case in order for the Stalnaker-Lewis approach to endorse 25.

* Fine asks us to consider Nixon’s role in the Cold War:
26. If Nixon had pressed the button, there would have been a nuclear holocaust.
On the highly plausible supposition that pressing the button would have had the effect intended by John Foster Dulles, or whoever else installed it in the Oval Office, 26 is true. But not according to the Lewis-Stalnaker approach, reckons Fine. To see why, let p be ‘Nixon presses the button’, and let q be ‘nuclear holocaust happens’.  To all appearances, surely, the nearest (p&q)-world of nuclear Armageddon resembles α very much less than those relatively sedate (p&¬q)-worlds, in which something intervenes between Nixon pressing the button and the ensuing holocaust to prevent the latter. To get from one of these sedate worlds to α - (¬p&¬q)-world - all that is required is a change to a ‘small’ fact about the pressing of a button; everything else about these two worlds already matches. Whereas, to get from the nearest (p&q)-world to α requires wholesale change, to ‘large’ facts about nuclear Armageddon as well as the fact about the button.

Lewis takes the Bennett-Fine complaint as an invitation to refine the notion of world-similarity. To this end, he claims that the assessment of world-similarity involves according weightings to various input factors. The factors in question concern the extent to which the worlds being assessed match in terms of the
*matters of fact obtaining at those worlds
* the presence/absence of ‘small’ miracles at those worlds.
* the presence/absence of ‘large’ miracles at those worlds.
Specifically, Lewis contends that a defensible notion of similarity can be given, with weightings accorded to matters of fact and miracles which make 26 come out true. To this end, consider the following worlds in a deterministic model, with the same laws holding at each world in the model. With p and q as before, let t be the ‘crucial time’ i.e. the moment at which Nixon either presses or does not press the button:-

w0 is the actualized world of the model, a perfect match with α, i.e. a no-miracle, (¬p&¬q)-world, and also a matter-of-fact (henceforth, ‘factual’) duplicate of α.  Thus, at t in w0, just as in α, Nixon does not press the button, and no holocaust ensues.

w1 is a small-miracle world. It matches w0 perfectly, right up until just before t, but differs quite radically from w0 after t. Just before t in w1, there is a small miracle, consisting in the spontaneous firing of a few neurons in Nixon’s brain which, with nature immediately thereafter reverting to deterministic type, cause him to press the button at t, and the rest is the end of history. Now, w1 is a (p&q)-world: at w1, both the antecedent and the consequent of 26 hold. So it would vindicate the Lewis-Stalnaker approach if, notwithstanding the factual differences between w0 and w1 after t, the pw weighting criteria for world-world closeness made w1 the closest world to w0.

w2 resembles w0 in being a no-miracle world. However, at t in w2 Nixon does press the button at t, and the holocaust duly ensues. Hence, w2 resembles w1 in being another (p&q)-world. Now, being as w2 is both a no-miracle world and a (p&q)-world, it is tempting to view w2 as even more similar to w0 than w1 is. However, this temptation should be avoided, for w2 proves on closer inspection to boast radical factual differences from w0 both before and after t. This comes about for the following reason. Given the determinism of the model, any two no-miracle worlds in the model must be either perfect factual duplicates or match factually not at all throughout their histories. That is, if two worlds sharing the same laws in a deterministic model have matching factual histories up to a time t’, their factual histories must continue to match after t’. Contrapositively, if two such worlds are not matching factually at t’, then they can have been matching at no time before t’. Ergo, since w0 and w2 clearly differ as to whether or not they hold it true that Nixon presses the button, ergo they have different factual histories before as well as after t. It might look like a nice result if the chosen weighting criteria made w2 the closest world to α. But they should not do this, because there are too many factual differences between w2 and w0 for them to be describable in any meaningful sense as resembling one another.

w3 is a two-small-miracles world. Like w1 but unlike w2, w3’s factual history matches that of w0 until just before t. Moreover, just before t in w3, as in w1, there is the small miracle of Nixon’s neurons spontaneously firing, duly and unmiraculously causing Nixon to press the button at t. However, just after t a second minor miracle intervenes, in the form of Alexander Haig’s neurons spontaneously firing, causing in him a sudden access of human feeling. Hague duly sabotages the process leading to Armageddon. For a short while after t, w3’s factual history superficially resembles that of w0. Eventually, however, certain small factual differences between the two worlds – Nixon’s fingerprint on the button, his developing post-traumatic stress etc – cause larger and larger factual divergences between w3 and w0. Now, w3 is a (p&¬q)-world, so it is important that the weighting criteria do not make w3 the closest world to w0. If they do, 26 will come out false in the pw theory.

w4 is a big-miracle world; it matches w3 right up until Hague’s intervention. However, there now occurs a big miracle, whereby all trace of the two minor miracles is erased. Soon after t, w4 reconverges with w0, and exactly resembles it thereafter. Now, w4 is, once again, a (p&¬q)-world, so once again it is important that the weighting criteria do not make this the closest world to w0.

The issue here is that 26 seems to be true, ergo the pw system ought to have a veridical account of 26. For this to come about requires the pw system to attach the right weighting criteria to the notion of world-world similarity. The right weighting criteria in this respect are the ones that make w1 be more similar to w0 – in effect the α of the model – than worlds w2-w4 are. To this end, Lewis eventually derives the following prognoses [ibid. p472], in descending level of importance:-
a. avoid large miracles.
b. maximise the spatiotemporal region throughout which perfect match of particular fact obtains.
c. avoid small miracles.
d. secure approximate similarity of particular fact.
To complete the picture: w2-w4 all fall foul of a and b, whereas w1 only falls foul of c and d. Ergo, with weightings criteria for world-world similarity as envisaged by Lewis, 26 comes out true in the pw system.


(iii) Intensionality Revisited
Earlier, we saw that the modalist analysis of modality, featuring as it does the ⎕ and ◊ of QML, is intensional. To recapitulate, there is sometimes evinced the fear that the intensional character of the modalist analysis commits the modalist to a non-compositonal account of modal sentences, whereas these are easily seen to be compositional. We also saw that the fear is not very well-founded, as the homophonic character of the modalist analysis preserves compositionality. However, for reasons already explained, the homophonic character of the modalist analysis leaves some residual mystery about the mastery ordinary speakers have of new modal sentences, and this latter consideration may well be taken to motivate a theory with an extensional analysis of modality. 

The pw analysis of modality is articulated in the language of the first order predicate logic. As is well-known, the language of the first order predicate logic is extensional. Melia in particular takes the extensionality of pw language as answering the intensionality-based objections against modalism mounted by Quine et al. And as Divers remarks,
“There is no doubt that the desire to provide a compositional account of the truth conditions of modal sentences, and the conviction that a substantial element of Quine’s critique of modality would thereby be defused, motivated the development of [pw theory].
pw is articulated in an extensional language, and provides an account of the truth conditions of modal sentences:
PW. a is a possible/necessary F iff a is F at some/all world(s) w.
Furthermore, the ability to understand new sentences can be inductively inferred from the ability to understand substitutivity and certain other relatively uncontentious principles. Hence, extensionality is a boon to pw theory.

(iv) Iteration Revisited
Earlier, we saw that, in view of its formation rules, modalism is committed to the indefinite iterability of modal operators. This is unfortunate, because modalism is also committed to homophonic analyses of modal sentences. This means that, although iterated modal sentences are well understood [Peacocke 1978], they are also somewhat cumbersome, in that the analyses inherit the prolixity of the analysands.

Some pw theorists, e.g. Lewis, provide an elegant solution to the iteration problem – the method of reduction alluded to earlier, whereby an iterated modal formula e.g. ⎕⎕◊◊◊⎕◊φ reduces to the fragment of that formula consisting in its rightmost operator and its non-modal element – viz. ◊φ. Call ◊φ the ‘reduced fragment’ of ⎕⎕◊◊◊⎕◊φ. Reduction is a characteristic of S5 – some might say its defining characteristic. Now a given pw theory typically models a structure with reflexive accessibility, but may incorporate into a reflexive structure any combination of transitivity and/or symmetry, as well as such other axioms as are seen fit for inclusion. However, if the theory models a structure boasting an accessibility relation which is an equivalence relation, i.e. all of reflexive, transitive and symmetric, i.e. if all worlds are mutually accessible, then any modal formula has the truth value of reduced fragment. Table 2 below illustrates this nice ‘collapsing propensity’ in the case of ◊◊p, which reduces to ◊p in S4 and therefore in S5. 

Tab. 2 Modelling Collapsing Iteration
α
w1 w2 w3
For each w, pw? p ¬p ¬p
Further assumptions Rw1w2 Rw2w3
Ergo the hypothesis:- ◊p ◊◊p
Assuming S4/transitivity: (Rw1w2→ Rw2w3)→ Rw1w3) ◊p
QED

(v) Modal Inference and Validity Revisited
Earlier we saw that, aside from the iteration problem, there is a range of intuitively correct modal inferences, the correctness of which is inscrutable and inexplicable to the modalist, reliant as he is on uninterpreted operators; even though in typical cases QML’s formation rules compel the modalist to accept as well-formed the strings constituting typical premises and conclusions. In the current subsection, I investigate a group of candidate theorems and axioms, originally introduced in the corresponding subsection on modalism. Each case proves significantly more tractable to the pw theorist than to the modalist.

Our first two cases are the necessitation rule and the distribution axiom:-
Nec.  ϕ is a theorem → ⎕ϕ is a theorem
Dist. ⎕(ϕ→ψ) → (⎕ϕ→⎕ψ)
These are both included within the weak system K. To be part of a weak system is to be relatively uncontentious, one would have thought, so arguably the modalist can be granted Nec and Dist. Nevertheless, even in these two uncontentious cases, the source of validity in each case remains inscrutable to the modalist. In contrast, the pw theorist knows exactly how to interpret Nec and Dist:-
PWNec. ϕ is a theorem → ∀w(ϕ is a theorem at w)
PWDist. ∀w(ϕw→ψw) → (∀wϕ → ∀wψw)
The PW theorist appears to be in a position to assert PWNec and PWDist, based solely on these results. This is a good result, since intuitively these are genuine theorems.

Our next four cases concern the modal axioms:-
T. ⎕ϕ → ϕ.
S4. ⎕ϕ → ⎕⎕ϕ
B. ϕ → ⎕◊ϕ
S5. ◊ϕ → ⎕◊ϕ
Which of these axioms correctly characterizes modality is a question the modalist is at a loss to even begin answering. In this respect, his fortunes contrast with those of the pw theorist. Although it is true that, except for the T axiom the pw theorist also cannot tell us which theorems correctly characterize modality, unless he follows Lewis’s cue and imposes on his system by fiat an accessibility relation which is one of equivalence, nevertheless he can give truth conditions corresponding to each axiom in terms of the inter-world accessibility relation R:-
PWT. ∀w(Rww).
PWS4. ∀w1∀w2∀w3((Rw1w2 & Rw2w3) → Rw1w3)
PWB. ∀w1∀w2(Rw1w2 → Rw2w1)
PWS5. ∀w1∀w2(Rw1w2)
The pw theorist appears to be in a position to assert PWT. This is a good result, since intuitively the metaphysical modality which is the typical subject matter of modal theory is indeed reflexive. Although the pw theorist is not in a position to assert the other three cases, unless by fiat, he is in a position to show how, within the pw system at least, their correctness is a matter of which secondary properties the accessibility relation has, i.e. what combination it boasts of reflexivity, transitivity and symmetry. I venture to suggest that this is also a good result; it would be very odd if we could discover which worlds were accessible from which worlds, merely by investigating our semantics.

Our final two cases concern the Barcan and Converse Barcan formulae. For an arbitrary 1-ary predicate F:-
BF. ◊∃xFx → ∃x◊Fx or, equivalently ∀x⎕Fx → ⎕∀xFx
CBF. ∃x◊Fx → ◊∃xFx or, equivalently ⎕∀xFx → ∀x⎕Fx
BF and CBF are highly contentious. As Reina Hayaki remarks:-,
“Jointly, these formulas require that the domains of all possible worlds be identical:
everything that could exist does exist, and everything exists necessarily. Since it seems clear that some objects are contingent ... quantified S5 with BF and CBF has never been popular as the correct logic for metaphysical modality’
The burden of contentiousness is often seen as lying with BF rather than CBF, for it is a straightforward matter to produce a counterinstance to BF, in the form of a model featuring mere possibilia, i.e. possible F’s existing at a world w distinct from α, but not existing at α. The friend of BF who allows mere possibilia is put to the unpleasant expedient of having to deny necessity of origin. Hayaki illustrates this with the case of Wittgenstein. Despite never having married, Wittgenstein could have fathered a child. If so, then by BF something actual is a possible child of Wittgenstein. But by the necessity of origin, nothing actual could have been fathered by Wittgenstein. Hence, BF can only be maintained at the cost of denying the necessity of origin. The modalist is presumably as likely as anyone else to appreciate the strength of Hayaki’s case, but is once again unable to formalize his intuition; unlike the pw theorist, as Table 3 shows.

Tab.3 BF Countermodel
α
w1 w2
d(w) {a,b} {a}
◊∃xFx Fb
¬∃x◊Fx

In the BF countermodel, b’s presence in w1 makes it the case that it is possible for something to be F. However, a is the only inhabitant of the world of evaluation w2; and a, being a non-F in all the worlds in the model, is not a possible F. Hence we have ¬∃x◊Fx. Ergo Tab.3 depicts a successful countermodel to BF.

(vi) Comparatives and Actuality
Earlier, we saw that modalism encounters difficulties with expressivity, specifically in relation to the formalization of a range of intuitively natural sentences, namely those featuring modally-embedded comparatives. As before, Melia’s example of such a sentence is the following:-
5. There could have been more things than there actually are.
Briefly to recapitulate, we found modalism not entirely lacking in the resources to express 5. In particular, Forbes has shown how to achieve this, by enriching the QMLA language of modalism with a denumerable range of subscripted operators. As we saw, the resultant theory successfully analyses 5 by means of
5c. ◊1((⎕∀x(AEx → A1Ex)) & ∃y¬AEy)
However, as Melia points out (q.v. §1.3.iii of the present chapter), the resultant theory generates analyses bearing striking structural and grammatical affinities to the corresponding analyses generated by pw theory. For example, 5c bears such affinities to
5d. ∃w1((∀w∀x(Exα → Exw1)) & ∃y¬Eyα)
To judge by Melia’s ‘linguist’ thought experiment, it is natural to conclude that subscripted-operator modalism is not a genuinely distinct modal theory in its own right, but rather a mere notational variant of pw theory.

Now, if a mere notational variant of pw theory, sc. subscripted-operator modalism, is shown to have at its disposal the resources to express modally-embedded comparative sentences, then the reasonable expectation will be that the original theory, sc. pw theory, will command the same or at least analogous resources. In fact, the pw-theoretical analysis of 5 has already been anticipated; it is of course none other than 5d.



Conclusion

In the current chapter, I have introduced modalism. Like actual counterpart theory, modalism purports to deliver a viable account of modality not involving reference to possible worlds or possibilia. Its several theoretical and expressive difficulties serve to motivate a rival pw-theoretical account of modality since, as the second half of the current chapter shows, pw-theory is largely free of these difficulties.


Chapter 2. Modal Realism

In modal philosophy, there is a distinction to be made among pw-theoretical accounts of modality, between those that are realist and those that are anti-realist about other worlds. Thus modal realism is the following claim:-
R. ∃w¬(w=α).
Anti-realism, which either negates or is agnostic [Divers] towards R,  is investigated in ch3. The present chapter is taken up with realism.

There is also a distinction to be made amongst the realists, between actualists and possibilists. Of the following pair of sentences, actualists accept
A. Everything is actual
and reject A’s negation
M. There are things which do not actually exist.
Possibilists do the opposite.

§§1-3 of this chapter concern an issue for actualists: how are they to account for plausible de dicto claims about ‘aliens’ and merely possible individuals, without conceding any ground to possibilism. One response – Domain-Inclusion – involves stipulating that d(α) ‘includes’ the domains of other worlds. This response generates problems in relation to the Barcan Formula. These problems are avoided by a second response – representation-functional (RF) actualism. However, RF-actualism is closely related to the ersatzism sharply criticized by Lewis [1986], and faces much the same objections, mainly concerning unreduced modality.

§4 investigates the dialectic between competing conceptions of possibilism. Briefly, Lewis’s counterpart-theoretical conception cct of possibilism gives a better account of the problems of actuality/indexicality and of transworld identity than do, respectively, the ‘Leibnizian’ and ‘Kripkean’ conceptions. However, in §5, cct is exposed to several further objections. Two of these – the Objections from Cardinality [Forrest & Armstrong, Nolan] and Epistemology [Richards, Skyrms, Chihara] – although not decisive against cct, serve to motivate the investigation into anti-realist pw-theories which I undertake in ch3.


(2.1) Actualism

Whereas possibilists, as we presently see, typically conceive of other worlds as concrete bounded spacetimes like α, actualists typically conceive of them as either abstract entities or as concrete representational devices; either way, as belonging to a different category to α. Lycan [1979 pp302-3] classifies actualist proposals into those which treat worlds as:-
i. Linguistic entities such as sets of sentences or state-descriptions [Carnap 1947]; 
ii. Intensional entities such as propositions [Adams 1979, Lycan ibid.] or states of affairs [Plantinga 1974]; 
iii. Properties [Stalnaker 2003]; 
iv. Combinatorial constructs [Armstrong 1989]; or, perhaps
v. Mental items, although as Lycan has pointed out and as continues to be the case, this latter option has yet to be the subject of extended proposals.

A major concern for the actualist project is revealed when we consider plausible de dicto claims about ‘aliens’ and merely possible individuals. Examples, respectively:-
1.  There might have been an individual other than those that actually exist.
2.  Wittgenstein might have fathered a son.
The problem for actualists is that, on the face of things, the pw analyses of 1 and 2 are possibilist. pw-semantics analyses 1 in terms of aliens – individuals which exist at some world w but not at α; and analyses 2 in terms of a son of Wittgenstein, which again exists at w but not, to the best of our knowledge, at α. The actualist problem, then, is how to do justice to the strong intuition that 1 and 2 are true, without committing himself to possibilia and possibilism. John Divers calls this ‘the D Problem’. The actualist
“does not believe in the existence of possible but non-actual individuals, so what account will she offer of the members of D?” [2002 pp211 et passim]

Karen Bennett [2004] identifies two actualist responses to the D Problem:-
a. What she calls Domain-Inclusion (DI) actualism avoids possibilism by taking the quantified-over domains of other worlds to be subsets of the domain of α:-
DI. ∀w(d(w) ⊆ d(α))
Conversely, in her terminology, d(α) includes d(w). Whilst DI-actualism certainly addresses the possibilist threat, it encounter serious problems in attempting to do justice to our intuitions about alien possibility claims and claims about merely possible individuals – e.g. 1 and 2 above. This is  explained in §2 presently following.
b. Non-DI actualism (§3) denies that d(α) includes d(w). Possibilism is avoided by enriching the semantics with a representation function, r in my formalism, which I believe to be innovatory. This brings to the fore the conception of worlds as representing-devices, sharply criticized by Lewis under the somewhat pejorative soubriquet of ‘ersatzism’.


(2.2) Domain-Inclusion Actualism

As before, DI is the claim that d(α) includes d(w); the individuals which exist at w actually exist:-
DI. ∀w(d(w) ⊆ d(α)).
If D is defined in the standard way as Uwd(w), then DI has d(α)=D as a logical consequence. As Divers explains, there are two cases of this, depending on whether the domain of quantification is fixed or variable; or, equivalently, whether each d(w) is a necessarily improper or possibly proper subset of d(α). The identification of d(α) with D
“supports two conceptions of the function [d] that assigns a local domain to each world. The default position is that no further constraint is imposed on [d] so that any local world-domain can be any subset of [d(α)]. Otherwise [d] may be so constrained that the local domain is invariant across worlds whereby all (and only) the actual individuals exist at every world, i.e. for every world w, [d(w)=d(α)].” [Divers 2002 p213]
The first conception described by Divers corresponds to the Kripkean system (KS) introduced in ch1 of the present work. The second conception corresponds to the Simplest Quantified Modal Logic (SQML) of Linsky & Zalta (L&Z) [1994, 1996]. “Either way”, as Divers further remarks,
“the threat looms of the validation of various modal theses that are at odds with prior modal opinion.” [ibid.]

In particular, both conceptions encounter problems in connection with the validation of the Barcan Formula (BF), and the SQML conception encounters further problems in connection with the validation of the Converse Barcan Formula (CBF) and the Thesis of Necessary Existence (NE). In §2.1 immediately following, I consider the BF problem, in particular how some actualists [sc. Plantinga, L&Z, Williamson] accommodate BF by positing a range of what Bennett calls ‘proxies’ – abstract actualia in the role of possible Wittgenstein-sons etc. As it happens, the three proxy actualist proposals identified by her each also addresses the NE problem in the spirit of SQML, as is explained in §2.2. In §2.3, I address an argument of Bennett’s against proxy actualism. Although I think her argument fails, I do not consider that its failure leaves proxy actualism much better off; since the proxies are still likely to be considered queer.

(2.2.1) DI-Actualism and the Barcan Formula
Bennett “notes right away” [2005 p300]  that DI-actualism implies the Barcan Formula
BF. ◊∃xFx→∃x◊Fx
Divers points out that DI-actualism with SQML validates CBF, but does not validate BF. A counter-example would be a model with a world at which nothing is a possible son of John, although some individual actually is, and a fortiori could be, a son of John. Nevertheless, there remains a serious difficulty in KS of
 “assigning ... satisfiable truth conditions to actual world counter-examples”
i.e. to BF, in relation to claims involving mere possibilia such as 
2.  Wittgenstein might have fathered a son.
This is explained as follows. By pw theory, BF’s antecedent is true iff an individual x exists at a world w and is F there. By DI, x also actually exists, and therefore exists simpliciter. Since x exists at w and is F there, by pw-conversion, x is possibly F. Conjoining the preceding two insights, we have BF’s consequent: there exists an x such that x is possibly F. Hence, DI-actualism implies BF. I capture this with the phrase DI→BF.

The consequences of DI→BF can be seen in relation to both 1 and 2.
* pw theory analyses 1 in terms of an individual x existing at w which does not exist at α. But the conjunction of 1 with BF – with F as the property of being an alien i.e. not actually existing – implies the existence of an actual individual which is a possible alien, i.e. a possible non-actual existent. This is what underlies Bennett’s complaint that BF itself is
“straightforwardly incompatible with the claim that there could have been something which does not actually exist.” [ibid. p301].
As she further explains, 
“feeding that claim – namely ◊∃xA¬Ex – into [BF] yields the incoherent ∃x◊A¬Ex, which says that there is (here in [α]) a thing which is possibly not identical with anything in α.” [ibid. note 7 p323].
*To see the problem with 2, consider as F the property of being a son of Wittgenstein. 2 then has the following QML translation viz. ◊∃xFx, which is the antecedent of BF. Application of BF itself in conjunction with its own antecedent generates an argument by modus ponens to BF’s consequent ∃x◊Fx. Translating this latter back into English, we obtain the result that something actual is a possible son of Wittgenstein.

It should be noted that the 2 problem is slightly less straightforward than the 1 problem. DI-actualism has the consequence, in virtue of  DI→BF, that 1 is false; but that either 2 is false or something actual is a possible son of Wittgenstein. Ruling out 2’s denial, what makes the alternative awkward is that it is not at all obvious what in our ontology of actual individuals could have the property of being a possible son of Wittgenstein. It seems highly unlikely that any concrete actualia have properties of this sort, and any choice we made would look arbitrary; what would make this teapot, but not that hatstand, a possible son of Wittgenstein? Nor do any of the abstract objects already recognized by us seem fit for this purpose. It seems that the only hope for the DI-actualist who aspires to do justice to our intuitions in respect of 2 is to enrich the ontology with a set of abstract individuals which actually exist and are possible sons of Wittgenstein, possible Jabberwocks etc etc. These, finally, are the individuals which Bennett calls ‘proxies’ [2004, 2006].

At least three proposals have been tabled in the proxy actualist spirit. The proponents of these proposals are:-
(i) Plantinga [1974, 1979], for whom proxies are necessarily existing individual essences, contingently not instantiated by existing individuals;
(ii) Linsky & Zalta [1994, 1996], for whom proxies are necessarily existing but contingently non-concrete individuals; and
(iii) Williamson [1998, 2000, 2001], for whom proxies are necessarily existing but ‘bare’ particulars, i.e. particulars contingently not clothed in properties such as being a golden mountain. 

Recall that the D Problem is in essence the problem of avoiding possibilism whilst doing justice to the fairly strong intuitions that 1 and 2 are true. 1 comes out strictly false under each of the three proposals, because in each case quantification is over things which exist necessarily, and therefore cannot be considered aliens, vindicating Bennett’s remarks, alluded to earlier, on the incompatibility of BF and 1. However, in each case the intuition that 1 is true is explained, and to this extent respected. Each account provides for things which play the alien role; it’s just that these things – concatenations of individuals and properties, more or less – aren’t quantified over:- For Plantinga, the alien role is played by the other-worldly instantiations of contingently uninstantiated individual essences. For L&Z, this role is played by contingently non-concrete individuals being concrete in other worlds – concretings, if you will. For Williamson, the role is played by actually bare particulars being clothed in properties at other worlds – clothings, if you will. In contrast, 2 comes out true, but at the cost of commitment in each case to the respective proxies.


(2.2.2) DI-Actualism With Fixed Domains
In the preceding paragraph, I alluded to a feature which all three proxy actualist proposals have in common, namely that in each case quantification is over things which exist necessarily – Plantinga’s individual essences, L&Z’s individuals, Williamson’s particulars. Earlier still, I alluded to the validation in SQML of the Thesis of Necessary Existence
NE. ∀x⎕∃y(y=x).
NE is validated in SQML because of the latter’s invariant domains. In SQML, the domain of quantification is D, the domain of individuals. Hence, for all worlds w, d(w)=D. Every individual exists at every world.

NE is also validated independently, but by a less direct route. In the earlier passage, I made another allusion, to the validation in SQML of the Converse Barcan Formula:-
CBF. ∃x◊Fx → ◊∃xFx
In SQML, the antecedent of CBF is true if an actual individual is F at a world w. But if some individual is F at w, then it is possible for an individual to be F, which is the consequent of CBF. Ergo, CBF is validated in SQML. The validation of NE follows as a consequence of this fact, if the thesis of Serious Actualism (SA) is assumed, as L&Z note. SA is the thesis that
“it is not possible for an object to have a property without existing, i.e. ... exemplification entails existence.” [1994 p437]
Formally,
SA. ⎕(φx →  (∃yy=x))
where φ is atomic and contains x free. L&Z [ibid.] explain this by recourse to the ⎕-formulation of CBF:-
CBF. ⎕∀xFx → ∀x⎕Fx.
Let F be a property such that ⎕∀xFx, e.g. the property of self-identity. Application of CBF to its own antecedent yields its own consequent ∀x⎕Fx, by modus ponens. From this latter, instantiation to a yields ⎕Fa. Instantiating SA yields ⎕(Fa → (∃yy=a)). Next, by applying the K axiom
K. ⎕(φ→ψ) → (⎕φ→⎕ψ)
to ⎕(Fa → (∃yy=a)) and ⎕Ra with the obvious substitutions – φ/Fa, ψ/∃yy=a – we arrive at ⎕∃yy=a. Finally, since a is arbitrary, application to this last of the Rule of Generalization in Classical Quantification Theory [L&Z 1994 p433]
RG. ├φ → ∀xφ
yields ∀x⎕∃yy=x, which is NE, QED.

On the face of things, the validation of NE – directly, or indirectly via CBF and SA – is an undesirable feature of SQML. We have the strong intuition that many individuals exist contingently. But the three proxy actualist proposals can be represented in the spirit of SQML, as aspiring to mitigate SQML’s undesirable-looking validation of NE. Again, the necessary existences which are quantified over are Plantinga’s individual essences, L&Z’s individuals, and Williamson’s particulars. Our intuitions about contingent existences are explained in the same way, by the contingent nature of instantiations, concretings and clothings, even though these are not quantified over in SQML. What might make all this trouble seem worthwhile, is the retention of SA. As we presently see in the case of representation/functional actualism/ersatzism (§3.2 below), abandoning the exemplification→existence thesis raises a plethora of issues pertaining to unreduced modality.

A natural response to this might be that, if SQML leads to such awkwardness with NE as well as BF, then why not adopt KS with its variable domains? However, this response amounts to the suggestion that NE be avoided by making the proxies exist contingently. But proxies must be abstract, we think, and it is hard to see how abstracta can be supposed to exist contingently. Anyway, even supposing a case could be made out that proxies are contingently existing abstracta – indeed, that (some) abstracta exist contingently – it would seem to be an arbitrary matter which worlds they exist at, and which worlds they fail to exist at. This consideration seems to motivate associating proxy actualism with a fixed-domain logic such as SQML.

(2.2.3) Against Proxy Actualism – Metaphysical Queerness, and Converse Meinongianism
There is unquestionably something strange-looking about the three proxy actualist proposals. As Divers remarks, in relation to the L&Z proposal,
“manifestly, the idea that there is a category of entities for which concreteness is contingent is metaphysically queer.” [2002 p215]
Peculiarity in Plantinga: contingently uninstantiated individual essences. Peculiarity in Williamson: contingently bare particulars.

Bennett develops a different argument against proxies in her [2006]. I call this the Argument from Converse Meinongianism. Its central allegation is that, whereas possibilists accept the fell Meinongian claim
M. There are things which do not actually exist
proxy actualism is committed to the equally undesirable
MC. There are actual things which do not exist.

Bennett’s explanatory mission is slightly complicated by differences between the accounts given by Plantinga and by L&Z. Whereas Plantinga thinks in accordance with commonsense that you and I and most other everyday individuals exist contingently, L&Z as we know depart from commonsense in holding that everything including everyday objects exists necessarily. Conversely, whereas L&Z quantify in accordance with commonsense over otherwise workaday individuals – although these sometimes happen not to be concrete – Plantinga’s account departs from commonsense in quantifying over individual essences – although of course these latter are supposed to exist necessarily. 

Bennett gets round these complications by dividing the entities of the target theories into:-
the stock; these are the proxies of the respective theory, the 
‘actually existing stand-ins ...  in some metaphorical sense waiting to be drawn upon to 
populate other possible worlds’ [Bennett 2006 p268],
 i.e. Plantinga’s individual essences, or Linsky & Zalta’s contingently non-concrete individuals; and
the display case of w; this is what ordinary speakers describe as existing in a world,
i.e. Plantinga’s instantiating individuals, or Linsky & Zalta’s (typically) (possibly contingently) concrete individuals, but crucially excluding the proxies.
Bennett’s complaint concerns the proxy actualist’s distinction between stock and display case. In brief, the contents of stock and display case are both actual. So the proxy actualist does not commit himself to the possibilist claim
M. There are things which do not actually exist.
But he does say that there are things in the stock which do not exist in the display case – the proxies. And since normal English speakers elide ‘exist in the display case’ as ‘exist simpliciter’, proxy actualism is in effect committed to
MC. There are actual things which do not exist.

I am doubtful of the success of Bennett’s complaint, because it seems to beg the question against proxy actualism. For proxy actualism is only committed to MC on the standard interpretation of ‘exists in the display case’ sc. ‘exists simpliciter’. And proxy actualism explicitly rejects this standard interpretation. According to it, there are things – Plantingan individual essences or Linskian-Zaltan individuals – which exist even outside the display case. So the only way for Bennett’s argument against proxy actualism to proceed is by begging the question against it. However, proxy actualism is still in not in the clear, in view of its metaphysically queerness.




(2.3) Non-Domain-Inclusion Actualism

Bennett, impressed by the force of her arguments against DI-ism [2005] and proxy actualism [2006], goes some way towards commending a Non-DI-ist approach. Non-DI-ism claims that some world w is such that
¬DI. (d(w) ⊆ d(α)).
As Bennett will put it, d(α) does not include d(w).

¬DI may strike some as straightforwardly possibilist. Surely, to say that there are things (in d(w)) which are not d(α) is exactly the same as saying that there are things which don’t actually exist – and that’s just Possibilism.

The approach advocated by Bennett involves, for some a and F, substituting the thesis of Exemplification by Existence
EE. If a is F at w then a exists at w
in favour of the thesis of Exemplification by Representation
ER. If a is F at w then w represents that a is F.

At this point, RF actualism may begin to look familiar. Worlds which are e.g. sets of sentences or propositions, in the manner of Lycan’s classifications, do not contain F individuals which exist at them. These worlds represent that individuals in their domains are F. So, that’s not Possibilism. RF actualism is substantially the same as the version of actualism criticized by Lewis [1986] under the pejorative soubriquet of ‘ersatzism’. Indeed, Bennett makes the connection explicit, referring to RF actualism as linguistic ersatzism [2005 p304].

(2.3.1) Representation and Formalism
The formalization of the notion of RF actualism involves enriching, say, KS with a non-factive representation function, r in my formalism. For a story s and proposition p, r<s>(p) intuitively means ‘according to s, p (is the case)’.

r can be used to modify the semantics in two ways, either with a wide scope corresponding to fictionalism, or with a narrow scope corresponding to ersatzism. Thus, let pw be a KS theory of modality. 
* The wide scope reading for a de re possibility analysis is then:-
f◊. ◊Fa ≡ r<pw>∃wFaw
which is tantamount to a fictionalist treatment of Kripkean pw-theory.
* The narrow scope reading retains the pw-theoretical character which is of interest to the ersatzist as modal realist:-
e◊. ◊Fa ≡ ∃w r<w>Faw

Ersatzism faces a very serious objection concerning unreduced primitive modality, for surely phrases like ‘representing that’ and ‘according to’ are modal notions. Indeed, this is the worry on which Lewis plays so effectively in his [1986] attack on ersatzism, which forms the subject of the subsection now following.

(2.3.2) Lewis vs. Ersatzism
Lewis distinguishes ersatzist accounts of modality into three types, depending on how the worlds featured in the accounts represent possibilities:-
Linguistic ersatzism features worlds which are like stories or theories written in some language, and represent possibilities in virtue of the meanings of the words and sentences of the language, in short by saying.
Pictorial ersatzism features worlds which are like pictures or scale models, and represent possibilities by being isomorphic to them.
Magical ersatzism features worlds which just represent possibilities without any explanation of how this comes about.

As Lewis remarks, such ersatzist proposals as have actually undergone serious development are generally either explicitly linguistic, or silent as to their modes of articulation. Moreover, it is not easy to envisage the future emergence of serious pictorial or magical proposals. Accordingly, I propose to concentrate on linguistic ersatzism alone.

Lewis objects to linguistic ersatzism on two principal grounds. Firstly, it does not do away with the need for primitive modality. Secondly, it lacks descriptive power.

(2.3.3) Lewis on Ersatzism and Primitive Modality 
The need for primitive modality arises in two ways:-
Firstly, the worldmaking language certainly needs to be maximally consistent. An
 inconsistent set of sentences begets an impossible world, and a non-maximal set risks “conflating ... possibilities that would be distinguished if ... extended to a larger consistent set” [Lewis ibid. p151]. And consistency and maximality are themselves modal notions; a set of sentences is consistent if its members could all be true together, and maximal if the addition to it of further sentences generates inconsistency, whereby the members of the larger set could not all be true together.
Secondly, there will almost certainly be the need for implicit representation
Quite generally, if a worldmaking language L contains a sentence p, then L represents explicitly that p. However, there is likely to be some p*, such that L does not contain p*, nor any set of sentences which jointly mean exactly that p*. In such a case, if p* describes a genuine possibility, there will be the need for implicit representation; L must contain a number of sentences which jointly imply that p*. And once again, implication is a modal notion; a set of sentences implies that p* iff it is impossible for those sentences to be true together with p* false.

As Lewis remarks [ibid. p151], implicit representation seems to be required in at least one special case. According to ersatzism, each world w represents itself is actual; i.e. that the world is as w and no other world v says it is. But for w to represent this, representation pretty much has to be implicit, for explicit representation would involve w representing the world, representing itself representing the world, and representing all the other worlds v misrepresenting the world ... which calls to mind a library, every book of which contains the full text of every other book in the library including itself.

Setting aside the special case, and supposing that implicit representation is needed in order to account as seems likely for other cases, the obvious way not to be put to this necessity is to enrich the language. However, doing so increases the risk of unwanted implicit representations, taking the form of inconsistencies:-
“In fact, an inconsistent set of sentences can simply be regarded as one that represents explicitly that so-and-so, while it also represents implicitly that not so-and-so.” [ibid. p152]

Now, whether the goal is to eliminate unwanted implicit representations – i.e. inconsistencies – from a rich language, or to eliminate the need for implicit representations from a poor language, the same solution presents itself. This is that the troublesome primitive modality be replaced with a syntactic surrogate, whereby consistency/implication is defined in terms of formal deduction rules. There now arises the threat of narrowly logical consistency/implication, i.e. of there being allegedly consistent ersatz worlds according to which there are married bachelors and suchlike. This threat is blocked by specifying certain sentences of the world-making language as axiomatic. Ultimately, however, it does not seem possible to specify the axiom set without relying once again on primitive modality.

Primitive modality, according to Lewis, crops up in two places: in the matter of the fundamental properties and relations of simple things, and in the relation of local to global descriptions.

Lewis’s example of the sort of inconsistency threat that crops up in relation to the fundamental properties of simple things is the case of particle charge. The threat with point particles is that the ersatzist may end up with narrowly logical consistency, whereby there are allegedly consistent ersatz worlds according to which there are particles which are both positively and negatively charged. The obvious way for the ersatzist to block this threat is by specifying that an axiom of unique charge be included in the axiom set. However, Lewis’s notional ersatzist is brought up short by the question, whether charge is unique contingently or necessarily. If the former/latter, then inclusion/exclusion of the axiom misrepresents the facts of modality. The only safe course of action is recourse to primitive modality.

The relation of local to global descriptions raises the spectre of primitive modality, by prompting the question, how does the ersatzist get from local matters of fact i.e. about the distribution of matter across spacetime points, to global matters of fact, e.g. about the existence of talking donkeys? Again, the obvious way to block this threat is by means of axioms, this time connecting the local facts to the global facts. But the obvious solution poses in turn an obvious problem; namely, the infinite and therefore uncompletable nature of the axiom set. Even supposing it completable, the herculean nature of the task involved seems, as Lewis points out [ibid. p156], a high price to pay simply to avoid taking modality as primitive.

In conclusion, either the ersatzist is simply stuck with primitive modality, or else he chooses to tolerate it in the face of an even more unwelcome alternative, his eventual hope being perhaps, as Lewis suggests [ibid.], that primitive modality can be made to appear a relatively trifling disadvantage compared to, say, the unwelcome ontologies of the possibilists.

(2.3.4) Lewis on Ersatzism and Descriptive Power
Lewis discerns two problems of descriptive power; a problem concerning indiscernible possible individuals, and a problem concerning alien properties.

Lewis reckons that distinct but indiscernible individuals are genuinely possible, for instance
if there were two-way eternal recurrence … or if the universe consisted of a perfect crystalline lattice, infinite in all directions.” [ibid. p157]
The linguistic ersatzist conflates the distinct but indiscernible individuals inhabiting such systems. For this sort of ersatzist, a possible individual just is its description. The problem is not that the ersatzist denies that there are many indiscernible individuals; rather, it is that
“the one ersatz individual is actualized many times over.” [ibid. p158]

Note that the second problem does not concern alien individuals. McMichael [1983] has argued that linguistic ersatzism cannot accommodate de re claims about alien individuals, illustrating his argument by way of iterated modalities. Suppose that Obama might have had a third daughter who actually worked in pest control but who – that very woman! – might have been a florist. Now, Obama’s extra daughter doesn’t actually exist, so we cannot possibly have a name for her. And without names for alien individuals, it does not seem that we distinguish between worlds that do not differ in the roles played in those worlds, but only in the individuals occupying those roles.

Karen Bennett [2005] claims that Lewis’s second problem of descriptive power is a special case of McMichael’s original problem. But this is not quite right. McMichael’s problem corresponds to an issue dismissed by Lewis as a harmless haecceitist’s problem, intuitions in favour of which 
“can be met by other means, [e.g.] by describing what kinds of individuals there are and how they are related” [Lewis 1986 p158-9]
What the second problem does concern is alien properties. It certainly seems genuinely possible for there to be alien properties. Consider a world w impoverished by comparison with α. Surely, the denizens of w ought to believe in the possibility of rich worlds like α, containing properties unknown to and not describable by them. By parity of reasoning, we ought to believe in worlds richer than α. To reject belief in such worlds betrays an unreasonable because arbitrary chauvinism. Such worlds would feature properties which the linguistic ersatzist could neither name nor (unlike the alien individuals above) describe.


(2.4) Possibilism

Possibilism, to repeat, is the following claim:-
M. There are things which do not actually exist.
On the usual possibilist understanding, worlds are ontologically on a par with α. Typically, they are themselves concrete objects, distinguished one from another by each being completely closed spacetimes, containing concrete objects and quite likely abstract ones too.

Like ersatzism, possibilism adopts the Leibnizian intuition which sees possibility in terms of goings-on at other worlds. However, unlike ersatzism, which compensates for doing so by reinterpreting the term ‘world’- to mean ‘maximal book’, or ‘maximal proposition set’, or etc – possibilism bravely retains the ordinary-language understanding of ‘world’, i.e. as a concrete entity, typically replete with flesh-and-blood individuals who are often (although often not) just like you and me.

Note that the possibilist typically elaborates the ordinary-language understanding of ‘world’ for the purpose of distinguishing worlds from one another. The standard means of achieving this is in terms of spatiotemporal relations. Thus, for a pair of individuals a and b:-
3. a and b inhabit the same world iff spatiotemporal relations obtain between a and b.

On the reasonable assumption that the facts about modality are much as they are commonly understood to be, the possibilist’s most blatant departure from commonsense now becomes stark: he is immediately committed to an infinite plethora of concrete worlds, in contrast to the ersatzer and the common man, who very reasonably consider there to be just the one, viz. α.

Possibilist theories of modality divide on two questions, concerning actuality and transworld identity. The possibilist is impelled, as the two subsections immediately following show, to give answers to both questions which involve him in further departures from commonsense. Moreover, it is in particular the possibilist response to the transworld identity question which has brought cct to its position of dialectical dominance intra possibilism.

(2.4.1) Absolute or Relative Actuality?
The first question concerns the status of actuality, and the locus of assessment of claims
both modal and non-modal. Here the question is, whether α’s actuality is absolute, and truth is truth at α; or whether α’s actuality is relative, truth is truth at w, and ‘actual’ is merely an indexical term, denoting a relation between w and its inhabitants. Correspondingly, are claims assessed from the perspective of someone in α, or from some transmundane perspective?
The actuality question divides what might be called Leibnizian possibilism from the Lewisian possibilism, which is what Lewis considers as modal realism, and which, in the present context, has yet to evolve its distinctive counterpart-theoretical plumage. The Leibnizian possibilist joins with the ersatzist and the common man in according absolute actuality to α. Truth is truth at α, and ‘actual’ is not indexical in any deep sense. The Lewisian takes the contrary views; actuality is relative, truth is truth at w, and ‘actual’ is deeply indexical.

Lewis raises two objections against the Leibnizian view [1986 pp92-94], an objection from epistemology or luck and an objection from contingency. Assume with the Leibnizians and the folk that truth is truth at α, and that claims are assessed from an α-ish point of view. If so, then there is
“some special distinction which [α] alone possesses, not relative to its inhabitants or anything else but simpliciter.” [ibid. p93]

Epistemology/luck. Out of all the infinitely many concrete worlds which
exist and which we could have inhabited, we have the immense good fortune to have ended up inhabiting the one and only actual world. But how can we know that α is the actual world? Epistemically, we’re in exactly the same position as all those miserably deluded wretches on other worlds w who mistakenly suppose w to be actual. If Leibnizian possibilism is true, then we are not justified in believing that we inhabit the actual world. Since we obviously are justified in believing this, Leibnizian possibilism must be false.

Adams reckons that a ‘simple property’ theory of actuality can account for our knowledge of our own actuality,
“for it can be maintained that actuality is a simple property which is possessed, not only by the actual world as a whole, but by everything that exists in the actual world, and that we are as immediately acquainted with our own actuality as we are with our own thoughts, feelings and sensations.” [Adams 1979 p200]
However, even if we grant immediate acquaintance, the problem is only postponed. Lewis could have had an elder sister,
“so there she is, unactualized, off in some other world getting fooled by the same evidence that is supposed to be giving [us] knowledge”. [Lewis 1986 p94]
So the simple-property supposition, far-fetched anyway, does nothing to justify our belief in our own actuality.

Contingency. Suppose actuality is absolute, i.e. α is the sole actual world. If α 
really is the actual world, then it turns out somewhat surprisingly that it is necessary that α is actual. That is to say, if only α is actual, if α is the only world with, say, the supposed simple-actuality property, then the truth of actuality does not vary from world to world; it is true at every world w that α is actual. But surely, it is contingent that α is the actual world, for surely other worlds could have been actual. How then do we square α’s actuality at w with w’s representing itself as actual? Actuality must be relative and not absolute, but Leibnizian possibilism says otherwise, ergo Leibnizian possibilism is not true.

(2.4.2) Transworld Identity?
The second question concerns transworld identity, the question whether or not individuals
can be present at or parts of distinct worlds. Let Kripkean possibilism be the claim that individuals can behave in this way. Lewisian possibilism claims that they cannot.

Kripkean possibilism faces a well-known objection of Lewis’s in respect of what he calls “accidental intrinsics” [Lewis 1986 pp198-202]. Lewis confesses that he finds
“problematic ... the way the common part of two worlds is supposed to have different properties in one world and in the other”.
Suppose that (as is presumably true) Humphrey is five-fingered in α, and that the Kripkean possibilist wishes to capture the intuition that
4. Humphrey might have been six-fingered.
He would be expected to offer something like the following analysis of 4, with h as Humphrey, and I and S as the relations/properties respectively of In-ness and being six-fingered, viz:-
5. ∃w(Ihw & Sh)
But how can this be? Humphrey’s size, shape and composition are intrinsic properties of his, and as Lewis remarks,
“there is no intelligible way for his intrinsic properties to differ from one world to the other”. [ibid. p201]
That is to say, how do we square the state of affairs in 4 with Humphrey’s being, as presumably he is, five-fingered in α? Call this the Doctrine of Intrinsic Invariance. The problem is: how should we characterize Humphrey? As five-fingered or as six? Note that this is not a problem for ersatzism; we have already established that, as far as the ersatzist is concerned, Humphrey himself is only in one world i.e. α, where of course he is five-fingered. With α as his locus of evaluation, the ersatzist finds himself in unproblematic accordance with the common intuition characterizing Humphrey as five-fingered. His other worlds just represent Humphrey as being six-fingered. By contrast, and to repeat the related prognosis, the Kripkean possibilist has available no privileged world to serve as the locus of evaluation. The worlds w are ontologically on a par with α. Short of producing an unconvincing denial of Intrinsic Invariance, it seems that he has no means of capturing 4.

It is the Problem of Accidental Intrinsics or, to speak in broader terms, the implausibility of supposing that one and the same individual can inhabit a plurality of numerically distinct worlds considered as concrete bounded spacetimes, which ultimately prompts the possibilist appeal to counterparts. The technical specifications of counterpart theory are spelled out in Lewis’s [1968]. The gist, as it pertains to de re modality, and continuing with the example of Humphrey’s possible electoral success, is as follows. Humphrey’s counterpart at w is an individual related to him by means of a similarity relation, which is somewhat analogous to the similarity relation obtaining between worlds as per the Stalnaker-Lewis account of counterfactuals (q.v. ch1).

In the typical case, Humphrey’s counterpart at w is the w-inhabitant most similar to him; although, as Lewis acknowledges, this is somewhat to oversimplify a complicated picture in two ways:-
(a) Gauging overall similarity is an often fraught process, typically involving the according of different weights to different respects in which individuals either do or do not resemble one another.
(b) In principle, an individual a can have several counterparts at w; and conversely, a can be the counterpart of several individuals, even if these individuals are world-mates of one another. The paradigm case of the first kind is one in which a’s closest resemblants at w are identical twins; conversely, the paradigm case of the second kind is one in which a is the closest resemblant at w of identical twins. 
In its classical manifestation i.e. cct, Lewisian possibilism captures 4 by means of the following, with h, I and S as before, and C being the relation of counterpart-hood:-
6. ∃w∃x(Ixw & Cxh & Sx).




(2.4.3) The Best Theory Argument
Lewis justifies possibilism – ‘modal realism’ for the remainder of this chapter - by means of what might be called a Best Theory Argument (BT ):-
“Why believe in a plurality of worlds? – Because the hypothesis is serviceable, and that is a reason to think that it is true. The familiar analysis of necessity as truth at all possible worlds was only the beginning. In the last two decades, philosophers have offered a great many more analyses that make reference to possible worlds, or to possible individuals that inhabit possible worlds. I find that record most impressive. I think it is clear that talk of possibilia has clarified questions in many parts of the philosophy of logic, of mind, of language, and of science ...” [Lewis 1986 p3]
Accordingly, much of Lewis’s [1986 ch1] consists in an account of some of the subject-matters in which modal realism proves serviceable. That is to say, he finds that, if modal realism is assumed, then systematic philosophy goes more easily in respect of the following subject-matters, in addition to modality, viz:-

Subjunctive (typically counterfactual) conditionals. Earlier [ch1§2.2 (ii)] I 
introduced the Stalnaker-Lewis account of counterfactuals. The fundamental idea underlying their approach is pw-theoretical. p⎕→q is non-vacuously true iff the closest p-world (i.e. the closest world where p holds) is also a q world. Assuming the Stalnaker –Lewis analysis, and assuming per modal realism that the p-worlds and q-worlds in question really exist, then it follows that p⎕→q is indeed non-vacuously true.

The characterization of the content of thought Modal realists can characterize the
content of a person s’s knowledge or system of beliefs about the world in terms of, respectively, his class of epistemically and doxastically accessible worlds. That is to say, a world w is epistemically or doxastically accessible to s iff s knows/believes nothing to rule out that w is s’s world, and whatever is true at some world epistemically or doxastically accessible to s is epistemically or doxastically possible for s.

Quantification over properties. As Lewis remarks [ibid pp50-51], the most
intuitive way of characterizing a property is to take it as the set of its instances. However, as he further remarks, this practice generates a notorious problem of conflation, because of the co-extensiveness of some properties. For instance, in α at any rate, the set of creatures with hearts and the set of creatures with kidneys are co-extensive, so on the popular characterization having a heart and having kidneys are one and the same property. But these are surely different properties. Modal realism rescues the properties-as-sets intuition from the conflation problem by identifying properties, not with their extensions at α, but with their intensions, i.e. their extensions at all worlds w. Continuing with the cordite/renate case, realists will postulate the existence of worlds containing creatures with hearts but not kidneys, and worlds where the opposite holds. Thus, having a heart and having kidneys are actually co-extensive properties, but can be distinguished by their differing intensions.

All of this is by way of saying that modal realism is our BT. Commitments to a range of entia non grata – whether these be modal abstracta, the ⎕→ relation, Fregean psychological items, or properties – are all replaced by a commitment to just one kind of entity, sc. the possible world. Thanks to Quine, we know that we ought to believe in the entities to which our BT commits us. Ergo, we ought to believe in worlds; we ought to be modal realists.


(2.5) Against Modal Realism

The present section surveys four objections focusing on cct’s modal realist element. These are:- Melia’s refutation of BT, Lycan’s Actuality Objection, the Cardinality Objection [Forrest/Armstrong, Nolan], and the Epistemology Objection [Richard, Skyrms, Chihara]. Of the first two of these objections, the following might be said:- firstly, Melia’s refutation of BT is not a direct attack on modal realism itself, and secondly I contend that Lycan’s objection fails. It is ultimately the latter pair of Objections, albeit presented as softening-up exercises rather than as decisive refutations, which are taken to motivate the anti-realist theories of modality introduced in ch3. 

(2.5.1) Melia’s Refutation of the Best Theory Argument
Melia’s [1995] is to be understood as attacking what could be thought of as BT’s major premise:-
BTP1. If BT commits us to the X’s, then we ought to believe in the X’s
That is to say, according to Melia, in some cases topic-specific BT’s commit us to entities which we are nevertheless not obliged to believe in. The basic idea is that sometimes our BT is distinct from the BT. On some such occasions, the BT will be free of the dubious commitments with which our BT saddles us, and then it is quite proper not to believe in the dubious entities of our BT. Such occasions come about in two types of case, namely in cases of ineliminable ignorance and in cases of inadequacy of linguistic resources. I illustrate cases only of the first type, since I regard them as sufficiently establishing Melia’s case against BT. My illustration is by means of an example adapted more or less wholesale from Melia [ibid.]:-

Ineliminable ignorance. Consider sentences about average F’s. Melia’s example:-
7. The average mum has 2.4 children.
On the face of things, 7 commits us to dubious entities such as average mums, so it is unlikely that the theorist – call him Joe – will want to include 7 in his BT. But suppose Joe has good evidence that there are 2.4 times as many children as mothers. So 7 seems to be getting something right, without which BT seems to be incomplete. The obvious first gambit: 7’s meaning can be captured, and the objectionable reference to average mums eliminated, by means of
8. The number of children divided by the number of mums is 2.4.
But 8 only succeeds in replacing a commitment to average mums with a commitment to numbers. Suppose Joe is a nominalist of the fairly stringent kind who finds even this commitment objectionable. His obvious second gambit is to count the mums and kids, and then give what Melia calls an intrinsic description, this being a sentence formed out of Quine’s beloved first order logic with identity, which captures 7/8 without committing Joe to either average mums or numbers. For Joe, the numbers might be small enough for this purpose. He might be tacitly restricting his quantifiers; he might be talking about some isolated hamlet where there are just twelve children and five mothers. Then we can picture Joe characterizing the appropriate intrinsic description:-
9. There are individuals a, b, c, d and e who are all mums and distinct from one another, and also individuals f, g, h, i, j, k, l, m, n, o, p and q who are all children and distinct from one another.

However, now consider Joe’s equally stringently nominalistic brother Mel, an astronomer with excellent evidence for contending that
10. The average star has 2.4 planets.
Plainly, just as Joe objected to 8, for analogous reasons Mel will object to
11. The number of planets divided by the number of stars is 2.4.
But the analogous second gambit is not an option for Mel. We can’t picture him saying, of the stars and planets, that there are individuals a, b, c, etc. Although he may have very good evidence for 10, it is pragmatically impossible for him to reckon up the celestial bodies. In this case the best thing he can do is add 10/11 to his BT, but absolve himself of the impending commitment to average stars or numbers, on the grounds that his BT is, as it were, the place-holder for the BT, this latter being of course free of the unwanted commitments thanks to the presence in it of the appropriate intrinsic description, whatever this latter is.

We should locate Melia’s argument in the dialectic. It is not directly an argument against cct itself. It is posed as a refutation – and makes a fairly convincing case in my view – of an argument purporting to justify belief in the modal realist component of cct

(2.5.2) The Actuality Objection
Lewis distinguishes the Actuality Objection into two parts [1986 pp97-98]. The first part of the Objection says that terms like ‘actual’ and ‘the world’ are ‘blanket’ terms, meaning that they apply to absolutely everything, without restriction. Hence, to quantify beyond the actual or the world, as possibilism aspires to, is to commit
“nonsense on a par with saying that some things do not exist, or with Meinong’s shocker: ‘there are objects of which it is true to say that there are no such objects’” [ibid. p97]
As Lewis remarks [ibid. p98], the Objection’s first part is urged nowhere more vigorously than in Lycan [1979]. Lycan distinguishes the original actuality-indicating quantifier from the modal realist’s Meingongian quantifier, which does not indicate actuality, and then professes to find ‘relentlessly’ Meinongian quantification ‘simply unintelligible … literally gibberish or mere noise’ [ibid. p290].

The second part of the Actuality Objection says that, given the actuality of everything, that is given the blanket nature of ‘actual’ and ‘world’, modal realism cannot account for mere/unactualized possibilities. For if, as per the Objection’s first part, everything unrestrictedly real is actual, then other worlds are likewise actualized. But modality, or to say the least a particularly salient fragment of modality, concerns mere or unactualized possibilities, i.e. alternatives to actuality. So a great deal of possibility has nothing to do with worlds, and modal realism cannot account for it. Lewis suggests that something like the second part of the Objection is what motivates Van Inwagen’s remark that ‘[Lewis] face[s] the problem of explaining what [worlds] would have to do with modality if there were any of them’ [Van Inwagen 1985 p119].

As Lewis remarks, the second part of the Objection depends upon the first part, so in answering the first part he answers the second. His response, although he does not himself use the exact terminology, is as much as to say that the Objection begs the question against modal realism. If ‘actual’ is a blanket term, then of course it follows that quantification beyond the actual is nonsense on stilts; and that quantification over worlds which are just more of actuality leaves pw-theory unable to account for the merely possible. But Lewis is explicit that in his own mouth ‘actual’ is not a blanket term. As we know from earlier, he thinks of it as a term used by the inhabitants of a given bounded spacetime to refer to the contents of that spacetime; furthermore, quantification over bounded spacetimes which do not overlap with our bounded spacetime, sc. α, amply allows pw-theory to account for the merely possible.

Helpfully, Lewis provides a diagnosis of the background against which the Objection is made. He admits that commonsense “adheres firmly” to the following three theses:-
CS1. Everything is actual.
CS2. Actuality consists of everything that is spatiotemporally related to us (give or take some abstract entities).
CS3. Possibilities are not parts of, but alternatives to, actuality.
Lewis finds that the difference between himself and his critics is as follows. His critics claim firstly that CS1 is analytic, and its denial is unintelligible or gibberish or etc; secondly that by contrast CS2 is “up for grabs”. Moreover, on the reasonable assumption that most criticism of his position emanates from actualist quarters, we should expect his critics to reject CS3. In contrast, Lewis puts all three theses on an equal footing – all three are up for grabs. He can then be fairly criticized for rescinding from commonsense in the case of CS1. However, this is less serious than Lycan’s original charge, that modal realism is unintelligible or gibberish or etc, especially given Lewis’s accordance with commonsense in the cases of CS2 and CS3.

(2.5.3) The Cardinality Objection(s) 
Lewis actually identifies two Cardinality Objections, his respective names for these being ‘All Worlds in One’ (AWiO) [Forrest & Armstrong 1984] and ‘More Worlds Than There Are’ (MWtX) [Davies 1981]. Both Objections exploit the Humean recombinatorial apparatus – appropriated by Lewis for the purpose of ensuring that modal realism delivers a plenitudinous account of possibility – in order to embed sets of worlds within single worlds. Paradoxes are then derived, similar in spirit to those afflicting Frege’s naïve set theory. However, the Objections differ in the forms the respective embeddings take. In the case of AWiO, the set is embedded simply by duplicating the contents of its members into a single world. With MWtX, the set is embedded by making it the set of worlds characterizing the content of a person’s thought. In what remains of the present subsection, consideration of MWtX is set aside, and the focus is on AWiO; in particular, on the reasons for thinking that AWiO, if not altogether decisive against modal realism, amounts to sufficiently formidable a worry to motivate investigation of the merits of anti-realist alternatives to modal realism.

The standard way of obtaining the AWiO paradox is by means of the following procedure, substantially attributable to Forrest & Armstrong [1984] and Lewis [1986]. Assuming modal realism, consider all the possible worlds as a class of possible individuals. Now the unqualified version of a principle of recombination, a qualified version of which is endorsed by Lewis [1986 pp87-88], claims the following:-
UPR. For any class C of possible individuals, some world contains any number of non-overlapping duplicates of all the individuals in C.
If UPR is applied to the class of all possible worlds (since these can be considered as individuals), we end up with as it were a ‘big’ world, called variously ‘WB’ [Forrest &Armstrong.] or ‘Giganto’ [Nolan 1996], being the world which per UPR contains non-overlapping duplicates of each of the worlds in C, ergo non-overlapping duplicates of all the individuals in each world. Since we began by considering all the possible worlds, clearly WB should itself be one of the worlds in C, whose contained individuals are duplicated by WB. As is presently shown, WB by this means ends up containing more individuals than it contains. Arguing back by modus tollens from the negation of this reductio, we are invited to reject either or both of modal realism and unqualified recombination.

In what follows, I investigate two responses on behalf of modal realism:-
* Accept that the Cardinality Objection is sound. Defend modal realism by sacrificing unrestricted combination. This is the approach adopted by Lewis, who qualifies the recombination principle with a proviso:- “size and shape of spacetime permitting” [1986]. However, it is not clear that spacetime-restrictions of the type suggested by Lewis escape accusations of arbitrariness; moreover, even supposing that Lewis is right about there being some upper limit on the number of point particles, cardinality problems can perhaps still be generated by selecting as ‘contained individuals’ entities which seem capable of existing in numbers greater than the cardinality of the Lewisian cut-off point for point particles (whatever that is), e.g. bosons [Pruss 2001], or angels [Usquiano & Hawthorne 2011].
* Reject the Cardinality Objection. Insist that modal realism’s truth does not imply the existence of worlds containing more individuals than they contain. In this spirit, Nolan argues in effect that UPR is only true if the class C to which UPR alludes is construed as a set. That is to say, there is a world corresponding to C only if C forms a set. However, as Nolan further argues, some classes of individuals are too large to form sets. In particular, the class of all possible individuals is as likely as any class is to be too large to form a set. Hence it does not follow from modal realism’s truth, that some world contains duplicates of all the individuals in all the worlds.

In what now follows, I explore the responses of Lewis and Nolan in a little more detail.

Lewis’s Response to the Cardinality Objection
Lewis [1986] follows the original Forrest & Armstrong formulation of the Objection, in selecting electrons to play the ‘contained individuals’ role. Suppose that the big world WB contains K electrons. Then there are 2K-1 non-empty subsets of the electrons in WB. Now apply UPR. Then, for every non-empty subset, there is a world containing duplicates of all and only the electrons in that subset. In particular, each subset, and accordingly each corresponding world, contains at least one electron, and is duplicated into WB. Ergo WB contains at least 2K-1 electrons, contradicting our original hypothesis that it contained just K electrons. Reductio ad absurdum.

Plainly, the Objection invites us to reject either modal realism or UPR. As a modal realist, Lewis naturally identifies UPR as the source of the trouble. He advocates modifying UPR, by adding to it the proviso ‘size and shape permitting’. 

Forrest &Armstrong [1984 p166] criticize Lewis’s proviso response for ‘seem[ing] to be ad hoc’. Lewis acknowledges that ‘any restriction whatever will seem at least somewhat ad hoc’ [1986 p103]. However, he notes that some size/shape cut-off points seem more arbitrary than others. For example, a restriction to finite-dimensional manifolds looks ‘much more tolerable’ than one to e.g. four- or seventeen-dimensional manifolds. The hope is then that some cut-off point exists which, like the finite-dimensional manifold, seems relatively natural. To head-off accusations of ad-hoc-ery, Lewis takes the proviso to be independently motivated. Without it, UPR would deliver proofs that there were unfeasibly large spacetimes. In particular, we would need a spacetime which was larger than continuum-sized, in order to accommodate the classes of more than continuum-many individuals which UPR would otherwise generate [1986 p101].

A second response to Lewis’s proviso takes the form of suggesting that there might be individuals of a kind capable of existing in numbers of greater cardinality than the cardinality of the point-particle cut-off, whatever that turns out to be. As before, Pruss [2011] and Usquiano & Hawthorne [2011] have argued that this could be the case with, respectively, bosons and angels. Indeed, no matter whether Pruss and/or Usqiano & Hawthorne are right about their particular cases, it might be thought possible for there to be entities of at least some sort or other, existing in sufficient numbers to enable the paradox to run its course without point-particle-related constraints. However, such suggestion would be highly speculative to say the least, and we should hesitate to take it as constituting grounds for proclaiming the refutation of modal realism.

Nolan’s Response to the Cardinality Objection
Nolan [1996] approaches his proposed solution to the Cardinality Objection, via a criticism of the standard Forrest/Armstrong/Lewis (FAL) formulation of AWiO, which he alleges does not establish its intended conclusion, sc. that modal realism plus UPR entails the existence of a ‘big’ world containing more individuals than it contains. According to Nolan, the flaw in the FAL formulation is that it does not exclude the possibility of the electrons in the ‘big’ world duplicating the duplicand worlds on an equivalence-class rather than a one-one basis. That is to say, the FAL formulation requires that each distinct 1-electron duplicand world be duplicated by a distinct 1-electron set in the ‘big’ world, that each distinct 2-electron duplicand be duplicated by a distinct 2-electron set in the ‘big’ world … that each distinct k-electron duplicand be duplicated by a distinct k-electron set in the ‘big’ world. However, it is compatible with the FAL formulation that just one 1-electron set in the ‘big’ world duplicate the equivalence class of all the 1-electron duplicand worlds, that just one 2-electron set in the ‘big’ world duplicate the equivalence class of all the 2-electron duplicand worlds … that just one k-electron set in the ‘big’ world duplicate the equivalence class of all the k-electron duplicand worlds. Finally,
“[t]his will only ensure at most that there are N worlds containing electrons, not 2N. And since adding N to N N number of times (that is, multiplying N by N) only yields N where N is any infinite cardinal, this does not even show that there are more than N electrons in the entire pluriverse.” [Nolan 1996 p244]

Nolan takes the foregoing to show that the FAL formulation of AWiO fails to establish that modal realism plus UPR entails the existence of a ‘big’ world containing more individuals than it contains. Nevertheless, he contends, there is yet a “quick and simple” argument from UPR to the desired conclusion [ibid p246]. This quick and simple argument asks us to suppose for reductio that there is a set of all possible objects with the cardinality C. Now there must be some greater cardinality than C. For example, the power set of the set of possible objects must have a greater cardinality than C. Call the cardinality of this power set C*. From reapplying UPR, it follows that, for some object, some world contains C* duplicates of that object. Ergo there are at least C* possible objects. Ergo, the set of possible objects has a larger cardinality than it has. Reductio ad absurdum.

Having shown that at least one formulation of AWiO has some purchase against modal realism, Nolan takes it upon himself to defend the latter against the former. In particular, his suggestion is that what is described in his formulation as the set of possible objects (and a fortiori, one supposes, the worlds) can be treated as together forming a proper class, proper classes being classes which do not form sets [ibid. p247]. To spell everything out, the idea is that there is only a world for every set-forming class – there are no worlds corresponding to proper classes.

Proponents of Nolan’s suggestion then have two possible gambits:-
*Gambit (1) Deny that proper classes have cardinality, on the grounds that cardinality is only well-defined for sets;
*Gambit (2) Argue that at least some proper classes are such that no class has a greater cardinality than them. Moreover, surely the class of possible objects is ‘large’ in this way if anything is.

Much of the remaining two thirds of Nolan’s paper answers the objection that Lewis has independent reasons for preferring that the possible objects form a set. For example, as Nolan remarks [ibid. p248], Lewis conceives of a property as being the set of its instances. To this end, Nolan concerns himself with showing that proper classes are an adequate theoretical substitute for sets.

Suppose we accept this latter contention of Nolan’s. On the face of things, the appeal to proper classes is not ad-hoc. Certainly, it is not lacking in historical precedent, most obviously in the case of mathematics. For, as is well-known, the paradoxes of naïve set theory invite the treatment of certain classes as proper rather than as forming sets. For example, a standard way of solving Russell’s Paradox is to treat the class of sets which do not contain themselves as proper classes rather than as forming a set; likewise in the case of the Burali-Forti paradox and the class of ordinal numbers. This suggests the following picture:-
(A) that certain ‘large’ classes, e.g. the class of all sets not containing themselves and the class of Von Neumann ordinals, generate paradox if they are understood as forming sets, and therefore should be understood as forming proper classes;
(B) that the class of possible individuals is ‘large’ in the same way that the classes of sets and ordinal numbers are ‘large’; per the above two gambits, proper classes either entirely lack cardinality or have a cardinality greater than or the equal of all other classes. Either method breaks the Nolan-style argument which, to reiterate from earlier, proceeds from the possible objects having a cardinality C to their having a cardinality C* such that C*>C.

Withal, we may yet consider Nolan’s appeal to proper classes to be ad hoc.The problem is that Nolan is appealing to a notion i.e. proper class, which was originally intended only for solutions to paradoxes in set-theory, not for the present heavy-duty metaphysical purpose of solving AWiO for the benefit of modal realism. Proper classes are unavoidably ad hoc solutions to the paradoxes of set-theory. Mathematicians want to carry on doing set-theory, and postulate whatever it takes for them to be able to do so – in the relevant case, proper classes. This does not mean that just any collection of individuals, saliently the possible individuals, can form a proper class. Moreover, the ad hoc-ness of proper-classes in mathematics may be a virtue – or at any rate not a vice given its inevitability; but when this ad-hoc-ness of proper-classes is carried over into modal realism, it is more likely to be considered vicious. In this latter case, proper classes are only appealed to as a way of addressing the Cardinality problem. Thereupon Nolan faces the task of showing that proper classes can do most of the theoretical work set aside by Lewis for sets. This besides the burden of showing that it is reasonable to suppose that collections of non-mathematical individuals can form proper classes. Nevertheless, I do not consider this to be a very pressing concern. Ad-hoc-ery would be a more pressing issue if the notion of proper class had no independent argument to sustain it; whereas, in this case the complaint is only that the notion has but one independent argument to sustain it.

If Nolan is wrong, and the FAL version of AWiO works, then modal realism + UPR entails the existence of a ‘big’ world containing more individuals than it contains. As we have seen, Lewis addresses this worry by proposing limitations on the size and shape of spacetime, which he takes to be independently motivated, and modifying UPR accordingly. Whether or not we are persuaded by his proposal depends on how persuasive we find the examples given by Pruss and by Uzquiano & Hawthorne of entities – bosons or angels – capable of existing in numbers greater than the cardinality of spacetime points. We may not consider that AWiO tells conclusively against modal realism. However, perhaps residual worries of the type raised by Pruss and Uzquiano & Hawthorne serve to motivate the investigations into anti-realism undertaken in ch3.

If Nolan is right, and the FAL version of AWiO does not work, his ‘quick and simple’ version of AWiO says that for any cardinality C of possible objects, there must be some greater cardinality C*, such that for some object some world contains C* duplicates of that object. Ergo there are at least C* possible objects. By way of a solution to the Objection, Nolan proposes that the possible objects form a proper class, which might be ‘large’ enough either to avoid having a cardinality or to be such that no class has a larger cardinality. The suspicion with Nolan’s suggestion is that it appeals to a notion – the notion of proper class – which may be only virtuously ad hoc in mathematics, but which will be considered viciously ad hoc when introduced for the purpose of heavy duty metaphysics, e.g. in order to sustain modal realism. Again, a worry over the possible ad-hoc-ness of proper classes in modal metaphysics does not amount to a definitive refutation of modal realism. Nevertheless, as with FAL the residual worry – this time over ad-hoc-ness – does seem to warrant investigating the anti-realist alternatives.

(2.5.4) The Epistemology Objection 
The Epistemology Objection is attributed by Lewis [1986 p108] and Chihara [1998 p87] to Tom Richards [1975]. Perhaps its briefest statement is that modal realism implies an implausibly pervasive scepticism about the epistemology of modality – if modality is a matter of truth at other possible worlds as per modal realism, then commonplace modal beliefs are by-and-large unjustifiable. As Richards writes:-
“While possible-worlds semantics does yield truth-conditions for possibility statements … the truth conditions are such that, for any given statement, it is impossible in general to determine whether they are met and hence whether the statement is true.” [ibid. pp109-110]

As is presently to be explained, the Objection turns upon the supposed universality of a causal principle of justification – CP below. The proponent of the Objection faces a dilemma, over whether or not to treat CP as universally applicable:-
If CP is universally applicable, then it applies a fortiori to the case of mathematics, and the threat looms of commitment to a pervasive scepticism about knowledge of sets and the like.
If CP is not universally applicable, and in particular does not apply to the case of mathematics, then there arises the problem of explaining how we do have knowledge of sets.
In practice, the second lemma is the Objector’s likely preference; an hypothesis which imposes a burden of explanation is easier to accept than one which implies a pervasive epistemological scepticism about an intuitively well-understood domain. The final part of the current proceeding consists in presentations, firstly of Lewis’s positive account of knowledge of modality and mathematics, which invokes a Theoretical Unity Thesis (TUT); and secondly of Chihara’s criticisms of TUT. In the final analysis, the Objection cannot be regarded as conclusive against modal realism until the matter of mathematical epistemology is adequately addressed; although it may well be felt that enough concern is raised about modal realism, and about Lewis’s reliance on TUT, to motivate investigating the anti-realist alternatives.

The Objection is also – a fact which Lewis remarks upon [1986 p108] and then seeks to exploit in his defence of modal realism – the formal analogue of one of the horns of a well-known dilemma in the philosophy of mathematics [Benacerraf 1973]. In essence, Benacerraf’s Dilemma is that mathematical Platonism is incompatible with the justification of mathematical belief. This follows, firstly from the causal inertia of platonic mathematical objects, and secondly from application of CP.

Since the Dilemma is a dilemma, it has the two obvious lemmas:-
lemma 1. mathematical Platonism no justified mathematical belief
lemma 2. justified mathematical belief not mathematical Platonism
Lewis is principally interested in lemma 1, because the structure of lemma 1’s associated dialectic is obviously and interestingly analogous to the structure of the dialectic of Richards’s objection against modal realism, since what that says is
modal realism no justified modal belief.
However, whereas Benacerraf’s Dilemma is intended by its author as a reductio against mathematical Platonism, Lewis subverts Benacerraf’s intentions, arguing from mathematical Platonism and the patent fact that much of our mathematical belief is justified, to the falsity of CP, or at least its less than universal applicability. Lastly, the inapplicability of CP in the modal case invalidates the argument from modal realism to the impossibility of justifying modal belief.

Here is the aforementioned causal principle of justification:-
CP. a’s belief that p is justified → p’s truth conditions are causally related to a’s belief that p.
To see why CP is integral to the Objection, consider its modal instance, obtained by substituting ◊q in place of p. Given the pw-semantics of modal realism, the truth condition for ◊q consists in q’s being true at some world. Now:-
(a) It is highly plausible that for A to cause B requires spatiotemporal relations to obtain between A and B. Whereas:-
(b) The worlds of modal realism are bounded spacetimes. So spatiotemporal relations do not obtain between distinct worlds, or between respective goings-on in distinct worlds. A’s in one world can’t cause B’s in another world. Moreover:-
(c) Crucially, if a’s belief that ◊q is at all typical, then the world of a’s belief that ◊q and the world where q holds are distinct.
It follows, providing again that the case is typical, that ◊q’s truth condition and a’s belief that ◊q are not causally related; and lastly, by application of modus tollens to CP’s modal instance, that a’s belief that ◊q is not justified. Since a and q are arbitrary, the result is a generalized scepticism about modal epistemology.

Lewis responds to the Objection by attacking CP, or rather the supposition that it is universally applicable:-
*Firstly, the case of mathematics shows that CP is not universally applicable. If CP was universally applicable, then our knowledge of mathematical facts would be called into question along with our knowledge of modal truths (assuming modal realism), because the salient objects in the mathematical case – entia such as numbers and sets – are “causally isolated from us and unavailable to our inspection” [1986 p109] in view of their abstract nature. 
* Secondly, CP is inapplicable in the particular case of modality. We do have knowledge of modal facts even though the salient objects in the case – the pw’s and pw-inhabiting individuals of modal realism – are causally isolated from us here in α, as per the Objection as stated.
* Thirdly, acknowledging that an explanation is owed as to how we come by any sort of knowledge absent CP, Lewis appeals to his Theoretical Unity Thesis TUT:-
“If we are prepared to expand our existential beliefs for the sake of theoretical unity, and if thereby we come to believe the truth, then we attain knowledge” [1986 p109].

Brian Skyrms [1976] concedes Lewis’s point about CP not being universally applicable, maintaining nonetheless that the Objection against modal realism survives this concession. Specifically, he agrees with Lewis that CP does not apply in the mathematical case, but disagrees with Lewis about the modal case, insisting that CP does apply there, to the detriment of modal realism. Skyrms’s basis for distinguishing between cases in which CP is/is not applicable, and the bearing this has on the epistemologies of mathematics and modality, can be inferred from the following passage:-
If possible worlds are supposed to be the same sorts of things as our actual world; if they are supposed to exist in as concrete and robust a sense as our own; if they are supposed to be as real as Afghanistan, or the center of the sun or Cygnus A, then they require the same sort of evidence for their existence as other constituents of physical reality.” [ibid. p326 Skyrms’s itals.]
CP is not applicable in the mathematical case, because the salient objects – numbers, sets etc – are abstract; but CP is applicable in the modal case, because the salient objects according to Lewis’s modal realism – possible worlds and their inhabitants – are typically concrete. In this way, the Objector manages to maintain the line that modal realism implies scepticism about modal epistemology, without committing himself to scepticism about mathematical epistemology as well.

Lewis [1986 pp110-112] acknowledges that Skyrms’s abstract/concrete analysis of the CP applicability distinction does indeed revive the Objection against modal realism. However, he contends that the abstract/concrete analysis is objectionable in itself, and that there is another plausible way to analyse the CP appplicability distinction, sc. in terms of the necessary vs the contingent, which makes both mathematics and modality into cases in which CP does not apply, defeating the Objection against modal realism..

Lewis prefaces his complaint against the abstract/concrete analysis with the somewhat surprising-looking claim that he does not
“really know what is meant by someone who says that mathematical objects are abstract whereas donkeys … are concrete” [p111].
At any rate, a plausible construction of what is meant is the so-called Negative Way, according to which donkeys are concrete in virtue of standing in spatiotemporal relations to eachother, whereas mathematical objects are abstract in virtue of not standing in spatiotemporal relations to eachother. All well and good as far as it goes, since
“the Negative Way does at least make a relevant distinction: the entities it calls abstract cannot be known by causal acquaintance.” [ibid.]
However,
“that gives us no help in understanding how else they can be known. To say that abstract entities alone are known without benefit of causal acquaintance seems unprincipled: they’d better be, or they can’t be known at all! Could ‘abstract’ just mean ‘don’t worry’?” [ibid.]

This establishes my earlier point, that the Objector who accepts CP’s non-applicability in the case of mathematics avoids scepticism about mathematical epistemology only by assuming the burden of explaining how mathematics is understood. Prima facie, this is not an impossible task. However, the attendant difficulty is exacerbated if the Objector rejects TUT, as presumably he is motivated to since accepting TUT risks engaging a commitment to modal realism!
As indicated above, Lewis proposes an alternative account of epistemology, which invokes a necessary/contingent analysis of the CP applicability distinction. In particular, the “department of knowledge” involving causal acquaintance is “demarcated by its contingency”. Instances of causal acquaintance “set up patterns of counterfactual dependence whereby we can know what is going on around us”.  For example, a’s visually-based knowledge involves visual experience which depends counterfactually on the scene presented before a’s eyes. By contrast, instances of non-causal acquaintance do not set up patterns of counterfactucal dependence:-
“Nothing can depend counterfactually on what mathematical objects there are, or what possibilities there are”. [ibid.]
In which light, consider the existence of donkeys:-
“Our contingent knowledge that there are donkeys at our world requires causal acquaintance with the donkeys, or at least with what produces them. Our necessary knowledge that there are donkeys at some worlds ... does not require causal acquaintance either with the donkeys or with what produces them.” [p112]
Presumably, our necessary knowledge of the existence of donkeys at some worlds is the result of our expanding our beliefs for the sake of theoretical unity, as per TUT again. Chihara [1998] objects against TUT on the grounds that the appeal to it is adhoc, that Lewis’s account of knowledge is undermined by Gettier-type problems, and that claiming to know about the existence of other worlds on the basis of TUT is “preposterous”. As will be clear, I regard only the Gettier issue as representing a potentially serious problem for Lewis:-

The ad hoc-ness issue
Chihara suggests that Lewis accepts TUT only because
“he [can] find no other plausible way both of explaining how we could have knowledge of the existence of sets and also of defending his analysis of possibility against the [Epistemology] Objection … Lewis gives us no justification or grounds for accepting his epistemological thesis: he just asserts it”. [1998 p91]
Lewis might complain that Chihara’s criticism is unfair. If TUT is a plausible way of explaining mathematical knowledge and defending the Lewisian analysis of possibility, it is not easy to see what the objection is against Lewis’s invoking TUT. However, in fairness to Chihara, it must be said that he does not pursue this particular line against TUT beyond the remarks quoted.

The Gettier issue
Chihara argues [pp91-93] that Lewis’s account of knowledge is undermined by Gettier-type problems in the case of our supposed knowledge of other worlds, even if Lewis is correct in arguing that such problems do not arise in the case of simple non-contingent matters. To see this, Chihara invites us to notice the striking similarity obtaining between Lewis’s thesis and the traditional Tripartite account of knowledge:-
TAK. a knows that p iff (i) it is true that p; 
(ii) a believes that p; and
(iii) a’s belief that p is justified.
As Chihara notes, we obtain what is “essentially” TUT from TAK by replacing (iii) with
(iii*) a accepts that p for the sake of theoretical unity. 

It may be helpful to have a Gettier case to hand:- Suppose that I am watching a BBC news broadcast, in which the newsreader announces that David Cameron has joined the Jehovah’s Witnesses. Hearing the announcement, I form the belief that it is true. Since the announcement is from a source hitherto renowned for its impeccable reliability – the BBC – my belief is justified. It is moreover true; at this very moment Cameron is to be found outside Ladbroke Grove tube selling the Watchtower. The fly in the ointment:- the newsreader really has no idea of Cameron’s political change of heart. His announcement was coincidental, the result of overwork and a surfeit of amphetamine sulphate. In short, I have the justified true belief, but not the knowledge, that Cameron has joined the Jehovah’s Witnesses.

Lewis’s gloss on what we learn from Gettier:-
“the analysis of knowledge is plagued with puzzles about truths believed for bad reasons: because you were told by a guru, because you made two mistakes that cancelled out, because by luck you never encountered the persuasions or evidence that would have misled you”. [1986 p113]
As intimated, Lewis thinks that Gettier problems do not arise in the case of simple non-contingent matters. Once you fully understand and accept statements such as “2+2=4” and “there are no true contradictions”, it is not clear that your acceptance of these statements would fail to amount to knowledge if you only accepted them because e.g. you happened to miss the “lecture by the persuasive sophist who would have changed your mind” [ibid.].

Chihara grants Lewis’s claim, but argues that Gettier problems still arise for Lewis’s account in the matter of other worlds:-
“Would a person’s acceptance of the claim that there are other worlds fail to be knowledge if he accepted the claim only because a guru told him to? Absolutely!” [p92]
This seems fairly obviously correct. Even if modal realism is true, believing in other worlds just because the guru told us to does not make it the case that we know there are other worlds.

Eventually, we seem to be left with an impasse. Lewis’s account of epistemology is good because it stops Gettier problems from arising in the case of simple non-contingent matters, but bad because Gettier problems still arise in the case of other worlds. Defenders of modal realism will emphasize the benefit, attackers the drawback. 

The ‘preposterousness’ issue
Chihara complains [p93] that 
“It seems … preposterous to claim to know that there are countless worlds containing planets similar to, but different from, the earth, and on which there live people very similar to us – and also to claim that we know such things not on the basis of scientific studies or experimentation, but rather because accepting such worlds yields a reduction in the diversity of the primitive notions we need for our total theory.” [p93]
However, this passage does not really amount to an argument against TUT, so much as a restatement of Chihara’s distaste for it.

This concludes the investigation of the Epistemology Objection. To recapitulate, the complaint is that modal realism implies an implausibly pervasive scepticism about modal epistemology. The success or failure of the Objection turns on the question of CP’s universality or restrictedness. If the Objector holds that CP is universally applicable, he risks committing himself to a pervasive scepticism about mathematical epistemology. Hence, his likely preference is to accept that CP has only restricted applicability – applying to modality to the detriment of modal realism, but not to mathematics as would be to the Objector’s detriment. However, in this case the Objector assumes the burden of explaining how we do come by knowledge of mathematics, given that we do not do so by invoking TUT. Ultimately, neither the Objection against modal realism, nor the associated objection made by Chihara against Lewis’s account of epistemology, is to be regarded as conclusive against modal realism, although sufficient concern may have been raised to motivate the investigation of anti-realism in ch3.


Conclusion

In the present chapter I have introduced modal realism, according to which modal statements are to be analysed as statements about possible worlds, and such worlds exist in sufficient numbers so as to support substantially all the modal claims people make. Modal realists divide into actualists and possibilists, the former believing that other worlds are actual, abstract and ontologically subsidiary to α, the latter believing that other worlds are non-actual, concrete and ontologically on a par with α.

Actualism further subdivides into variants known here and elsewhere as proxy actualism and ersatzism, these differing in terms of their respective stances on the question of domain-inclusion. However, proxy actualism is committed to the existence of abstract actual individuals which are possible jabberwocks, sons of Wittgenstein etc; whereas ersatzism is justly criticized by Lewis for its reliance on primitive modality and for its lack of descriptive power.

Possibilism further subdivides into Leibnizian, Kripkean and, thanks to Lewis’s endeavour, counterpart-theoretical variants. In some respects, the Leibnizian and Kripkean variants accord better with commonsense than does Lewis’s cct. Nevertheless, Leibnizian possibilism is undone by its exposure to a pair of objections from epistemology/luck and contingency; and Kripkean possibilism by Lewis’s problem of accidental intrinsics. The defects of Leibnizian and Kripkean possibilism have served to establish cct’s dominance of the intra-possibilist dialectic. However, several objections have been raised against the latter’s possibilist/modal realist element. It is very much to be doubted whether even the most pressing of these – the Objections from Cardinality and Epistemology – amount to conclusive refutation of modal realism. Nonetheless we may grant that sufficient doubts have been raised against modal realism, to motivate investigating the anti-realist alternatives. Thus ch3 immediately following.

Chapter 3. Modal Anti-Realism.

In the present chapter, I present the two main strands of pw-theoretical modal anti-realism:-
Naive Atheism denies the existence of other worlds. When combined with pw-semantics, atheism comes in a form which is attractive to determinists, but leads to modal collapse – the collapse of positive contingencies into necessities, and of negative contingencies into impossibilities.
Agnosticism [Divers] neither affirms nor denies the existence of other worlds. Agnosticism avoids the problems of atheism – i.e. rescues contingency – by insisting on a moderate version of agnosticism, permitting actual counterparts, and instigating a policy of rational dispensation.

I contend that atheism has equal entitlement to the actual-counterpart apparatus. Moreover, to recognize this is, not just to undermine agnosticism, but also to call into question the entire project of analysing modality by means of possible worlds. Agnosticism is undermined, because a sophisticated atheism taking up its entitlement to actual counterparts gets results in terms of the assertibility of contingency claims, which bear comparison with those obtained by agnosticism. The recognition of this causes anti-realists to revert to their default atheism – no need any longer to pretend we’re not sure about worlds: they don’t exist! The pw project is called into question, because of the ability of sophisticated atheism to assert a very wide range of salient modal claims. If actual counterparts are permitted, then world-locutions are only required for generally very weird and esoteric modal claims, which we do not need to accept – and thus the role of possible worlds is marginalized. In summary, these considerations pave the way for the actual-counterpart-theoretical atheism sc. act, the formalization of which is the subject-matter of ch4 following.



(3.1) PW-Theoretical Atheism.

Assume firstly the prognosis arising from ch1, that modal semantics is or ought to be pw-theoretical; and secondly the prognosis arising from ch2, that realism is objectionable. Then it is as Divers remarks
“natural to pursue the anti-realist thought that one can secure at least some of the benefits associated with talking in terms of other worlds without incurring a commitment to the existence of any other worlds.” [Divers 2004 p660]
Now the obvious way to implement an anti-realist programme is simply to deny the existence of other worlds. In this spirit, modal atheism denies the realist thesis of chapter 2
R. ∃w ¬w=α
and affirms its negation:-
¬R. ∀w w=α.

(3.1.1) Atheism’s Modal Collapse
Divers notes that atheism, which he calls “error theory about other worlds”, when conjoined with a pw-theoretical analysis of modality e.g. 
PWS. ◊/⎕A iff ∃/∀w (at w, A),
“leads to error theory about modality.” [ibid. p661] He means by this that atheism collapses the distinction between possibility and necessity, denying contingency and making all truths (and indeed all possible truths) necessary, and all falsehoods (and indeed all possible falsehoods) impossible. 

That atheism leads to such a modal collapse is easily shown. We begin with an arbitrary true modal claim:-
1. ◊A
Application to 1 of the left-right direction of the PWS biconditional yields 1’s pw-theoretical translation:-
2. ∃w at w, A
From 2 and ¬R, we obtain
3. ∀w at w, A
Application to 3 of the right-left direction of the PWS biconditional yields the modal claim of which 3 is the pw-theoretical translation:-
4. ⎕A
Application to 1 and 4 of the conditional introduction rule yields the necessity of possibility:-
5. ◊A → ⎕A
Finally, application to 4 of the A/¬A substitution rule yields the impossibility of the possibly false:-
6. ◊¬A → ⎕¬A QED

(3.1.2.) Atheism and Determinism
To do modal atheism justice, it cannot be said to be entirely lacking in motivation, modal collapse and all. To see this, we have only to reflect on those periods during which determinism has been taken very seriously indeed. Even now, with determinism at least at the micro level seemingly consigned to history by the quantum revolution, restricted versions of it linger in the dialectics of many macro-level sciences, for instance in the social sciences, (e.g. notably in the works of B.F.Skinner) and in e.g. Marxism. The underlying point here is of course that, to some determinists, modal collapse is an attractive consequence of atheism.

Determinists believe that events are effects. But cause and effect can be understood in different ways. A certain kind of determinist, call him a weak determinist, has a Stalnaker-Lewis-style account of causation [q.v. ch2], whereby causes are not necessary; X causes Y if, inter alia, the nearest X-worlds are also Y-worlds – which leaves open the possibility of there being remoter X-worlds which are not also Y-worlds. Clearly, modal collapse is not an attractive consequence of atheism for the weak determinist. However, consider the strong determinist, who has a necessitarian conception of causation. For her, things literally couldn’t have turned out differently: ∀p(p→⎕p). Modal collapse most expeditiously captures her view of modality.

Ultimately, of course, the strong determinist intuition runs slapbang into the opposing very powerful intuition that there just is contingency. Given that this latter intuition is both very strong and latterly sustained by the quantum revolution, the remainder of this chapter – and indeed the remainder of the entire opus – proceeds on the assumption that modal collapse and the denial of contingency are consequences which an adequate theory of modality will avoid.

(3.1.3) Atheism, Assertibility Gaps, and Subsidiary Norms
Throughout his investigation, Divers refers to the atheist as suffering an assertibility deficit in relation to the folk and the realist. He means by this that there is a wide range of modal claims, sc. contingency claims, which the folk assert, and which the realist can join the folk in asserting in view of his belief in worlds, but which the atheist cannot assert; as we learnt just now from the proof in 1-6 above. At this juncture, it is worth alluding briefly to a minor omission on the part of Divers. To whit, he does not mention the fact that the atheist is also fairly described as suffering an assertibility surfeit. This is exactly the converse of the deficit mentioned by Divers, insofar as for every contingency claim which the atheist cannot join the folk/realist in asserting, there is conversely a necessity/impossibility claim which the atheist cannot join the folk/realist in resisting. Ultimately, Divers’s deficit and the corresponding surplus are equally valid descriptions of the atheist’s failure to halt the slide from diamond to box. With this understood, however, I propose for the purposes of simplification largely to maintain Divers’s focus on the deficit cases. That said, this scruple is what underlies my occasional propensity hereafter to speak neutrally of residual assertibiltiy gaps rather than of deficits.

Divers concedes charitably enough that the atheist need not draw a Quinean anti-modalizing moral from the looming modal collapse. A pro-modalizing atheist may still vindicate our modal practices by appealing to some norm which is subsidiary to the norm of truth, in the manner of Hartry Field’s error-theorist appealing to conservativeness in the case of mathematics [Field 1980, 1989], or of van Fraassen’s agnostic appealing to empirical adequacy in the case of physical unobservables [van Fraassen 1980].

To illustrate how subsidiary norms work in the mathematical case, let us attend to a definition of mathematical conservativeness:-
“[According to Field] applied mathematics is conservative: when we add mathematics to a nominalist theory we get nothing new about the non-mathematical world. More formally, mathematical theory S is conservative over non-mathematical N iff, for any A in the language of N, if A is a consequence of S+N, then A is a consequence of N alone.” [Melia 2006 p203]
Thus, although as far as Field is concerned, mathematical sentences are strictly false, since they concern platonic entities which do not exist, nevertheless the conservativeness of mathematics allows him to extract a pro-mathematizing moral. He can use mathematical sentences without committing himself to consequences to which his dialectically antecedent non-mathematical theories do not commit him. 
To illustrate how subsidiary norms work in the case of physical unobservables, let us attend to a definition of empirical adequacy:-
“A hypothesis is empirically adequate when it predicts or explains the phenomenological regularity that it was proposed to predict or explain. This means that an empirically adequate hypothesis is one that—together with certain auxiliary assumptions—deductively [implies] the phenomenological regularity as an observation.”
On something like this basis van Fraassen, reasonably as it may be supposed, maintains an agnostic attitude towards the hypothesis that unobservable physical entities exist. He is disposed to use as part of his discourse sentences featuring unobservable physical entities, since these are often very successful predictors and explainers of phenomenological regularities in the way described. And this fact about such unobservables-sentences allows van Fraassen to extract a pro-unobservables moral. He can continue to use unobservables-sentences just when doing so leads to predictive or explanatory success.

Likewise, in the modal case, Divers’s (charitable) hope is that, in order to distinguish ‘good’ modalizing from ‘bad’, the pro-modalizing atheist can exploit, if not an alleged conservativeness or empirical adequacy on the part of modality, then some other suitable norm. Divers suggests in passing that such a norm might take the form of ‘actualistic adequacy’ [2004 p677], by which he means, if I understand him correctly, consistency with truth in α. However, he produces no fleshed-out proposals to this end; as indeed he is entitled not to, his main point being that the agnostic does an adequate job of closing the assertibility gap between himself and the folk/realist, without needing to appeal to subsidiary norms, whatever forms these take [Divers 2004 p661 et passim]. Accordingly, I turn my attention to agnosticism in the section immediately following.


(3.2) Agnosticism

Earlier, I described atheism as the obvious way to implement the anti-realist programme, since it expressly denies the existence and affirms the non-existence of other worlds, i.e. denies R and affirms ¬R. A less obvious way of implementing anti-realism is to refuse to accept the existence of other worlds, but to stop short of affirming their non-existence. Thus Divers’s modal agnostic, who refuses to affirm or deny either R or ¬R.

(We may think of atheism and agnosticism as the modal special cases of, respectively, strong and weak forms of sceptic. That is to say, the strong sceptic about X denies X, whereas the weak sceptic only refuses to affirm X.)

The main feature of atheism proved to be its tendency to generate modal collapse (q.v. §1), which Divers refers to as comprehensive error theory about modality. Agnosticism is motivated, as far as Divers is concerned, as a means of avoiding this mishap without committing oneself to the existence of other worlds.

It is all very well motivating agnosticism as the non-collapsing alternative to atheism. But if agnosticism is to be motivated in this way, it had better not itself collapse the modalities. At least, it had better not do so too extensively. For, as Divers concedes, a partial collapse is inevitable in the form of an analogous agnosticism about modality:-
“Worldly agnosticism when allied to the Lewisian analyses does commit one to agnosticism about a substantial range of modal claims about which the realist and the folk are not agnostic. The underlying and decisive logical point is as follows. Anyone who holds that an arbitrary biconditional, P iff Q, is true, and who is agnostic about the right-side, Q, of the biconditional is thereby bound to be agnostic about the left-side, P, as well. For one who is not agnostic about P either holds P true or holds P false. And given that she holds the biconditional true, she must, on pain of irrationality, hold that Q has whatever truth-value she takes P to have. But to hold of either truth-value that it is the truth-value of Q is contrary to agnosticism about Q. Thus, given commitment to the truth of the biconditional, sustained agnosticism about the right-side, Q, requires agnosticism about the left-side, P.” [Divers 2004 p673]
Divers devotes a considerable part of his [2004] to the devising of various measures and pretexts to diminish or explain the assertibility gap between the agnostic and the folk/realist. Accordingly, following Divers let the ‘disputed cases’ be those instances where agnosticism about worlds does lead to a corresponding agnosticism about modality. Divers argues that the residue of disputed cases is both much smaller than might be anticipated on two plausible assumptions, and (hopefully) substantially disposable by means of a policy of rational dispensation.

The two plausible assumptions which diminish the number of disputed cases are moderate agnosticism, and the permissibility of (same-world, and a fortiori) actual counterparts. The assumption of moderate rather than radical variants of agnosticism, conjoined with the technique of recasting pw-analyses as claims with unrestricted negative existential content, permits the assertion by the agnostic of a range of absolute and relative necessities, and also of a range of counterfactuals. 

Moreover, and as ultimately proves crucial to my purposes, the assumption that actual individuals can stand in as counterparts for other actual individuals permits the assertion of a very wide range of contingencies. All of this is explained in the subsections now following, together with the operation of Divers’s policy of rational dispensation over the remaining disputed cases.

(3.2.1) Moderate Agnosticism and Necessity/Impossibility
Divers asks us to consider dialethons, as a way of distinguishing between radical and moderate forms of agnosticism. [Divers 2004 p669] Dialethons are supposedly unobservable entities which are simultaneously F and ¬F. Radical agnosticism is agnostic about absolutely anything unobservable, even intuitively impossible entities such as dialethons. In contrast, moderate agnosticism is agnostic about unobservables, unless further characterization gives ground for disbelief in the entities in question, as is the case with dialethons – the existence of dialethons would constitute counter-examples to the Law of Non-Contradiction. In particular, then, the radical agnostic is agnostic about the existence of dialethons, whereas the moderate agnostic denies their existence. Hereafter Divers’s focus, reasonably enough, is wholly upon moderate agnosticism.

Serendipitously, the focus on moderate agnosticism brings within the agnostic’s assertorial purview a range of plausible necessity/impossibility claims. In order to bring this about, Divers analyses necessity and impossibility claims as claims with unrestricted negative existential content, thus:-
7. It is necessary that A iff there is no world at which ¬A
8. It is impossible that A iff there is no world at which A
With necessity and impossibility viewed in terms of unrestricted negative existential content, and assuming moderate atheism, it seems that the agnostic will be able to assert a battery of:-

(i) Necessitated truths of first order logic such as that nothing can be both F and ¬F; thus the way in which dialethons are characterized gives grounds for disbelief in them;

(ii) Necessitated analytical truths such as that there can be no male vixens, on the basis that the meaning of “vixen”, viz. female fox, gives grounds for disbelieving in the existence of male vixens; and, 

(iii) On many accounts, necessitated metaphysical truths such as that nothing is both water and devoid of hydrogen, on a basis such as that the essential property water has, of consisting in molecules of H2O, gives grounds for disbelieving in the existence of water which is devoid of hydrogen.

The same conjunction of moderate agnosticism with negative existential content can also be adapted so as to enable the assertion by the agnostic of claims as to relative necessity and impossibility. In order to achieve this, we modify the respective schemata for necessity and impossibility, 7 and 8. Thus, for an arbitrary modality R, we have:-
7R. It is R-necessary that A iff there is no world at which R and ¬A
8R. It is R-impossible that A iff there is no world at which R and A.
Amongst the more interesting applications of the negative existential apparatus in sub-domains of relative necessity are claims pertaining to the laws of nature. Thus, to describe A as nomologically necessary/impossible is to make a claim, the analysis of which is that there is no world at which the laws of nature hold and ¬A/A.

The apparatus of unrestricted negative existential content can also be adapted so as to enable the assertion by the moderate agnostic of both strict and subjunctive (typically counterfactual) conditionals. In the latter case, Divers asks us to consider the intuitively true counterfactual
9. If there had been no humans, there would have been no golfballs.
Here, as Divers points out [ibid. p672], the realist is likely to start out at something of a disadvantage. If, as certainly seems likely, he follows Lewis in incorporating into his account a principle of recombination, he will take himself as knowing a priori that there exist worlds containing golfballs but not humans. For this reason, Lewis and those following him appeal to a similarity relation, as a way of privileging some worlds over others. The hope is then that, with the right sort of similarity relation, all the human-lacking worlds otherwise most similar to α will turn out to be devoid of golfballs as well. This suggests the following analytical schema for counterfactuals, which incorporates the idea of unrestricted negative existential content:-
10C. A ⎕ C iff there is no world which is an (A&¬C) world and a selected world.
The schema for a strict conditional A⇒C falls out quite naturally as the limiting case for counterfactuals:-
10S. A⇒C iff there is no world which is an (A&¬C) world.
The point, according to Divers, is that the agnostic can join the realist in asserting counterfactuals analysed by means of 10C because, thanks to the conditional nature of the analysis, there is nothing especially realistic about the underlying story. For 10’s analysis
11. There is no world which is an (A&¬C) world and a selected world
is equivalent to the conditional claim
12. Any world which is an (A&¬C) world is not a selected world.
And 12 has the form ∀x(Fx→¬Gx), which does not bear any commitment to the existence of x’s. Which is as much as to say, the agnostic is quite as entitled as the atheist to describe what makes other worlds similar to α if there are any.

(3.2.2) Actual Counterparts & Contingency
The apparatuses of moderation and of unrestricted negative existential content cannot be put to service in enabling the assertion by the agnostic of contingency claims. Nevertheless, the agnostic does appear to have the warrant to assert many contingencies. That is to say, he has
  “a warrant to assert that it is possible that A, in certain cases, by courtesy of a warrant to assert that at the actual world, A”. [ibid. p674]
This has implications which are obvious for de dicto claims, but rather more interesting for de re claims, in respect of which the theory of counterparts comes into play. Divers illustrates the operation of such an ‘actualistic’ warrant with the case of Quine, who is actually a philosopher, but who is also a possible non-philosopher in virtue of having a legion of actual counterparts, sc. the very many white male Republican-voting Americans who are non-philosophers. To extrapolate from Quine’s case, quite generally an arbitrary a is a possible F if an actual counterpart of a is F. By this means, there is brought within the agnostic’s assertorial purview an exceedingly wide range of contingency claims; namely all the very many de re claims concerning individuals in respect of which α furnishes worldmates which can do duty as counterparts.

(3.2.3) The Residual Assertibility Gap & Rational Dispensation
Thus far, Divers has established that the agnostic can join the folk/realist in asserting ranges of absolute and restricted necessity claims, ranges of strict and counterfactual conditional claims, and a range of contingency claims. However, as Divers acknowledges, there yet remains a residual assertibility gap between the agnostic and the folk/realist. As explained earlier, this comprises, firstly an expressive deficit which is Divers’s main focus, and secondly the corresponding surfeit identified earlier in this chapter. The deficit consists in all those de re contingency claims of the form ◊Fa, in respect of which α does not provide counterparts of a’s which are F, and which the folk/realist but not the agnostic can and typically do assert. The surfeit consists in the corresponding necessity/impossibility claims which the folk/realist but not the agnostic can and typically do deny. The claims comprising the assertibility gap are referred to as the Disputed Claims (DC’s).

Divers disposes of the DC’s by means of a policy of rational dispensation. Rational dispensation can be thought of as a sort of error-theory-lite, in that it is conceded that the DC’s do indeed constitute a class of claims which the folk/realist can and typically do assert, but which the agnostic cannot assert. However, this difference between the folk/realist and the agnostic does not matter, because the DC’s are rationally conservative with respect to an agent’s ordinary non-modal beliefs. That is to say, adding the DC’s to an agent’s antecedent belief set makes no difference to his practical or intellectual conduct; alternatively, adding them in does not give him reason to Φ if he did not have reason to Φ in the first place.
Divers illustrates the operation of the rational dispensation policy with two cases, one de dicto and the other de re. In practice, the implementation of the policy involves replacing the DC’s believed by the realist but not by the agnostic with functionally equivalent beliefs acceptable to the agnostic.:-

The De Dicto Case
The de dicto case concerns how the agnostic can join the realist in resisting the “eccentric hypothesis” [p679]:-
13. Necessarily donkeys exist
The realist easily resists 13, by asserting 13’s negation
14. It is possible for donkeys not to exist
on the basis of his belief that:
15. There is some world lacking donkeys.
On the face of things, 15 gives the realist a reason to Φ, i.e. to resist 13, which the agnostic lacks. Moreover, there is certainly an assertibility gap, in the form of 14, which the realist can and the agnostic cannot assert. However, on closer inspection the agnostic is as well-placed as the realist to resist 13. His grounds are simply that he has been presented with no a priori grounds for 13. Having no grounds for asserting 13 makes the agnostic, as required, practically indistinguishable from the folk asserting 14 and the realist asserting 15. So 14 counts as rationally dispensable. Nothing is gained, as it were, by being able to assert it.

The De Re Case
The de re case concerns whether we have reason to Φ, with Φ-ing presented as an attempt to ψ. Often, having reason to φ amounts to no more than the requirement that the agent should not believe that ψ-ing is impossible. For instance, we ordinarily take a person s to have perfectly good reason to attempt to solve Goldbach’s Conjecture (GC), so long as s does not believe that solving it is impossible. The ‘agnostic’ who does not believe in the impossibility of a solution to GC is practically indistinguishable from the ‘realist’ who positively believes in the possibility of a solution to GC. Again, the belief is rationally dispensable, in the sense that nothing is gained by being able to assert it.

However, there are other cases, ones in which the requirement of rationality exerts a stronger force, when non-belief that ψ-ing is impossible is not practically indistinguishable from the belief that φ-ing is possible. In particular, the rationality requirement very often exerts a stronger force in life-and-death cases, when too much rides on ψ-ing’s likelihood to rely on ψ not being impossible. In such cases, the rational dispensation strategy becomes more difficult for the agnostic to implement. However, Divers argues that nevertheless in many such cases rational dispensation still can be implemented.

Divers’s own example, of a life-and-death case permitting the implementation of rational dispensation, concerns the reasonableness or otherwise of car-drivers overtaking [ibid. p681]. In the appropriate circumstances, the folk do consider it reasonable to attempt overtaking, i.e. when the attempt is likely to succeed.  Moreover, intuitively the realist can partake in the folk’s optimism, i.e. on the basis as that overtaking is likely to succeed because it is successful in the nearest possible worlds in which it is attempted.

And here, finally, the agnostic hoping to implement rational dispensation faces something of a challenge, because it will not do to execute a move analogous to the move in the de dicto case: the agnostic cannot argue that there is no discernible difference between the behaviour of the folk/realist as a result of believing that overtaking is likely to succeed, and his own behaviour as a result of not believing that overtaking is impossible. If his behaviour, when he did not believe that overtaking was impossible, matched that of the folk/realist when the latter believed that overtaking was possible, the long-run consequences would obviously be unpleasant for the agnostic.

Divers contends that the rational dispensation policy can meet the enhanced challenge, suggesting three ways of achieving this:-

*The first way involves dispensing with the realist’s belief in nearby successful-overtaking worlds, and replacing it with the corresponding actual-counterpart-theoretical belief. Many potential-overtaking occasions will have as counterparts other relevantly similar actual occasions when overtaking has been successful.

*The second way involves dispensing with the realist’s belief in nearby successful-overtaking worlds, and replacing it with the corresponding counterfactual belief:-
16. If overtaking were attempted (now), it would succeed.
Doing this allows the agnostic to take advantage of the conditional, non-realistic analysis of counterfactuals, which was expounded earlier. Then the agnostic can view overtaking as reasonable because, if there are nearby worlds, then they are successful-overtaking worlds.

*The third way involves dispensing with the realist’s belief in nearby successful-overtaking worlds, and replacing it with the corresponding non-modal belief, that in most such cases attempts to overtake will succeed. The basic idea here is that the agnostic can view overtaking as reasonable because, given that it succeeds in other cases, it will succeed this time too.

In each case, the point is that the agnostic can assert the relevant belief, and in doing so maintain an attitude which is practically indistinguishable from the attitude manifested in the folk/realist belief that overtaking is possible.

The two sections which presently follow, sc. §3 and §4, pursue a pair of objections against agnosticism. The §3 objection has only a limited bearing on the case which I am building for act, and I mention it mainly for the sake of dialectical completeness, and because it is novel. The objection argues that agnosticism is conceptually unstable, collapsing into a weak possibilism: realistic about other worlds, pallidly agnostic about their number and nature. The §4 objection bears more directly on the motivation for act. This objection, which to the best of my knowledge is novel in written form, exploits the actual counterpart machinery previously mothballed by Lewis’s 5th postulate [Lewis 1968 p114], and now brought out of retirement by Divers [2004 p674]. The objection points out that a resurgent atheism is as entitled as agnosticism to this machinery, and uses this point to undermine the motivation for agnosticism.

The pursuit of the §4 objection leads to a further undermining, of the entire pw edifice. This undermining takes the form of a redundancy objection, the subject of §5, which exploits Divers’s actual counterpart machinery, and drastically marginalises the role of other worlds in modality. It is the demise of pw-theory, and specifically because this comes about by exploiting the actual counterpart machinery to its full potential, which eventually motivates the actual counterpart theory act, which is introduced in ch4.




(3.3) Agnosticism Collapses Into ‘Weak Possibilism’

To summarise proceedings thus far, Divers’s main thesis is that atheism is subject to modal collapse, and that a significant advantage of agnosticism is that it avoids modal collapse. Modal collapse has the consequences that whenever A is possibly true then A is necessarily true, and that whenever A is possibly false then A is necessarily false. We agree with Divers that modal collapse is to be avoided where possible.

pw-theoretical atheism is, as before, the conjunction of pw-semantics and strong scepticism about worlds:-
PWS. ◊/⎕A iff ∃/∀w (at w, A) …
… and …
¬R. ¬∃w(¬w=α)
Let us accept Divers’s reasonable-looking claim, that
C. Atheism entails modal collapse.
Then, contraposing, if there is no modal collapse then either PWS or ¬R is false. Both the agnostic and the atheist agree that PWS is true, therefore it is ¬R that is at issue between them. However, the negation or ¬R is equivalent to Realism:-
R. ∃w ¬w=α.
Hence, if the agnostic is to avoid modal collapse then they must be committed to the existence of at least one non-actual world. This turns the agnostic from someone who suspends judgement about the existence of non-actual possible worlds, to someone who suspends judgement about how many and which non-actual possible worlds exist. This considerably undermines the stability of the agnostic’s position, collapsing it into the weakest kind of possibilism.

Of course, the agnostic may reject C, and indeed I presently argue against C. C is true when we restrict consideration to de dicto modal claims, and C is true for de re modal claims as well in the context of Kripkean semantics i.e. if transworld identity is used for claims such as that Quine might not have been a philosopher. However, with an actual-counterpart-theoretical semantics for de re modal claims, it is plausible that C will be false since, as I presently argue, the atheist can generate examples where a is F but not necessarily F (since a has an actual counterpart which is not F), and where a is not F but possibly is F (since a has an actual counterpart which is F). So the agnostic faces a dilemma:-
Horn 1. He continues to maintain, pace our arguments above, that atheism entails modal collapse, but then he must admit that agnosticism faces either modal collapse or collapse into the weakest form of possibilism; or
Horn 2. He accepts that atheism does not entail modal collapse. Doing so allows agnosticism both itself to avoid modal collapse and to remain distinct from possibilism, but promises to leave agnosticism demotivated by the rehabilitation of atheism.


(3.4) Actual Counterparts and the Demotivation of Agnosticism

Agnosticism, at least the way Divers shapes it, is motivated by the claim, seemingly established in §1, that the main anti-realist alternative to it sc. atheism leads to modal collapse; or as Divers more or less puts it, error theory about worlds leads to a thoroughgoing error theory about modality. In contrast, Divers’s agnostic escapes a thoroughgoing agnosticism about modality, the analogue of the atheist’s modal collapse. Moreover, this escape is substantially attributable to the agnostic’s appeal to actual counterparts, which brings within the agnostic’s assertorial purview a wide range of de re contingency claims.

In the current section, I argue that atheism does not after all lead to modal collapse, precisely because the atheist can lay claim to the very same apparatus of actual counterparts for the very same purpose of making possible the expression of de re contingency. Divers’s motivation for agnosticism is that, thanks in large measure to the contingency-enabling apparatus of actual counterparts, it is expressively superior to atheism. Since the adoption of this very apparatus makes atheism comparable to agnosticism in terms of expressivity, I find that the motivation for agnosticism is undermined. Why merely suspend belief in a range of intuitively objectionable entities viz. other worlds spatiotemporally unrelated to our own, when we can get comparable results when actively disbelieving in them?

The notion of actual counterpart has a strangely controversial pedigree. The early Lewis prohibited (same-world and a fortiori) actual counterparts by means of his 5th postulate [1968 p114], but without explaining the reasons for his prohibition. However, since then he at least flirted with the idea of rescinding the ban, as the ‘poor Fred’ and eternal recurrence cases attest, albeit without explaining why he imposed the ban in the first place:-
“Here am I, there goes poor Fred; there but for the grace of God go I; how lucky I am to be me, not him. Where there is luck, there must be contingency. I am contemplating the possibility of my being poor Fred, and rejoicing that it is unrealized. … He is my counterpart under an extraordinarily generous counterpart relation, one which demands nothing more of counterparts than that they be things of the same kind. … It is not some other world, differing haecceitistically from ours, which represents de re of me that I am Fred; it is Fred himself, situated as he is within our world.” [Lewis 1986 pp231-232]

In the course of the passage from which the above exerpt is taken, Lewis offers a second context in which we might naturally expect to find a role for same-world counterparts. If modal realism is true, then it is plausible to suppose that there is a world w of eternal recurrence, consisting of successive époques which duplicate one another qualitatively in terms of their histories. And then it is natural to construe the relation obtaining between mutual duplicates from different époques of (the same) w as one of counterpart-hood.

I contend that Divers is right to defy Lewis’s ban on same-world and actual counterparts, for the ban is counter-intuitive and arbitrary. The ban is counter-intuitive, because surely a and b’s being worldmates is if anything yet another point of similarity between them. Since the counterpart relation has its basis in similarities between individuals, surely being b’s worldmate ought to make a more likely to be b’s counterpart, rather than entirely precluding the possibility of a being b’s counterpart. The ban is arbitrary, because there is not on the face of it any reason why counterpart-hood should be allowed to obtain between non-world-mates but not between world-mates.

I contend that Divers is right to flout the ban on actual counterparts, but also that he does not digest the full implications of doing so. The salient point here is that the atheist can be reasonably supposed to share the agnostic’s entitlement to the actual-counterpart apparatus. Indeed, Divers acknowledges this in at least two footnotes, which contrast atheism’s propensity to engender the collapse of de dicto modalities, with its more congenial treatment of de re modalities once it avails itself of the actual-counterpart apparatus:-
“I emphasize that the modal collapse that has been described concerns only the de dicto modalities. The error theorist will be entitled to assert de re possibilities and contingencies in some cases where it is false that A, and entitled to deny de re necessities in some cases where it is true that A.” [Divers 2004 p667]
Later elaboration of this point [2004 §5.2] makes plain that actual counterparts are what he has in mind in the first passage. The second passage requires no elaboration:-
“If we were considering the case of quantified modal logic, the prospect of counterpart-theoretic truth-conditions would come into play and so, thereby, would the prospect of Fa being true even though Fa is not true. That prospect is opened up by counterpart relations that obtain between worldmates” [Divers 2006 p203]
The acknowledgement of his entitlement to the actual-counterpart apparatus puts the atheist in a position to assert substantially the same range of de re contingencies as the agnostic. This being so, Divers is no longer in a position to contrast atheism’s supposed propensity to lead to a comprehensive error-theory about modality, with agnosticism’s propensity to avoid comprehensive agnosticism about modality. The two theories are comparable in terms of their expressivity. Either each engenders ‘comprehensive’ problems with modality, or neither does.

The expressive comparability between atheism and agnosticism has further implications which undermine the motivation for being agnostic. For I take it that agnostic antirealism is motivated by the perceived problems of the atheist alternative. Implicitly, that is to say, atheism is the default within the anti-realist community. In general, those who entertain doubts about the existence of worlds – presumably for reasons of the sort aired in ch2 – entertain strong doubts. Intuitively, other worlds are objectionable. However, according to Divers, atheism leads to a comprehensive error-theory about modality. With atheism thus anathematized, anti-realists form reluctant agnostics. With atheism rehabilitated as I contend it should be, anti-realists revert to their default atheism.

In the current section, I have argued that the actual counterpart apparatus, adapted by Divers for the purpose of bringing a range of de re contingencies within the assertorial purview of the agnostic, can be commandeered by the atheist in order to bring substantially the same de re contingencies within his assertorial purview. I have argued that this piggy-backing behavior on the part of a resurgent atheist serves to undermine the motivational basis for agnosticism. In the section now following, I argue that the atheist’s piggy-backing behavior further serves to undermine the entire pw-theoretical enterprise. 



(3.5) Actual Counterparts and the Redundancy of Worlds 

In brief, given the size of α, the possibility of accounting for a great deal of modality in terms of actual counterparts promises drastically to marginalize the modal role of possible worlds. For α is a strikingly large and well-populated place. It contains an enormous number of individuals, festooned with an exceedingly rich array of properties. Suppose we take it that the arguments given earlier have established the propriety of analyzing de re contingency by means of actual counterparts, where the latter are available. Then a plausible conjecture is that very many actual individuals are in fact related to one another by means of the counterpart relation. This being so, it seems overwhelmingly likely that enough individuals in α are available to serve as counterparts in order to ground the analysis of substantially all the modal claims that can be regarded as salient, i.e. just about all the modal claims that a reasonably practical individual will ever feel himself or herself called upon to express. To invoke the entire pw architecture, merely for the sake of expressing a smallish number of comparatively weird and esoteric modal claims, seems akin to taking a sledgehammer to a nut. Indeed, one begins to wonder whether such a small, and weird and esoteric nut is better treated as illusory. In this way, modalizing with actual counterparts threatens to marginalize the modal role of possible worlds.

Earlier, I castigated Lewis’s 5th postulate prohibition of same-world and actual counterparts for being counter-intuitive and arbitrary, and also remarked upon the strange omission by Lewis of any reasons for including the 5th postulate. There is now scope at this point in the dialectic for a third objection against the 5th postulate. In short, the best and indeed the only reason I can think of for including the 5th postulate is that, with actual counterparts out of the way, possible worlds resume their central role in the analysis of modality. Inclusion of the 5th postulate begins to seem like a desperate ad hoc measure to ensure that possible worlds do indeed retain this central role.

The pw-theorist persuaded of the wrongness of the 5th postulate might seek to accommodate actual counterparts in one of three ways, none of them satisfactory:-

I. Mixed Theory. Actual-counterpart-theoretical sentences analyse the salient modal claims. pw-theoretical sentences analyse only the non-salient modal claims. However, the mixed-theory response is unsatisfactory, because it involves relinquishing a unified analysis of modality; and because it appears to make the difference in the analysis of salient and non-salient claims dependent in a bizarre way on the interests of us inhabitants of α. It is an odd supposition that nature would carve the metaphysics of modality in this way.

II. Double-Grounding. Actual-counterpart-theoretical sentences analyse the salient modal claims. pw-theoretical sentences analyse all modal claims, both salient and non-salient. Double-grounding fares little better than the mixed theory. On the plus side, there is a unified pw-theoretical analysis of modality as a whole as well as the actual-counterpart-theoretical analysis of the salient modal claims. However, it is not clear that double-grounding amounts to more than window-dressing; a stipulative, ad hoc means of making the redundancy of possible worlds less immediately stark, without altering the fact that possible worlds are only invoked in order to enable the expression of non-salient modal claims.

III. Epistemic/Semantic Disequivocation. Actual counterparts justify (many of) the salient modal claims, perhaps by means of a causal connection principle of the sort familiar from ch2. In contrast, pw’s and their contents make (all true) modal claims true, whether salient or non-salient. Epistemic/semantic disequivocation is unsatisfactory, because it generates a dilemma according to which either all those salient modal claims which are justified are systematically over-justified, or the non-salient claims are not justified at all. For what are we to make of the justification of non-salient claims?

Suppose it is possible to justify s’s belief that p without whatever epistemological principle is operative in the salient actual-counterpart-theoretical cases. This would afford scope for the justification of non-salient claims. However, by parity of reasoning it would also afford scope for the justification of salient claims, (many of) which are already justified, according to the hypothesis. Thus, all those of our salient claims which are justified by e.g. the causal connection method are also justified by the principle governing the non-salient cases as well. Ergo, all justified salient claims are over-justified. This is not to say that over-justification is completely inadmissible. That would be too strong a claim. Epistemology should allow for cases of over-justification, as when s comes to believe that p as a result of being informed that p by multiple witnesses of p’s occurrence. However, epistemology should not make over-justification systematic, as it does on this horn of the dilemma.

Alternatively, suppose it is not possible to justify s’s belief that p without the epistemological principle operative in the salient actual-counterpart-theoretical cases. This would leave the non-salient claims unjustified. In which case, it occurs to wonder why we should continue to theorize about non-salient claims at all, if our beliefs about them are not justified. Ultimately, the project of analysing modality in terms of possible worlds does not seem to be well-motivated, if it is granted as it should be that de re contingency is susceptible of analysis within counterpart theory, by whatever actual individuals are available to serve the counterpart-theoretical roles that need filling.



Conclusion

In the present chapter, I have introduced the two main strands of pw-theoretical anti-realism, namely modal atheism and the modal agnosticism originally presented by John Divers. I have exhibited the fairly well-known propensity of a naive atheism – the conjunction of pw-semantics and strong scepticism about worlds – to collapse the modalities, and have also shown how to leverage Divers’s pro-agnostic appeal to actual counterparts into objections both against agnosticism itself and against the wider project of accounting for modality in terms of pw-theory. These insights, relating to the actual counterpart notion and its role in demotivating rival anti-realisms and pw-theory as a whole, can be regarded as paving the way for the introduction of act in the chapter immediately following.
Chapter 4. Actual Counterpart Theory

This chapter is concerned with the formalization of an actual-counterpart theory of modality - act. Unlike its sibling counterpart theory cct, act is not pw-theoretical (although it is model-theoretical in the manner of traditional pw-theories, in that it implements a Kripkean model theory just as they do). act differs from pw-theories in at least two ways:-
pw-theories implement Kripke’s [1963] remark, that the members of the K element of a model structure ms “intuitively relativize” to worlds; whereas in act the members of the K element relativize to spatiotemporal regions of α.
The G element of ms, which serves in ms as the locus of assessment of claims, is understood in pw-theories as denoting the whole bounded spacetime α; but understood in act as denoting a contextually-defined spatiotemporal sub-region of α, roughly corresponding to what ordinary speakers mean in using indexical terms such as ‘here’ or ‘now’ or ‘the vicinity’.

In §1, I develop act somewhat in the manner of the development of cct undertaken in [Lewis 1968], considering and forming final judgments where appropriate about:-
* The composition of the primitive predicates and postulates of the theory;
* Which clauses are to comprise the scheme for translating from modal sentences already formalized by means of QML/QMLA into sentences of counterpart theory;
* The nature and status of the counterpart relation Cxy.
In §2, I evaluate act’s response to the technical and theoretical objections raised against its non-pw-theoretical rival modalism in ch1. This section also includes some specifications of act’s translation of counterfactuals, and brief prognoses concerning whether act translates certain modal principles as theorems. §3 is concerned with act’s expressivity, and specifically concerns the expression of claims about modal actuality, the comparison being with modalism’s confrontation with modal actuality, also q.v. ch1. In §§’s 4-5, I test act’s resistance to analogues of the Cardinality and Epistemology Objections, which were raised against cct in ch2.


(4.1) act

We begin with the unapplied model theory of Kripke [1963]. In the original Kripkean notation, a model structure ms is an ordered triple <G,K,R>, such that K is a set, G is a member of K, and R is a reflexive accessibility relation on K. For simplifying purposes and where relevant, I follow Lewis in taking R to be transitive and symmetric as well as reflexive –i.e. R is understood as an equivalence relation on K.

Now G is singled-out by applications of Kripkean model theory as that member of K which is treated as the locus of assessment of claims both modal and non-modal. In a standard pw application such as cct, K is understood as the set W of all possible worlds, and G as our world α. (Indeed, recall that in ch1 I dispensed altogether with the Kripkean G, replacing it in ms with α.) Matters run differently for act, in which K is understood as the set of spatiotemporal sub-regions r comprising α. Thus G as a member of K is a sub-region of α. More specifically, and keeping faith with the standard mode of application, G is treated as that particular region of α which serves as the locus of assessment of claims. Intuitively, G corresponds to the region of α to which an assessing agent refers in using indexical terms like ‘here’ and ‘now’, generally terms suggestive of the immediate spatiotemporal vicinity of the utterance. I call this region ‘@’. The differences between the two interpretations are therefore as per Table1.


Tab. 1. cct vs. act interpretations of ms elements.
G K
locus of assessment space of possibility

cct α worlds in W
act @ regions in α

An important difference between ‘@’ and ‘α’ is that the denotation of the former but not the latter varies with context. To illustrate this contrast, consider the reasonable-sounding claim that donkeys might not have existed. The standard pw-theorist, i.e. who follows Kripke in understanding G as being α, grounds the possible non-existence of donkeys in their non-existence at other worlds. In contrast, the act-theorist who uses ‘@’ to mean something like ‘here’ might ground the possible non-existence of donkeys in their non-existence on e.g. the surface of Mars. Alternatively, the act-theorist who uses ‘@’ to mean something like ‘now’ might ground the possible non-existence of donkeys in facts about the relatively recent evolution of the species equus asinus, or in facts about its eventual extinction.

Subject to the caveat directly following, most of the remainder of the present section consists in a formalization of act, which shares much of the structure and indeed some of the phrasing of the formalization of cct to be found in Lewis’s seminal [1968].

Caveat A particular consequence of the structural parasitism on Lewis’s paper is that, just like cct, the formalization of act is based on a modal logic which is Kripkean rather than SQML-based. And it needs emphasizing that this basis of formalization is provisional, pending proceedings in ch6. Quantification is Kripkean rather than SQML-based, in the sense that quantifiers range over variable domains of individuals – d(w) of worlds in the case of cct, d(r) of α regions in the case of act – rather than ranging over the domain D of all possible objects. The reasons for a proponent of an actualist theory of modality such as act to prefer Kripkean logic to SQML are taken to be as described by Linsky & Zalta [1994], Bennett [2005] et al, as well as in ch2 of the present work. Briefly, Kripkean but not SQML quantification avoids validating controversial principles such as the Barcan Formula and the Necessitation of Existence:-
BF. ◊∃xFx → ∃x◊Fx
NE. ∀x⎕∃y x=y.
That the present choice of Kripkean logic over SQML is only provisional is because an appeal to SQML is an integral feature of Ulrich Meyer’s promising solution to the Hazen/Fara/Williamson Actuality Problematic, which forms the subject-matter of ch6. 

I now turn as promised to a formalization of act, based like cct on variable domains of quantification.

(4.1.1) The Theory
(i) Primitive Predicates 
The following are primitive predicates of act:-
Rx. x is a spatiotemporal region of α.
Ixy. x is in spatiotemporal region y.
@x. x is in the locus of evaluation.
Cxy. x is a counterpart of y.

The Ixy specific to act plays a role analogous to that of the Ixy specific to cct [q.v. Lewis ibid. p113], with the difference that the y now picks out regions rather than worlds. The Cxy of act is much the same as the Cxy of cct, modulo the modificatory approach discussed below. Rx and @x play in act the roles played respectively in cct by the Wx and Ax predicates. Just as the domain of quantification in cct is to contain α and everything in α, whatever α is in the particular context, so analogously the domain of quantification in act is to contain @ and everything in @, whatever @ is in the particulat context.

An interesting feature of the contextual variability of @ is that, in its limiting case, it expands to the size of α. Cases in which this feature of @ may become relevant include 
* Actualist claims about the whole of reality [q.v. Bennett, Lycan, the present ch2 et al];
* Cosmological discourse;
* The Cardinality problem investigated in ch2, in its application to act, where intuitively α plays the role analogous to the role played by the so-called ‘big’ world in the Forrest/Armstrong formulation. [q.v. ch4 §4]

(ii) Counterpart Properties?
I turn now to the question of whether the list of primitive predicates presented above can be considered complete, or should be extended. In particular, it may be thought expedient to augment the list with a fifth predicate asserting a second order counterpart relation between properties:-
Cxy. x is a counterpart property to y.

The purpose of introducing Cxy is to diminish the scale of an alien property problem faced by act. For obvious reasons, the Cxy solution is only well motivated for property realists. Nominalists are likely to address the problem by means of modifying the condition on counterpart-hood. All of this is explained as follows.

Intuitively, some property F exists without being instantiated – which is to say that F is an alien property. Call this the alien property intuition. As things stand, act cannot endorse the alien property intuition, since ex hypothesi F is uninstantiated in α, where the truthmakers for modal analyses all reside according to act.

Note that the present alien property objection is distinct from the objection that act cannot endorse the intuited possibility of alien properties. This latter objection is unsound, because act can endorse the possibility in question. All that is required is that there be properties instantiated elsewhere in α but not in @, which is a reasonable supposition. But this latter solution does not answer the present alien property problem, which (to reiterate) concerns the possibility of properties not instantiated anywhere in α.

The addition of Cxy to the list of primitive predicates allows act to endorse the alien property intuition, so long as F is not too weird – so long as some of the modal truthmakers in α do have some property G which is similar enough to F to be considered its counterpart. Adopting this measure has the effect of reducing the alien property problem to a much more manageable deeply alien property problem, concerning the existence of properties which are unlike any actually instantiated properties.

The inclusion in the predicates list of a predicate inviting quantification over properties will displease those of the nominalistic persuasion. However, these latter need not abandon the project of easing the alien property problem, for the same result can be achieved by modifying the familiar counterpart predicate Cxy. On the standard interpretation, x is y’s counterpart in the context of property F only if x shares F with y. On the suggested interpretation, x is y’s counterpart in the context of F, so long as x either shares F with y, or has properties G such that C(G,F). The offensive reference to quantified-over properties is then eliminated by means of some nominalistically-acceptable paraphrase – most obviously by treating properties as classes of objects.

The Cxy-modifying approach may appeal to others besides the strictly nominalist. In particular, the followers of Armstrong [q.v. his1989b] hold that universals (including properties) are ontologically dependent upon their instances. Armstrongians purposing to ease the alien property problem now have a choice, between adding Cxy to the list of primitive predicates in accordance with their property realism, or modifying Cxy and quantifying over classes such as the classes F and G. To tidy up loose ends, platonists and nominalists, occupying their respective poles on the realism-antirealism spectrum, do not enjoy such liberality. Platonists hoping to dispose of the alien property problem should choose the Cxy option; whereas, for reasons already discussed, nominalists should choose Cxy-modification.

(iii) Postulates
The primitive predicates are to be understood according to their English readings, and the postulates of the theory. Amongst these latter are the following.
P1. ∀x∀y(Ixy Ry)
(Nothing is in anything except a spatiotemporal region of α.)
P3. ∀x∀y(Cxy ∃zIxz)
(Anything which is a counterpart is in a region of α.)
P4. ∀x∀y(Cxy ∃zIyz)
(Anything which has a counterpart is in a region of α.)
P6. ∀x∀y(Ixy Cxx)
(Anything in a region of α is a counterpart of itself.)
P7. ∃x(Rx & ∀y(Iyx iff @y))
(Some region contains all and only things which are here/now.)
P8. ∃x@x
(Something is here/now.)
P9. ∀x∀y∀z∀v((Ixz & Iyv & @x &@y) → z=v))
(Everything which is here/now is in the same region of α)

There is a striking isomorphism between act and cct, in that in most cases the postulate for the one is got from the corresponding postulate for the other by straightforward substitution of terms, between r/w and between @/α. The act postulates are numbered so as to correspond to the cct postulates [q.v. Lewis 1968 p114] of which they are the respective analogues.

However, note the presence in the above list of P9, which does not correspond to any of the cct postulates; and note also absence from the above list of the following pair, which do correspond to postulates of cct:-
P2. ∀x∀y∀z((Ixy & Ixz) → y=z)
(Nothing is in two distinct spatiotemporal regions of α.)
P5. ∀x∀y∀z((Ixy & Izy & Cxz) → x=z)
(Nothing is a counterpart of anything else in its spatiotemporal region.)

In act, P9 is needed in order to secure the uniqueness of the locus of evaluation @. As Lewis explains [1968 p114], the uniqueness of the locus of evaluation α in cct follows from the cct P2 and P8, whereas in act P2 is of course absent as just mentioned, hence the need in act for an explicit postulation of  the uniqueness of @, viz. P9. The uniqueness of @, secured either explicitly as in the case of act or implicitly as in the case of cct, avoids problems with e.g. the law of contradiction. The having of two or more loci of evaluation would otherwise generate problems with logic, e.g. in relation to existential claims – it is reasonable to suppose that for some property F, ∃xFx would hold in one but not in another locus of evaluation; to illustrate, let F be the property of being a donkey, and consider as loci of evaluation Earth and Mars.The following several paragraphs explain the reasons for excluding P2 and P5.

(iv) Distinct Same-Region Counterparts?
Would it have been plausible to postulate that nothing had a distinct same-region counterpart? Recall that in ch3 I argued quite vehemently against the cct P5, complaining that its prohibition of same-world counterparts is ad hoc – it’s only in situ amongst Lewis’s postulates, because permitting same-world counterparts would undercut the motivation for cct’s modal realist element, since (absent P5) α contains most of the individuals required to serve as counterparts in a viable modal theory. Analogous considerations do not apply in the case of the putative act P5. In very many contexts, @ is too small to contain nearly all the required individuals, ergo permitting same-region counterparts would not undercut the motivation for the act project, ergo no complaint is to be raised on this basis about the ad-hoc-ness of prohibiting same-region counterparts. Equally, the act P5 cannot itself be said to be very well motivated. In very many contexts, @ is certainly big enough to contain individuals which are distinct but nevertheless sufficiently similar to one another to be considered prima facie as one another’s counterparts. Conclusion: the cct P5 may be ad hoc, but the act P5 seems both arbitrary and unmotivated.

(v) Overlap, Nesting and Scatter?
Would it have been plausible to postulate that nothing is in two distinct regions of α – in Lewis’s mereological terminology [1968 p115], would it have been plausible to postulate that no two distinct regions overlap? Lewis notes [ibid.] that others [Carnap, Kanger, Hintikka, Kripke, Montague et al] have proposed interpretations of QML which allow overlap. In contrast, his own conception of worlds as concrete bounded spacetimes militates against overlap, this being what motivates the cct P2 – intuitively, how could this individual in this concrete bounded spacetime be identical with that individual in that concrete bounded spacetime? To head off any complaint about P2 being ad-hoc, Lewis produces [1986] an independent argument against overlap, namely the Objection from Accidental Intrinsics [q.v. my ch2]. To recapitulate: given that Humphrey is five-fingered in α, but six-fingered in some other world w, we seem to lack any basis for contending that he is intrinsically five-fingered, as intuitively he is. Additionally, Lewis adverts [1968 ibid.] to the generality he enjoys over the overlap conception, in view of the fact that the counterpart relation pertaining to his conception is not in general an equivalence relation, unlike the identity relation pertaining to the overlap conception.

The regions r of α in act share some but not all properties with the functionally analogous worlds w of cct. The notion of region is as standard in topology – most saliently, r is bounded, just as Lewis conceives w to be. The salient difference between the act r and the cct w is that spatiotemporal relations obtain between distinct r, but not between distinct w. Unless cct-like granularity is imposed by fiat on act, regions may overlap with one another, be nested within one another, and may themselves be scattered. Moreover, the imposition of granularity is not well motivated, for overlap in act does not occasion the same sort of worry over accidental intrinsics as was identified by Lewis in the case of cct. Humphrey – Humphrey himself! That very man! – may very well be in two overlapping regions of α. Wherever he is in α, he is five-fingered, answering the intuition that his five-fingered-ness is intrinsical to him. And yet, he is also possibly six-fingered, in view of his six-fingered counterparts, the men in α who are most like him except in the matter of their six-fingered-ness. Conclusion: it would not have been plausible to postulate that no two regions of r overlap.

(vi) The Status of the Counterpart Relation
To quote Lewis [ibid. p114], Cxy is “our substitute for identity between things in different worlds”. Now there are two senses in which Cxy may be considered to substitute for identity. That is to say, Cxy may substitute for identity by means of:-

Non-replacement. Where X and Y are literally identical, they share all and only eachother’s properties and are eachother’s counterparts. For example, you are your own counterpart.

Replacement. Where X and Y are literally distinct, they may nevertheless share a great many properties in common, and thereby be counterparts of one another. For example, Quine and Mitt Romney are distinct, but are both white, American Republican voters, and may be considered eachother’s counterparts.

Two factors – the prohibitions of overlap (P2) and distinct same-world counterparts (P5) – determine that in cct intra-world substitution of identity by Cxy is by non-replacement, and inter-world substitution is by replacement. In contrast, in act these two prohibitions are absent. Substitution by non-replacement may and often does come about in the inter-regional case when overlap is operative, and substitution by replacement may and often does come about in the intra-regional case when counterparts are distinct. 

As mentioned earlier, Lewis holds that, unlike the identity relation, Cxy is not a relation of equivalence, and I concur. Like the identity relation, Cxy is reflexive – everything is its own counterpart. However, as Lewis contends, it would not have been plausible to postulate that Cxy was either transitive [1968 p115] or symmetrical [p116].
* Not transitive, because your counterpart in w1 may have a counterpart in w2 which resembles you less closely than something else in w2 does, in which case your counterpart’s counterpart is not your counterpart.
* Not symmetric, because you and your brother may share a counterpart which however more closely resembles your brother than it resembles you, in which case you are not its counterpart.

In addition, I concur with Lewis’s contention that it would not have been plausible to contend that an individual has exactly one counterpart in every world, or that no two things have a common counterpart.
* An individual may have twins as its closest resemblants, in which case they are both its counterparts.
* Twins may have a common closest resemblant, in which case it is their counterpart.
* Some individual may not resemble anything else, in which case nothing is its counterpart, and it is not a counterpart of anything.

Finally, I concur with Lewis’s description of Cxy as 
“a relation of similarity … the resultant of similarities and dissimilarities in a multitude of respects, weighted by the importances of the respects and the degrees of the similarities”. [p115]
However, note that my concurrence with Lewis is modulo the Cxy modification discussed earlier in this chapter.

(4.1.2) The Translation
Lewis points out [ibid. p116] that there already exists a formalization of modal discourse – QML. The existence of a well-developed logic of modality is convenient, because it means that the counterpart theorist need only give directions for translating QML sentences into counterpart-theoretical sentences, rather than undertaking the tedious task of formalizing modal discourse directly by means of counterpart theory. The act-theorist has the same entitlement as the cct-theorist to the strategy of translating QML sentences into counterpart-theoretical sentences. The purpose of the present subsection is to provide such a translation. As much as is convenient, I persist with Lewis’s notation and, again, much of his phrasing.

Before embarking on the general translation scheme, like Lewis although going slightly further than him, I provide certain general procedural rules for translating 0-place closed sentences, 1-placed open sentences, sentences containing modal operators which are not initial, QML sentences containing no modal operators, and QML sentences embedding modal operators within the scope of other modal operators. The general procedural rules follow from the characterisation of AT1-AT2j given below, and are provided for illustrative/pedagogical purposes only.

General Procedural Rules
*First are the ⎕ and ◊ cases of a closed (0-place) QML sentence with a single initial modal operator, together with the corresponding translations from QML into act-ish, and from act-ish into English:-
⎕φ ∀γ(Rγ φγ) φ holds in any spatiotemporal region γ
◊φ ∃γ(Rγ & φγ) φ holds in some region γ
To form the sentence φγ from the sentence φ, we restrict the range of each quantifier in φ to the domain of things in the region denoted by γ. We bring this about by replacing occurrences of ∀β with ∀β(Iβγ ...), and by replacing occurrences of ∃β with ∃β(Iβγ & ...).
*Next are the ⎕ and ◊ cases of an open 1-place sentence with a single initial modal operator, together with the corresponding translations:-
⎕φβ ∀γ∀δ((Rγ & Iδγ & Cδβ) φγδ)) φ holds of every counterpart of β in
every region of α
◊φβ ∃γ∃δ(Rγ & Iδγ & Cδβ & φγδ) φ holds of some counterpart of β in
some region of α

*If the modal operator is not initial, we translate the subsentence it governs. A quantifier lying outside the scope of any modal operator is restricted so as to range only over the objects in @. We bring this about by replacing occurrences of ∀δ with ∀δ(Iδ@ → ...), and by replacing occurrences of ∃δ with ∃δ(Iδ@ & ... ):-
∀β⎕φβ ∀β∀γ∀δ(Iβ@ →((Rγ & Iδγ & Cδβ) φγδ)))
φ holds of every counterpart in every
region of every β in @
∃β◊φβ ∃β∃γ∃δ(Iβ@ & Rγ & Iδγ & Cδβ & φγδ)
φ holds of some counterpart in some
region of some β in @

*The limiting case of a quantifier lying outside the scope of any modal operator is when in fact a QML sentence contains no modal operators. This explains why such a sentence is also translated by restricting its quantifiers so as to range over objects in @, as in:-
∃βφβ ∃β(Iβ@ & φβ)

*Finally, iterated modalities are sentences featuring modal operators embedded within the scope of other modal operators. In such cases, we both restrict any quantifiers in φ and translate any subsentences of φ with initial modal operators.

The Scheme tsα
Again my scheme tsα for act closely follows Lewis’s scheme tsl for cct [ibid. p118], save for the labelling differences as described in the earlier footnote. Like Lewis’s scheme, my scheme is conceived as a direct definition of the translation of a sentence φ of QML:-
tsα1. The translation of φ is φ@ (φ holds here/now/in the spatiotemporal vicinity) 
together with a recursive definition of φγ:-
tsα2a. φγ is φ if φ is atomic.
tsα2b. (¬φ)γ is ¬φγ
tsα2c. (φ & ψ)γ is φγ & ψγ
tsα2d. (φ v ψ)γ is φγ v ψγ
tsα2e. (φ → ψ)γ is φγ → ψγ
tsα2f. (φ ≡ ψ)γ is φγ ≡ ψγ
tsα2g. (∀βφ)γ is ∀β(Iβγ → φγ)
tsα2h. (∃βφ)γ is ∃β(Iβγ & φγ)
tsα2i. (⎕φβ1...βn)γ is ∀γ1∀δ1 ... δn((Rγ1 & Iδ1γ1 & Cδ1β1 & ... & Iδnγ1 & Cδnβn) → φγ1δ1 ... δn)
tsα2j. (◊φβ1...βn)γ is ∃γ1∃δ1 ... δn(Rγ1 & Iδ1γ1 & Cδ1β1 & ... & Iδnγ1 & Cδnβn & φγ1δ1 ... δn)

Using these two definitions, we find that the following sample QML sentences have the following translations:-
∀xFx. ∀x(Ix@ → Fx) Everything here/now is an F
◊∃xFx ∃y(Ry & ∃x(Ixy & Fx)) Some region of α contains an F
⎕Fx ∀y1∀x1((Ry1 & Ix1y1 & Cx1x) → Fx1)
Every counterpart of a in every region
of α is an F
∀x(Fx → ⎕Fx) ∀x(Ix@ → (Fx → ∀y1∀x1((Ry1 & Ix1y1 & Cx1x) → Fx1)))
Anything in any region of α which is a
counterpart of an F here/now is also an F

(4.1.3) Necessity De Dicto and De Re
act shares in common with cct the ability to to describe an attribute of an object as an essential attribute of that object. To describe the attribute expressed by the one-place sentence ϕ as an essential attribute of the object denoted by the singular term ζ is to assert the translation of the QML sentence ⎕ϕζ [Lewis ibid. p120]. However, we have not yet considered the translation of modal sentences containing singular terms, mainly because  we know thanks to Russell that a singular term can be treated as a definite description and eliminated from tsα and kindred schemes [ibid.].

When it comes to the restoration of eliminated singular terms, there is as Lewis remarks [ibid.] a hitch, in the form of scope, insofar as different choices of scope will lead to different translations. Let be ζ be the description (the β: ψβ). Then the translations of ⎕ϕζ will have the following pair of interpretations and translations, the second of which corresponds to the expression of necessity de re:-

Narrow scope. ⎕ϕζ is interpreted as ⎕∃β(∀ε (ψε ≡ ε=β) & ϕβ), and receives the translation ∀γ(Rγ → ∃β(Iβγ & ∀ε(Iεγ → (ψγε ≡ ε=β) & ψγβ)))
(Any α region γ contains a unique β such that ψγβ; and for any such β, ψγβ.)

Wide scope. ⎕ϕζ is interpreted as ∃β(∀ε(ψε ≡ ε=β) & ⎕ϕβ), and receives the translation ∃β(Iβ@ & ∀ε(Iε@ → (ψ@ε ≡ ε=β)) & ∀γ∀δ(Rγ & Iδγ & Cδβ) → ϕγδ)))
(@ contains a unique β such that ψ@β; and for any counterpart δ thereof, in any α region γ, ϕγδ.)


(4.2) act and Modalism

Recall that in ch1 I set out modalism, and reviewed certain objections levelled by Melia in particular against it, regarding supposed technical inadequacies and failures of expressivity. In the same chapter, I showed how pw-theory meets analogues of these objections. In the current section, I run the same procedure in relation to act, of guaging its response to analogues of the objections against modalism. Modalism, as before, is understood as an account of modality, based on QML or QMLA.



(4.2.1) Necessity and Essence.
According to Melia, modalism does not (easily) capture the distinction between necessity and essence. Intuitively, necessity is a property of dicta whereas essences are properties of res. A sentence is necessarily true or necessarily false if it must be true or must be false, whereas on the common understanding an essential property of an object is a sine qua non property of that object, a property which that object must have in order to exist. As we saw in ch1, Melia thinks that modalism conflates necessity and essence; allegedly, it has to use the same formula sc. ⎕Fa to capture both “’a is F’ is necessarily true’ (de dicto) and ‘a is essentially F’ (de re).

In the event, as we also saw, this turned out to be a non-problem. Contrary to Melia, modalism very easily captures the necessity/essence distinction. ⎕Fa can be reserved for the attribution of necessity to the dictum Fa; and, helping ourselves once more to the existence predicate ‘E ...’, familiar from the development of modalism in ch1, ⎕(Ea → Fa) captures the notion of a’s being F essentially.

Even though capture of the necessity/essence distinction turned out to be easy for the modalist, it behoved me to ensure that capturing the distinction was similarly unproblematic for pw-theory. And so it proved. According to pw theory, ‘a is F’ is necessarily true iff ‘a is F’ is true in all possible worlds; and a is essentially F iff a is F in every world in which a exists.

Of course, cct is pw-theoretical. Some pw theories permit transworld identities; in such theories, one and the same object can and typically does exist in more than one world. cct is a variant of pw theory which does not permit transworld identities, but which resolves the resultant issue of how to accommodate de re modalities by postulating a counterpart relation. It would have behoved me to ensure that capturing the necessity/essence distinction was as unproblematic for cct as it proved to be for standard pw theory. Luckily, Lewis did this in his [1968]. Briefly, ‘a is F’ is necessarily true iff every world contains a counterpart of a, and every such counterpart is F; whereas a is essentially F iff, in every world containing a counterpart of a, every such counterpart is F.

We have already seen above how, formally speaking, act captures the necessity/essence distinction. For the record, an informal account proceeds as follows. Briefly, ‘a is F’ is necessarily true iff every region contains a counterpart of a, and every such counterpart is F; whereas a is essentially F iff, in every region containing a counterpart of a, every such counterpart is F. 

(4.2.2) Counterfactuals
Recall Melia’s complaint that modalism does not provide a unified account of modality and the intuitively related notion of subjunctive conditionals, the typical cases of the latter being counterfactuals. Recall also the response suggested on behalf of the modalist, inspired by Peacocke: homophonic analyses do provide an account which is ‘unified’ to the extent that no extra ontology need be appealed to; the only cost to the theory being the additional connective ⎕→. Nevertheless, I ventured the suggestion that there is something unsatisfactory about the notion of homophonic analysis, in that it doesn’t seem to be very illuminating or ampliative, or indeed analytical. In contrast, of course, pw theory provides a unified and genuinely analytical account of modality and counterfactuals. Both are analysed in pw terms, in ways that I trust are by now familiar. The basic schema, adjusted for cct, is:-
Counterf. cct. Fa⎕→Gb iff the most similar world in which a’s counterpart is F is also a world in which b’s counterpart is G.
The act version of the schema is obtained by the simple expedient of substituting ‘region’ for ‘world’.
Counterf. act. Fa⎕→Gb iff the most similar region in which a’s counterpart is F is also a region in which b’s counterpart is G.

(4.2.3) Intensional Failure
According to Melia, modalism is prone to intensional failure. That is to say, the intensional character of the modalist analysis of modality precludes the modalist from giving an account of the truth conditions of modal sentences, or of explaining the ability of ordinary speakers to understand and use new modal sentences.

Recall, again, the Peacocke-inspired response. If as seems plausible Melia’s complaint is grounded in an inference from intensionality to non-compositionality, the modalist’s homophonic semantics seems to be precisely an example of a semantics which is intensional yet compositional. And yet, as I remarked, there is still some savour to the second part of Melia’s complaint, insofar as the patent ability of speakers to understand and use modal sentences seems uncanny and inexplicable, given the intensionality of the modalist’s analysis. Finally, recall that pw-theory gives an account of the truth conditions of modal sentences, and explains the ability of ordinary speakers to understand and use new sentences. The possibility/necessity of a’s being F is analysed in terms of a’s being F at some/all possible worlds. The ability of ordinary speakers to understand and use new sentences is then presumably explained by their ability to grasp the pw truth conditions for those sentences.  

Again, what is sauce for the goose is sauce for the gander. The act theorist borrows the structure of the pw-theoretical account, but supplies his own terminology. In this way, act gives an account of the truth conditions of modal sentences, and explains the ability of ordinary speakers to understand and use new sentences. The possibility/necessity of a’s being F is analysed in terms of a’s counterparts being F in some/all regions of α. The ability of ordinary speakers to understand and use new sentences is then presumably explained by their grasp of region-theoretical truth conditions for those sentences.

(4.2.4) Iteration
Recall once more Melia’s fourth theoretical worry, which concerns the account that modalists give of the meanings of iterated modal formulae – formulae featuring iterations of the modal operators  and ◊. The problem is that modalism is committed to the meaningfulness of such formulae, yet as I wrote “this commitment sits awkwardly with the intuition in some quarters that they are meaningless”. Viewed pragmatically, there is something right about Melia’s complaint. Even though Peacocke’s followers will argue that his homophonic account provides a perfectly good understanding of iterated modal sentences, there is still at some pragmatic level something baffling about formulae like ◊◊⎕⎕⎕◊⎕⎕◊◊⎕◊◊⎕◊◊ϕ.

In pw theory, ⎕ is analysed as pertaining to all possible worlds, ◊ as pertaining to at least one possible world. The pw-theorist is likely to wish to commit to the meaningfulness of long-string formulae, if only because the formation rules dictate this. Fortunately, he can give an account of their meanings, although this depends on what sort of accessibility relation is assumed to obtain between the worlds. 

To attack pw-theory, on the basis is that it does not give a definitive answer as to the meaings of iterated strings, is to attack the wrong target. For pw-theory is not so much a theory as an unapplied semantics of modality. It is only applied theories, such as modalism, or indeed cct or act, which are properly inquired into as to their tendencies to propagate meaningfulness without being able to say what the meanings in question are. In the case of cct, of course, Lewis resolves the issue by simple fiat as he is entitled to do, stipulating that the cct accessibility relation is an equivalence relation answering to an S5 modal logic. For all worlds w, v, w is accessible to v. No doubt it is possible to construct variants of cct with accessibility relations which answer to different modal logics. As ever, S4 and B represent the immediately obvious ways of restricting the R term in ms, not to speak of the numerous other less obvious ways of achieving this. However, there would have to be some non-arbitrary basis for instituting such a restriction. Until such a basis is forthcoming, the pull is towards an S5 logic with pervasive accessibility. And serendipitously, it makes things simpler.

At any rate, the main point to be made here is that cct, or at least the S5-ish stipulation Lewis builds into cct as an applied pw-theory, makes it possible to say what long-string formulae mean. For, as ch1 made clear, S5-theories reduce the meaning of any long string of operators to the meaning of the innermost member of the string. For instance, ⎕⎕◊φ has the same truth conditions as ◊φ, and ◊φ is satisfied by φ holding at some possible world.

In short, the iteration objection exerts no hold over pw-theory because of pw-theory’s unapplied nature, and exerts no hold over cct, because cct thoroughly analyses ⎕ and ◊, albeit stipulation is integral to the procedure..

I want to be able to say that the iteration objection exerts no hold over act either. Although act is an applied theory which we take to be committed to the meaningfulness of long-string formulae, nevertheless I want to be able to say that act thoroughly analyses ⎕ and ◊ just as cct does, and so is in a position to say what long-string formulae mean. Can I say this? I think so. As in the case of cct, we get what we want by fiat. We stipulate, in the absence of a basis for supposing otherwise, that the act accessibility relation is an equivalence relation answering  to an S5 logic – for all actual spatiotemporal regions r, s, r is accessible to s. Again, with regard to iterated strings, the meaning of the string is given by the meaning of the rightmost operator in the string. In this way, for instance, ⎕⎕◊◊⎕◊◊◊◊⎕◊φ has the same truth conditions as ◊φ, and ◊φ is satisfied by φ’s holding in some spatiotemporal region of α.


(4.2.5)Modal Inference and Validity
In ch1, we saw how the modalist’s insistence on taking the ⎕ and ◊ of QML as primitive leads to a second objection on the part of Melia. This is that modalism affords no opportunity of testing for soundness, completeness or validity the many competing systems of modal logic (again S4, B, S5 etc), as the necessary precursor for deciding which of these systems is the correct logic of modality, or of explaining the intuitive correctness of certain inferences.

For instance, we would naturally view as correct the inference from ⎕φ to ◊φ, and yet, thanks to his disinclination to interpret ⎕ and ◊, the modalist must view the correctness of this inference as inexplicable. pw-theory, and a fortiori cct, generate ready explanations of the correctness of ⎕φ→◊φ. If φ (or φ’s cct translation) holds at all worlds, then φ (or ditto) holds at some worlds. But pw-theory, and a fortiori cct, are model-theoretical, and the extensionality driving the inferential tractability of pw-theory (or cct) is a feature of model-theoretical analysis in general. Thus, in act too, the correctness of ⎕φ→◊φ is susceptible of explanation. If φ (or φ’s act-theoretical translation) holds at all spatiotemporal regions of α, then φ (or ditto) holds at some such regions. Indeed, act’s extensionality strongly suggests that the lesson can be applied across the board. In general, like pw-theory and cct, act renders greatly more tractable the modal inference and validity problems encountered by modalism.

In addition, translation into counterpart theory can settle questions about other modal principles. In particular, it settles disputes about whether the following principles are to be considered as theorems in act, much as does in cct [q.v. Lewis 1968 p123]:-

The S4 principle ⎕φ→⎕⎕φ is not a theorem in act unless φ is a closed sentence, as the counterpart relation Cxy is not transitive.
The B principle φ→⎕◊φ is not a theorem in act unless φ is a closed sentence, as Cxy is not symmetric.
The principle of the necessity of identity (δ= ε)→⎕(δ=ε) is not a theorem  in act, as there is no postulate prohibiting an individual from having more than one counterpart in a given region.
The principle of the necessity of distinctness ¬( δ= ε)→⎕¬(δ=ε) is not a theorem in act, as there is no postulate prohibiting distinct individuals from sharing a counterpart in a given region.
The Barcan Principle ∀δ⎕φδ→⎕∀δφδ is not a theorem in act, as there is no postulate mandating that an individual must be a counterpart of anything else.
The principle ∃δ⎕φδ→⎕∃δφδ is not a theorem in act, as there is no postulate mandating that an individual must have a counterpart.
The Converse Barcan Principle ⎕∀δφδ→∀δ⎕φδ is a theorem in act.


(4.3) act and Subscripted-Operator Modalism

§4.2 investigated the act-theorist’s response to a number of theoretical problems, which following Melia I posed against modalism in ch1. In the same chapter, I also sympathetically reviewed yet another objection made by Melia against modalism. This objection proceeded from a concern about expressive rather than theoretical adequacy. Specifically, it has its source in the question of modal actuality, i.e. how we are to account for what might have actually been the case.

In some contrast to the theoretical objections against modalism, in which respect Melia’s role has been more or less to give an exposition of what are already fairly well-known objections against modalism, he has played a significant role in the dialectical evolution of the objection from modal actuality. Indeed in his hands, this objection has ultimately become, as should be familiar from ch1, the objection that modalism cannot account for modal actuality without appealing to a denumerable number of subscripted operators, thereby collapsing into a mere notational variant of the pw-theoretical language in which cct and other pw-theories are articulated.

Unsurprisingly, as we see in the subsection following, modalism can also be treated as a notational variant of the region-theoretical language in which act is articulated.

(4.3.1) SO-Modalism:- A Recapitulation
The early period of the modal actuality dialectic is substantially bestrode by Hazen, and culminates with the enrichment of modalism by means of the actuality operator A, which facilitates the expression of sentences about what might have actually been the case, as well as more complex modal actuality sentences featuring comparatives and the like.

The secondary dialectic is conducted between modalism’s defender Forbes, and attacker Melia. Melia complains that even A-enriched modalism lacks the resources for expressing certain of the modal comparative claims, including some quite commonplace ones. For instance, cutting to the chase, A-enriched modalism lacks the facility for expressing
1. There might have been more things than there actually are.

Subsripted-operator modalism is Forbes’s suggestion for dealing with recalcitrant modal comparatives such as 1. Forbes introduces to QMLA, for each number n, the operators ⎕n, ◊n and An. Whereas the original actuality operator of the Hazen era picks out the inhabitants of α, a given subscripted actuality operator picks out the inhabitants of the world introduced by the corresponding subscripted possibility operator, sc. the operator subscripted with the same n. In particular, 1 is captured by means of
2. ◊1((⎕∀x(AEx → A1Ex)) & ∃y¬AEy)
2 says that it might have been the case that necessarily, all the actual things – all the things that really are in this world – actually existed, and there also existed some mere possibilia. 2 captures the sense of 1, because the subscripted operator A1 causes us to evaluate what is actually the case from the point of view of the world introduced by the correspondingly subscripted ◊1. Call this world w1. All the individuals in our world α are also in w1, along with some individuals not in α. So some world contains more things than actually exist from our point of view. Translating the previous sentence’s pw-sprach back into the ordinary modal language, we have the desired result, sc. that there might have been more things than there actually are.

(4.3.2) Is SO-Modalism a Notational Variant of PW-Theory?
Melia’s complaint, as before, is that subscripted-operator modalism is a mere notational variant of pw-theory. Continuing with the same case, 2’s obvious pw-theoretical translation is
3. ∃w1((∀w∀x(Exα → Exw1)) & ∃y¬Eyα).
Melia’s complaint takes its source from the striking similarities obtaining between 2 and 3 in terms of their grammar and syntactic structure. 3 can be obtained from 2 firstly by retaining all the same connectives in the same places, and secondly by replacing, from left to right, 2’s ◊1 with 3’s ∃w1, 2’s ⎕∀x with 3’s ∀w∀x, 2’s AEx with 3’s Exα, and so on. Surely a field linguist encountering a tribe who used boxes and diamonds in the mode of 2 would conclude that the language spoken by the tribe was a mere notational variant of pw-theory. That means that we ought as field linguists to view subscripted-operator modalism as a mere notational variant of the language of pw-theory.

(4.3.3) Is SO-Modalism a Notational Variant of Region-Theory?
So far so familiar from ch1. The question now is: does 2’s obvious region-theoretical translation betray striking similarities to 2 in terms of its grammar and syntax? If so, it follows by parity of reasoning that a field linguist encountering a tribe using boxes and diamonds in the mode of 2 would be entitled to conclude that the tribal language was a mere notational variant of the act language. Now 2’s obvious region-theoretical translation is
4. ∃r1((∀r∀x(Ex@ → Exr1)) & ∃y¬Ey@).
The reader will readily note the striking resemblance between the pw-theoretical 3 and the region-theoretical 4. This is unsurprising. A theme throughout the present chapter has been the tendency for sentences in act-sprach to be obtainable from sentences in cct-sprach, simply by substituting r’s for w’s, α’s for spaces of possible worlds, @’s for α’s etc. Clearly, subscripted-operator modalism is indeed construable as a mere notational variant of the region-theoretical language of act.


(4.4) act and Cardinality

Recall from ch2 the Cardinality Objection against modal realism. To recapitulate, the Objection’s standard version [Forrest & Armstrong 1984,  Lewis 1986] (hence, FAL) has it that modal realism plus an unqualified recombination principle – UPR immediately below – entails the existence of a ‘big’ world WB containing more individuals than it contains.
UPR. For any class C of possible individuals, some world contains any number of non-overlapping duplicates of all the individuals in C.
Nolan argues that FAL fails to establish this conclusion, because of the possibility of WB’s duplicates duplicating the individuals in the duplicand subset, not on the basis of one-one correspondence, but instead by means of equivalence classes based on the cardinality of the subsets.

Recall further that Nolan argues that there is another quick and simple form of Cardinality Objection, based on the cardinality of the set of possible individuals. Suppose there is such a set, with a cardinality C. Then there is some other cardinal C*, say the cardinality of C’s power-set, such that C*>C. It follows by UPR that, for some possible individual, some world contains C* duplicates of that individual. Ergo, the cardinality of the set of possible individuals is at least C*.

Nolan’s solution involves appealing to the set-theoretical notion of proper class, and proposing that the collection of possible individuals forms a proper class rather than a set. The thought is then that the class of possible individuals either lacks cardinality, or has a ‘large’ cardinality, i.e. a cardinality greater than that of any other class. Either way blocks the argument from C to C*.

Recall also that in ch2 I evinced a suspicion that Nolan’s proposal is tainted with ad-hocery, but that I found this not to be a very pressing concern. The suspicion arises because of the provenance of the notion of proper class appealed to by Nolan. Proper class is as before a set-theoretical notion, postulated in order to extricate mathematicians from the paradoxes of set-theory. The ad-hocery such as it is may not be vicious in mathematics; to some extent, it may be said to be constitutive of solutions in set-theory – perhaps the paradoxes show us that there have to be proper classes. Nevertheless the ad-hocery may be considered vicious when the notion of proper class is appealed to for the purposes of a heavy duty metaphysics such as the modal realism integral to cct. However, to repeat my earlier prognosis, the issue of the ad-hoc-ness of a notion  is altogether less pressing when, as in the present case, the allegedly ad hoc notion already has at least one independent argument to sustain it, than when the notion is not sustained by any independent argument at all.

In the subsections now following, I expose act to a region-theoretical form of the Cardinality Objection.

(4.1) act and FAL
Assuming act, consider all the regions constituting α as a class of possible individuals. Now the region-theoretical analogue of UPR is the following claim:-
UPRr. For any class C of possible individuals, some region contains any number of non-overlapping duplicates of all the individuals in C.
If UPRr is applied to the class of all the regions in α (since these can be considered as individuals), we end up with as it were a ‘big’ region, which in imitation of Forrest & Armstrong we may call ‘RB’ [Forrest &Armstrong], being the region which per UPRr contains non-overlapping duplicates of each of the regions in C, ergo non-overlapping duplicates of all the individuals in each region. Since we began by considering all the regions in α, clearly RB should itself be one of the regions in C, whose contained individuals are duplicated in RB. Let us follow FAL in selecting electrons to play the ‘contained individuals’ role. Accordingly, suppose that the big RB contains K electrons. Then there are 2K-1 non-empty subsets of the electrons in RB. Now apply UPRr. Then, for every non-empty subset, there is a region containing duplicates of all and only the electrons in that subset. In particular, each subset, and accordingly each corresponding region, contains at least one electron, and is duplicated into RB. Ergo RB contains at least 2K-1 electrons, contradicting our original hypothesis that it contained just K electrons. Reductio ad absurdum.

The act-theorist may be tempted to rescind entirely from recombination. However, doing so will generate serious problems with his aspirations to plenitude, the endorsement within act of substantially all the possibility claims which intuitively should be endorsed. Anyway, α seems to contain enough material for the job –continuum many spacetime points, in fact. Accordingly, the act-theorist is well-motivated to follow Lewis in modifying UPRr. Firstly, act unlike cct does not prohibit overlap, so there is little reason for the act-theorist to feel any compunction to commit himself to the non-overlapping duplicates envisaged by FAL. Secondly, he is just as entitled as Lewis to help himself to the latter’s spacetime-limiting proviso, and for the same reasons. In short, it does not seem that the Objection gains much traction against act, at least not as it is formulated by FAL.

(4.4.2) act and Nolan
The notional region-theoretical follower of Nolan shares the anti-FAL prognosis, albeit for a different reason – he contends that act can get round the relevant analogue of the Cardinality Objection by means of RB duplicating the subsets on an equivalence-class basis rather than a one-one basis.

However, this leaves the act-theorist with the task of addressing the appropriate analogue of Nolan’s quick and simple version of the Objection. This analogue asks us to suppose for reductio that there is a set of all the objects in α, and that this set has the cardinality C. Now there must be some greater cardinality than C. For example, the power set of the objects in α must have a greater cardinality than C. Call the cardinality of this power set C*. From applying UPRr, it follows that, for some object, some region contains C* duplicates of that object. Ergo there are at least C* objects in α. Ergo, the set of objects in α has a larger cardinality than it has. Ergo, there is no set of all the objects in α.

The obvious – and correct – response on the part of the act-theorist is to avail himself of the ‘proper class’ response which Nolan offers to the cct-theorist. Much as in the case of cct, this involves accepting Nolan’s argument against the existence of a set of all possible objects, and also his proposal that the collection of all possible objects – parsed by the act-theorist as the collection of objects in α – forms instead a proper class. Then as in the case of cct, the class of possible individuals either lacks cardinality, or has a ‘large’ cardinality, i.e. a cardinality greater than that of any other class. Finally, as before, we advert to the worry over ad-hoc-ery, but find this worry not too pressing given the mathematical precedent.


(4.5) act and Epistemology

In ch2, I addressed the Epistemology Objection against cct [Richards, Skyrms, Chihara], which argues that cct is incompatible with knowledge of modality. I extracted a prognosis which, if not decisive against cct, was at any rate troubling for it. In contrast to cct, as I will now explain, act is serenely untroubled by the Objection.

As before, the Objection turns on a causal principle of knowledge:-
CP. a’s belief that p is justified → p’s truth conditions are causally related to a’s belief that p.
To see why CP represents a problem for cct’s epistemology, recall some remarks of mine in ch2. These were to the effect that for A to cause B seems to require spatiotemporal relations to obtain between A and B, whereas cct’s worlds are bounded spacetimes; spatiotemporal relations do not obtain between their respective contents, which precludes A’s in one world from causing B’s in another world. Hence, on the face of things, cct implies that a’s belief that p is not justified – because typically p’s truth conditions are not causally related to a’s belief that p, in view of the fact that the putative causal relata pertaining to the modal case typically inhabit distinct worlds.

To cut to the chase, the entia which in act play the role played by worlds in cct are of course spatiotemporal regions of α. In philosophically stark contrast to cct’s worlds, between which spatiotemporal relations do not obtain – engendering cct’s troubles with modal epistemology – spatiotemporal relations very obviously do obtain between distinct regions of α. It follows that, to the extent that CP is applicable in the modal case – i.e. modulo Lewis’s arguments that it is not – its applicability represents no obstacle to act’s aspirations to a viable epistemology of modality.

Indeed, to our already optimistic prognosis about the reasonableness of act’s epistemology, we might add another. This is a prognosis about the phenomenology of the acquisition of modal belief. Again, the contrast with cct as a modal realism is instructive. Consider my belief, which we may suppose to be reasonable, that I could have been a builder of dry-stone walls. How taken aback I am, to be told by the cct-theorist that my modal belief is actually based upon a belief, justified a prioristically by a theoretical unity thesis as Lewis’s cct-theorist would have it, about counterparts of mine who are dry-stone-wallers in a world spatiotemporally isolated from our world! In contrast, how familiar it feels, to be told by the act-theorist that my modal belief is based upon there being actual individuals – e.g. the Stevenson brothers of my direct personal acquaintance – who really are dry-stone-wallers!

To summarise this subsection, no epistemological worries arise for act as they do for cct. This is because all the action takes place within the single bounded spacetime α in act, so CP – modulo Lewis’s arguments against its applicability in the modal case – is compatible with a reasonable epistemology for act. In addition to this important result, act promises a much more plausible phenomenology than that offered by cct of the acquisition of modal belief.


Conclusion

In this chapter, I have formalized act in a manner reminiscent of Lewis’s formalization of cct. No surprises there, both being variants of counterpart theory. The principle differences between act and cct consist
in their respective interpretations of the G and K of Kripkean model structure. 
In cct, the locus of evaluation G is understood as the actual world α; whereas in act it is understood as a privileged region of α, namely the contextually variable vicinity of utterance, to which I have attached the label @. In cct, the space of possibility K is understood as the set of possible worlds W; whereas in act it is understood as the set of spatiotemporal regions comprising the actual world α.
In their respective lists of primitive predicates and postulates. I have proposed 
augmenting act’s list of primitive predicates with a predicate describing a second order counterpart relation between properties, anticipating further discussion of the merits of adopting this measure in ch5 immediately following. In addition, I have argued against postulating a ban on overlap in act – the fact that spatiotemporal relations obtain between the α regions of act indicates that overlap should be permitted
Notwithstanding these differences, the QML-counterpart theory translation scheme and semantics for act very closely resembles that for cct.

In the subsequent sections of this chapter, I have shown firstly that act addresses the objections against modalism highlighted in ch1 as felicitously as does cct; and secondly that act answers two important objections that have proved particularly problematic for cct, these being the Cardinality and Epistemology Objections. act’s response to the second objection is particularly promising, because the resultant phenomenology is revealed to be more plausible than it would be with cct assumed.
Chapter 5. Does act Deliver a Complete Account of Modal Truth?

This chapter addresses what may well be reckoned the obvious complaint against act. The Completeness Objection, as I call it, contends that act delivers an incomplete account of modal truth. Allegedly, there are many true modal claims relating to properties which intuitively exist but which are not instantiated in α. Truths about such claims are preserved under analysis by pw theories of modality such as cct, but not by theories which rely on the resources of α, as act does.

The Objection is set up in §1. §§2-3 describe two responses to the Objection, which tend to reduce but not eliminate the range of its application:- 
* In a concessionary spirit, modify the schemata by which act analyses modal sentences. In particular, introducing second order property-counterparts as per ch4, either by augmenting the list of primitive predicates or by modifying the counterpart relation Cxy – call these two approaches collectively the C-method – has the effect of reducing the scope of the Objection.   The Objection then becomes a matter only of accounting for the possibility of individuals with properties unlike any actually instantiated properties – I refer to the associated residual possibilities as the deeply alien possibilities. Thus §2.
* In a robust spirit, defend the completeness of act’s account of modal truth. In this spirit, Knobe, Olum & Vilenkin (KOV) [2006] argue that a consequence of the plausible and popular theory of inflationary cosmology is that all physical possibilities are realized. If KOV’s insight is sound, adopting an inflationary cosmology enables the reduction of  the scope of the Objection to a matter of accounting for the metaphysical possibility of things which are physically impossible – the merely metaphysical possibilities, as I call them. The KOV-based response, which I call Physical Actualism forms the subject matter of §3.

In a codicil to §3, I note that KOV appear to base their arguments on a relatively conservative model of inflationary cosmology, featuring island universes in which the cosmological constants are preserved; whereas other more liberal models permit island universes in which the cosmological constants are not preserved. Adopting one of these more liberal inflationary cosmologies has the effect, depending on how we construe it, either of realizing a great many merely metaphysical possibilities, or of bringing such possibilities within the purview of the physically possible.

Adopting one or both of the C-method and Physical Actualism, both of which I claim to be fairly well-attested, considerably increases the plausibility of a modal error theory in respect of the residue of deeply alien/merely metaphysical possibilities, truths about which are not preserved under analysis by act. Thus §4. In §5, I advert to some cases of alleged priori necessity (APN) attributed to Kripke, Putnam, and Alexander Bird, contending that the possibility claims consisting in denials of such APN’s constitute plausible candidates for the kind of error-theoretical treatment outlined in §4.

In §6, I acknowledge that the prosecution of error-theory in relation to APN-denials involves refuting the mode of argumentation  from X’s conceivability to X’s possibility (CP arguments). Although it is not clear that APN-based refutations are proof against CP arguments for possibility claims not involving APN, I evince the not unreasonable hope that these latter can in the main be disposed of by renewing the earlier appeal to the C-method or to Physical Actualism. Finally in §7, I defend APN against Jackson’s and especially Chalmers’s notorious two-dimensionalism (2D)-based attack. 


(5.1) The Completeness Objection

In a nutshell, the Completeness Objection is the complaint that act delivers an incomplete account of modal truth. That is to say, there are certain genuine modal truths which the relevant act schemata fail to analyse as true. As we have seen in previous chapters, realist theories such as cct typically accommodate modal truth e.g. about individuals being possibly F, by so arranging things that there are worlds containing F-ish counterparts of the possible F’s. But act eschews other possible worlds. And then (the complaint continues) regions of α, which do the theoretical work for act which worlds do for cct, no matter how numerous and diverse, cannot realistically aspire to provide F-ish individuals to serve as truthmakers for all the intuitively true claims about possible F’s. Another way of stating the complaint is by reverting to ch4’s talk of alien properties, the present sense of which is properties not instantiated in α.

The Objection can be expected to press acutely on, for example, various claims about:-

Possible Existence. Consider that for some F it is possible that some individual x be F when no individual is actually F. To illustrate, let F be the property of being a solid gold sphere with a 1bn mile diameter. For the sake of brevity, we can refer to anything with comparable dimensions as ‘huge’. Then we have 
1. ◊∃xFx
1 is arguably true. Perhaps it really is possible for there to be huge gold balls. At any rate, let us suppose that 1 is true. cct analyses 1by means of 
2. ∃w∃x(Ixw & Fx).
There are worlds containing huge gold balls. All is well; cct’s analysis preserves the intuited modal truth. In contrast, act analyses 1 by means of
3. ∃r∃x(Ixr & Fx).
Since it is very likely that no region of α boasts huge gold balls, patently all is not well; act fails to preserve the intuited truth of 1 under analysis. 

De Re Possibility. Consider that, for some individual a and property F, it is possible for a to be F when no actual counterpart of a is F. Now an interesting subclass of de re possibilities, where we might expect such a state of affairs to arise – to the detriment of act is the class of those possibilities de re, which differ from actuality only in very small ways, but which nevertheless fail to be instantiated by actual counterparts of the res. To illustrate, firstly let F be the property, relevant to jugs, mugs and like receptacles, of containing exactly n+1 molecules of beer, where n is the number of molecules of beer actually in the glass I have just finished. Secondly, let a be the glass on the table in front of me at this very moment. Then we have 
4. ◊Fa
This glass could have contained an extra molecule of beer. 4 is highly plausible. It could very easily have turned out that way. cct analyses 4 by means of 
5. ∃w∃x(Ixw & Cxa & Fx).
There are worlds in which there are glasses containing exactly n+1 molecules of beer. Again, all is well; cct’s analysis preserves the intuited modal truth. In contrast, act analyses 4 by means of
6. ∃r∃x(Ixr & Cxa & Fx).
Since it is very likely that no region of α boasts a glass containing exactly n+1 molecules of beer, act does not preserve the intuited truth of 4 under analysis.

(A digression:- The claims forming this subclass raise a problem for act which bears comparison with the problem which the Missing Shade of Blue (MSB) poses for Hume, at least under a popular construal. Both problems concern possibilia which are uninstantiated despite differing from actuality only in very small finely-grained ways. In Hume’s case, the MSB is very close in hue etc to shades of blue which are copies of impressions. Which is to say, the MSB differs from actually experienced shades of blue only in small, fine-grained ways, thereby putting acute pressure on the Humean thesis that ideas are copies of impressions – i.e. of actual experiences; analogously, there is very likely to be a property F, in relation to which there is any number of properties F1...n which likewise differ from F in fine-grained ways – e.g. glasses containing exactly n+2 molecules, n-6 molecules etc etc – and which yet fail to be instantiated – thereby, as before, posing a problem for α-reliant theories such as act.)





(5.2) From Counterparts of Properties to Deeply Alien Properties

The first way of reducing the scope of the Completeness Objection involves implementing the C-method adumbrated in the previous chapter. As before, this is achieved either by adding Cxy to the list of primitive predicates, or by modifying the counterpart relation Cxy bequeathed to us by Lewis. If we choose Cxy-modification, then for the contextually salient set of properties F, we replace
Cxy iff (Fy & Fx)
... with ...
Cxy iff (Fy & (Fx v (Gx & C(GF)))
The C-method can be construed as conceding the force of the Completeness Objection as decisive against the letter of act (at least, with the latter conceived in the pre-C-method sense), although the spirit of act is sustained by adopting the C-methodl. The effect of adopting the C-method is that the Objection no longer arises for all modal claims involving alien properties, but only for those involving a subset of the alien properties, which I call the deeply alien properties. The implementation of the C-method and the residual problem of deeply alien properties are both explained in the paragraphs now following. Explanation is with reference to the problematic cases of possible existence and de re possibility raised earlier.

(5.2.1) Implementing the C-Method
Possible existence. Instead of analysing 1 by means of 3, we analyse 1 by means of
3a. ∃r∃x(Ixr & (Fx v (C(GF) & Gx))).
It is possible that something is F iff, in some region of α, something is either F or G such that G-ness is a property-counterpart of F-ness. Thus equipped, we revisit the possibility of there being huge gold balls. Exactly as before in ch4§4.3, we let G be the property of being a huge iron ball, under the not unreasonable suppositions, firstly that such objects abound in α even if huge gold balls do not, secondly that G-ness is C-related to F-ness with F and G given these assignments. Finally, it is easily seen that, given the modified schema and the forgoing suppositions, act preserves the truth of 1 under analysis.

De re possibility. Instead of analysing 4 by means of 6, we analyse 4 by means of
6a. ∃r∃x(Ixr & Cxa & (Fx v (C(GF) & Gx))).
a is possibly F iff, in some region of α, some counterpart of a is either F or G such that G-ness is a property-counterpart of F-ness. Thus equipped, we revisit my tale of a pot. With F as before the property of containing n+1 molecules of beer, discovering an instantiated property G to serve as the property-counterpart of F is simply a matter of exercising the memory. Ex hypothesi, I just drank a glass containing n molecules. The property of containing n molecules is very similar indeed to the property of containing n+1 molecules. In short, let G be the property of containing n molecules of beer. Finally, it is easily seen that, given the modified schema and the forgoing suppositions, act preserves the truth of 4 under analysis. Something, sc. the actual glass is G such that C(G,F), so the actual glass functions as its own counterpart in accounting for the truth of ◊Fa.

(5.2.2) Why Property-Counterparts Are Not Ad Hoc
The complaint may be raised that property-counterparts constitute an ad hoc solution to the Completeness Objection. Intuitively, such a complaint is ill-founded.

Firstly, generally speaking counterpart relations between entia are relations of similarity, and it seems quite natural to think of properties as standing in relations of similarity or dissimilarity to one another. To revisit an earlier example, being scarlet is more like being red than being pea-green is, and surely containing n molecules of beer really is more similar to containing n+1 molecules than e.g. being a bull is.

Secondly, if counterpart relations are permitted between individuals, parity of reasoning suggests that they should be permitted between properties. Indeed, it is tempting to pose this consideration as the counter accusation against the cct-theorist, that postulating counterparts for individuals but not for properties is itself ad hoc. Of course, we must grant that this consideration will seem question-begging to the cct-theorist, who defines properties as sets of possible worlds, and the counterpart relation as a relation between individuals and not properties. Nevertheless, his stance may seem slightly odd to the neutral observer, well-habituated to grading properties in terms of their similarity to one another as per the previous paragraph.

(5.2.3) The Residual Problem of Deeply Alien Properties
The C-method does not enable the truth of absolutely all intuitively true modal claims to be preserved under analysis. In particular, it may be claimed that there are possible existences with properties which are too unlike actually instantiated properties to count as the latters’ property-counterparts – there are, so to speak, deeply alien possibilities, but not according to act+C-method. The scenario lends itself to the following formalization:-
7. ◊∃xFx & ¬∃r∃x(Ixr & (Fx v ∃G(C(G,F) & Gx)))
Correspondingly, in regard to de re possibilities, some individual a may be considered a possible holder of deeply alien properties. The formalization:-
8. ◊Fa & ¬∃r∃x(Ixr & Cxa & (Fx v ∃G(C(G,F) & Gx)))

In the section immediately following, I turn attention to a second way for act to respond to the Completeness Objection. This second response, which involves adapting KOV’s argument from inflationary cosmology to what I call Physical Actualism, also leaves a residue of modal claims the truth of which is not preserved under analysis. For the record, these latter are the physically impossible metaphysical possibilities, which I call merely metaphysical possibilities. I take up the resolutions of the residual problems facing both responses in §4.


(5.3) From Inflationary Cosmology to Merely Metapysical Possibilities

The C-method, as we have seen, may be construed as a concessionary response to the Completeness Objection. In modifying the (pre C-method) act schemata whereby modal sentences were supposed to be analysed, the force of the Objection is conceded. That is to say, modification – e.g. by means of the C-method – is a way of saving the spirit of act by conceding its letter.

Another way of responding to the Objection is to be robust, to concede little or nothing, to defend act to the letter. In a nutshell, the Objection complains that there are (intuitively true) possibilities which are unrealized in act’s analyses. A robust response simply denies this, insisting that all possibilities are realized. In this spirit, Knobe, Olum & Vilenkin’s (KOV’s) [2006] Argument From Inflationary Cosmology can be adapted as a component of a robust but only partial defence of act. The defence of act is partial rather than full, because it does not say that all possibilities are realized, only those of a certain sort – the physical possibilities. KOV argue that all physical possibilities are realized, in the strikingly counterpart-theoretical sense that everything physical that could happen to an individual does happen to an individual of the same type – call this thesis Physical Actualism. Physical Actualism has a very short way with problematic possible existences and finely-grained de re possibilities: big gold balls abound, and so do glasses containing finely-graded quantities of beer.

(5.3.1) KOV’s Argument
Physical Actualism comes about, according to KOV, because
“It follows from inflationary cosmology that the universe is infinite and can therefore be divided into an infinite number of regions of any given size. But it follows from quantum theory that the total number of histories that can occur in any one of these regions in a finite time is finite ... [ergo] ... all possible histories are realized in some region of the universe” [ibid. p48]
The final sentence in the passage just quoted is meant in the strikingly counterpart-theoretical sense that
“everything that can possibly happen to you will actually happen to some qualitative duplicate of you”. [ibid]
KOV also note [ibid.] that the word “possible” is reserved for physical possibility.

In order to explain the inferential link between (i) quantum theory and (ii) there being a finite number of possible histories in a region, KOV ask us to consider a region of space and interval of time, together defining a region of spacetime. By dividing the space in this region into sub-regions, we can define a history as a total specification of the contents of each sub-region at successive moments of time [p49].

KOV grant that, if the sub-regions and intervals between times could be made arbitrarily small as they can in classical mechanics, and the contents specified arbitrarily precisely, then the number of possible histories would be infinite. However, this is not possible in quantum theory. Sub-regions and intervals of time have to be large enough, and the contents specified sufficiently coarsely, to preclude interference between histories whereby they lose their distinctness from one another. Two histories which do not interfere with one another are said to be (mutually) decoherent, or “consistent”. Thus the set of possible histories is to be identified with the set of decoherent histories. Finally, it is this lower limit on the size which histories need to be in order to decohere (from eachother), which guarantees that the number of possible histories in a region is finite.

KOV’s statement “it follows from inflationary cosmology that the universe is infinite” slightly misrepresents what they then go on to say. Inflationary cosmology features a form of matter known as false vacuum, which is characterized both by the high energy and strong repulsive field which do the inflating work of the theory, and by instability and a tendency to decay into ordinary (true) vacuum. As KOV put it, “post-inflationary regions like ours – α if you will – form ‘island universes’ in the inflating sea”. [p50] Then it is not the entire ‘big’ universe containing all the sea and islands, which is infinite. This big universe, as KOV explicitly remark, may be finite. Rather, it is our island universe which is 
“spatially infinite, in the sense that the volume of space where the time since the Big Bang is the same as that time here goes on forever and so is infinitely large”. [ibid]

Now if our island universe is spatially infinite, then it is divisible into an infinite number of regions of any given size. Finally, 
“we have an infinite number of regions and only a finite number of histories that can unfold in them. Since the regions develop independently, every possible history has a non-zero probability, and will therefore, with probability 1, occur in an infinite number of regions”. [p51]

(5.3.2) But Is Inflationary Cosmology True?
Since it was proposed ‘as a speculative hypothesis’ [p50] by Alan Guth [1981], inflationary cosmology has become widely accepted as part of a ‘standard’ physics. It promises answers to the Horizon Problem and various other concerns. To quote a demotic source
“Inflation answers the classic conundrum of the Big Bang cosmology: why does the universe appear flat, homogeneous and isotropic in accordance with the cosmological principle when one would expect, on the basis of the physics of the Big Bang, a highly curved, heterogeneous universe? Inflation also explains the origin of the large-scale structure of the cosmos. Quantum fluctuations in the microscopic inflationary region, magnified to cosmic size, become the seeds for the growth of structure in the universe.”
The fly in the ointment: what if inflationary cosmology turns out to be wrong? The physics community is notoriously febrile, and physical theories move in and out of fashion. Inflationary cosmology could go out of fashion. On the other hand, what are we to rely on, if not current best physical theory? I can admit that act is hostage to refutations of this sort, but to concede that such refutations have any force is to undersell the merits of Physical Actualism. Current best physical theory says that Physical Actualism is true!

(5.3.3) Liberal Inflationary Cosmology:- a Codicil
Moreover, it should be noted that the choice of inflationary theory here attributed to KOV is relatively conservative. Other inflationary theories postulate the existence of island universes with different values of their cosmological constants to those observed here by us in our island universe, if such it be. The philosophical implications of these more liberal cosmological regimes can be construed two ways:- either a lot of merely metaphysical possibilities are realized in common with all physical possibilities, or the range of the physically possible is expanded to include what was hitherto regarded as merely metaphysically possible. In some island universes, ones with different cosmological constants to ours, a’s counterpart exceeds lightspeed. On either construal, the Completeness Objection is more or less neutralized. The Objection only gains any substantial purchase on act by my attributing to KOV the more conservative inflationary hypothesis, with its ‘constant’ constants.


(5.4) Deeply Alien Possibilities, Merely Metaphysical Possibilities, and Error-Theories

As we have seen, adoption of the C-method enables act to preserve the truth under analysis of modal claims not involving deeply alien properties. Likewise, adopting Physical Actualism enables act to preserve the truth under analysis of modal claims not involving merely metaphysical possibility – with the option perhaps of taking up a liberal inflationary hypothesis with variable cosmological constants, and thereby expanding the range of possibility that comes within the act-theorist’s assertorial purview.

The drawbacks of all this are firstly that, as mentioned earlier, the C-method has nothing to say about deeply alien possibilities; and secondly, that Physical Actualism has nothing to say about merely metaphysical possibilies. To exemplify the latter, pace events in CERN and modulo the liberal inflationary regime mentioned in §5.3.3, let us suppose that 9 and 10 now following state recalcitrant merely metaphysical possibilities. In the case of possible existence, we have
9. ◊∃x(x exceeds lightspeed).
In the case of de re possibility, we have
10. ◊(a exceeds lightspeed).
For example, let a be light itself; we conceive of light from the sun reaching us after e.g. seven minutes, even though we know that it actually takes more than eight minutes for light from the sun to reach us. Assuming that in this case at least conceivability implies possibility, we have indeed the metaphysical possibility of something e.g. light exceeding lightspeed, even though this is not physically possible – 9 and 10 state merely metaphysical possibilities.

This is a suitable point at which to take stock. In particular, the Completeness Objection has four possible outcomes, depending on which combination of the C-method and the method of Physical Actualism is adopted:-

(i) act adopting neither method secures the worst outcome, vis à vis the Completeness Objection, failing to preserve the truth under analysis of modal claims featuring uninstantiated properties – these were the ‘huge gold balls’ and ‘glass of beer’ problems from §1.
(ii) act adopting the C-method alone does somewhat better, failing to preserve the truth under analysis only of claims about deeply alien possibilities.
(iii) act adopting the method of Physical Actualism alone compares with (ii), failing to preserve the truth under analysis only of modal claims about mere metaphysical possibilities, e.g. the possibility of exceeding lightspeed.
(iv) act adopting both methods secures the best outcome, failing to preserve the truth under analysis only of modal claims about deeply alien merely metaphysical possibilities.

(5.4.1) Error Theory
Whichever of act per (i)-(iv) is adopted, the resultant version faces some version of the Completeness Objection, corresponding to the range of modal claims considered contextually residually problematic, i.e. residually problematic for the chosen version of act. Less trivially, the different versions of the Completeness Objection have some structure in common. Let T be a given version of act, i.e. as per one of (i)-(iv), and let p be some contextually residually problematic modal analysand. Then Completeness Objections typically have the following modus tollens structure:- p & (T→¬p)├ ¬T. T’s failure to preserve p’s truth under analysis invites T’s rejection. For example, in the case of (iii), the failure of act augmented with Physical Actualism to preserve the truth under analysis of claims about mere metaphysical possibility, invites the rejection of this version of act; or so the relevant version of the Completeness Objection would have it. The crucial point is that in each case, the relevant instance of (T→¬p) is very difficult to argue against, as the discussions in §§1-3 have surely established. Therefore, by process of elimination, in each case the defender of the relevant version of act is motivated to deny the relevant instance of p, i.e. is motivated to adopt an error theory in regard to p. Thus we have:-

(i*) Unaugmented act motivated to adopt an error theory with regard to claims about the possibility of individuals with uninstantiated properties;
(ii*) act augmented with the C-method motivated to adopt an error theory with regard to deeply alien possibilities.
(iii*) act augmented with the method of Physical Actualism motivated to adopt an error theory with regard to claims about merely metaphysical possibilities.
(iv*) act augmented with both methods motivated to adopt an error theory with regard to deeply alien merely metaphysical possibilities.


(5.5) A Posteriori Necessity

The act-theorist aspiring to prosecute an error-theory in relation to a particular range of intuited possibilities may garner some support for his project from a fairly persuasive precedent, in the form of  the arguments for a posteriori necessity (APN) given by Kripke et al, since these latter can be construed as error-theoretical treatments of the corresponding APN denials. The remainder of this section  consists in a rehearsal of four cases of argument for APN, relating to mathematical truths, identity, material constitution, and dispositional properties. 

(5.5.1) Mathematical Truth and A Posteriori Necessity
The Goldbach Conjecture (GC) [Kripke 1980 pp36-37] claims that any even number n: n>2 must be the sum of two prime numbers. Given the necessity of mathematical truths, GC is necessary if true, and impossible if false. Likewise, GC’s negation ¬GC, according to which some even number n:n>2 is not the sum of two primes, is necessary if true, and impossible if false. Until a proof of either GC or ¬GC is forthcoming, GC and ¬GC are of equal epistemic status. That is to say, either both are conceivable, or neither is. The oddness of supposing that we cannot conceive of the content of our conjectures suggests that GC and ¬GC are both conceivable. So: either GC is necessary and ¬GC is impossible, although we can conceive that ¬GC; or ¬GC is necessary and GC impossible, although we can conceive that GC. Now we cash out the error-theoretical construal in the obvious way: one of the intuited possibilities, that GC or ¬GC, is false.

(5.5.2) Identity and A Posteriori Necessity
The necessity of identity is the subject of a well-known proof given by Kripke [1979 pp478-479]. This proceeds as follows for individuals a and b:-

NID1. a=b Assumed
NID2. a=a ∀x(x=x)
NID3 (a=a) → ⎕(a=a) Necessity of self-identity
NID4. ⎕(a=a) NID2, 3, →-elim
NID5. (a=b) → ∀F(Fa≡Fb) Leibniz
NID6. Let Fx=def ⎕(a=x) Instantiates F
NID7. ⎕(a=b) NID1, 4, 5, 6, →-elim
NID8. (a=b) → ⎕(a=b) NID1, 7, →-intro QED

The crucial point of this is that, to judge by the proof just given, identities really are necessary, yet they are not typically discoverable a priori. You cannot discover just by thinking about it that Cicero=Tully, or that George Orwell=Eric Blair, or that the Hesperus you observe at x o’clock is one and the same heavenly body sc. Venus, as the Phosphorus you observe at y o’clock. Again, we cash out the error-theoretical construal in the obvious way. You may intuit that it is possible that George Orwell is not Eric Blair, but it is not possible that George Orwell is not Eric Blair.

(5.5.3) Material Constitution and A Posteriori Necessity
The idea underlying this notion of APN is that individuals and natural kinds have whatever material constitution they have of necessity; yet how they are constituted is not typically discoverable a priori.

In relation to individuals, consider the table at which I am now sitting. You have never visited me at home, and cannot discover just by thinking about it that this table is made of wood. For all you know, it is made of glass. That is, you intuit the possibility that it is made of glass. But I assure you that it is made of wood; and necessarily so, for if I was sitting at a table made of glass, it wouldn’t be the same table as this one.  Again, we cash out the error-theoretical construal in the obvious way: the intuited possibility that the table is made of glass is false.

Thanks in large part to Putnam’s ‘Twin Earth’ thought experiment [Putnam 1975 pp139-143], it has become more or less standard to take water as the paradigm natural kind in discussions about material constitution and a posteriori necessity. Briefly, water is composed of H2O. Any substance with an atomic structure other than H2O is not water, therefore it is necessary that water is H2O. However, the material constitution of water is not something which can be discovered just by thinking about it, but can only be discovered a posteriori, as it in fact was i.e. by Gay-Lussac and von Humboldt, around the beginning of the nineteenth century.

Since Putnam’s experiment figures prominently in discussions of a posteriori necessity, especially in relation to the 2D defence of Conceivability-Possibility (CP) argumentation which forms the subject matter of §7 presently following, it is as well to give a briefexposition of Putnam’s experiment here. Twin Earth is thought of as a planet qualitatively identical to our own, and (some of) whose inhabitants speak a language (Twinglish perhaps) which is functionally identical to our English, in the sense that Twinglish-speakers use the same phonemes as English-speakers use in response to the same kinds of qualia: Twinglish speakers say “beer”, “boat”, “bishop” and “botulism” in circumstances relevantly similar to the circumstances in which English-speakers utter those words. Now although Earth and Twin Earth are qualitatively identical they are not, as it were, materially identical. In at least some cases, different material substances cause the same kinds of qualia. In particular, on Earth the material substance H2O fills the role of being the potable stuff which falls from the sky, flows down rivers, forms lakes etc – call this role being the watery stuff. On Twin Earth the role of being the watery stuff is filled by a different substance, which Putnam famously calls XYZ. Finally, as before we cash out the error theoretical construal in the obvious way: the intuited possibility that water is composed of XYZ is false. 

(5.5.4) Dispositional Properties and A Posteriori Necessity
The idea underlying this notion of APN is that certain natural kinds have certain of their dispositional properties of necessity; and yet these dispositional properties are not discoverable a priori. Yet again, we cash out the error-theoretical construal in the obvious way: the intuited possibility, that the natural kinds in question lack the dispositional properties in question, is false.

Alexander Bird [2001] contends that this is the case with salt. In particular, as most of us know, salt has the dispositional property of dissolving in water. As Bird argues, salt has this property essentially – anything lacking this property would not be salt. However, the fact that salt has this property is only discovered a posteriori. 

A central role in Bird’s exposition is played by Coulomb’s Law (CL), which governs the magnitude of the electrostatic force of interaction between charged objects. That is to say, CL determines certain structural properties both of salt and of water, which Bird contends are essential properties of the respective substances:- 
* Firstly, the crystalline nature of salt is a function of the electrostatic force of attraction between the sodium cations and chlorine anions composing it.
* Secondly, the liquid nature of water is a function of the electrostatic force of attraction between positively charged and negatively charged poles of the dipoles into which water molecules are formed out of covalently bonded atoms of hydrogen and oxygen.
* Thirdly, when a salt crystal is suspended in water, the forces attracting a sodium atom on the surface of the crystal into the solution are also electrostatic in character, and hence also subject to CL.

Since the properties of salt and water which are determined by CL are essential properties of salt and water, it follows that the existence of salt and that of water each necessitates the truth of CL. If we now suppose for reductio that salt does not necessarily dissolve in water, we are supposing that there is a world in which salt and water both exist, and salt does not dissolve in water. But such a world would be a world in which CL was false. Finally, there cannot be a world in which CL is both true (because salt and water exist there, and their existence requires CL’s truth) and false (because salt does not dissolve in water there). Ergo, that salt dissolves in water is necessary. As before, we do not discover that salt dissolves in water just by thinking about it, so it is a posteriori as well as necessary that salt dissolves in water. In short, as Bird may be construed as arguing, the intuited possibility that salt lacks this dispositional property is to be analysed error-theoretically.

A final point in relation to APN. Several commentators have remarked that any number of APN’s can be generated automatically, just by affixing the actuality operator A or Kaplan’s ‘dthat’ operator, or some other rigidifying operator, to the front of a sentence expressing ordinary posteriori contingency.


(5.6) Conceivability and Possibility

Integral to the argument for APN is the refutation of a venerable mode of argumentation from X’s conceivability to X’s possibility (CP argumentation), i.e. from intuited possibility to genuine possibility. The remainder of this section consists in,
(a) presentations of the structure of CP argumentation and of APN-based  refutations of CP argumentation; 
(b) discussion of the applicability or otherwise of CP argumentation in non-APN cases.

CP arguments, e.g. for the possible existence of huge gold balls, for the possibility of things exceeding lightspeed etc typically have a modus ponens structure whereby, letting ● denote a conceivability operator, we have:-
CP maj. prem. ●X → ◊X
CP min. prem. ●X
CP conclusion. ◊X

Consider the case of unaugmented act (i* in §5.4.1 above) in relation to possible existence claims. This version of act is required to address e.g. the following CP instance:-
CP(i) maj. prem. ●(∃x x is a huge gold ball) → ◊(∃x x is a huge gold ball)
CP(i) min. prem. ●(∃x x is a huge gold ball)
CP(i) conclusion ◊(∃x x is a huge gold ball)

Alternatively, consider act augmented with the method of Physical Actualism in 
relation to merely metaphysical possibility. This version of act must address the following CP instance:-
CP(iii) maj.prem. ●(a exceeds lightspeed) → ◊(a exceeds lightspeed)
CP(iii) min. prem. ●(a exceeds lightspeed)
CP(iii) conclusion ◊(a exceeds lightspeed)

Analogous considerations apply in the other cases. 

Given its modus ponens structure, a typical CP for X’s possibility is a valid form of reasoning, and can only be refuted by denying either its major or its minor premise. That is, either
(a) deny that X is conceivable; or 
(b) accept that X is conceivable, but deny that X’s conceivability implies X’s possibility.

In practice, much of the literature on the topic of CP’s coalesces around a strategy of accepting that X is conceivable in a certain sense, and then arguing over whether or not X’s being conceivable in this sense implies that X is genuinely possible. CP proposals and refutations prosecuting such a strategy are susceptible of interpretation under either of the (a)/(b) readings above.

The crucial point in all of this is that, in each case, the act theorist who addresses the Completeness Objection by proposing an error theory with respect to some range of analysand modal beliefs, assumes the burden of refuting the corresponding CP instances by means of which, the analysand modal beliefs comprising the target of the proposed error theory are acquired. 

I now turn, as promised, to the presentation of the structure of APN-based refutation of CP argumentation. Accordingly, let KX be ‘X is knowable a priori’. Then the Argument from APN is set out as follows:-

APN1. ⎕X & ¬KX. APN hypothesis
APN2. ●¬X APN1, ¬KX→●¬X
APN3. ¬◊¬X APN1, ⎕X definition
APN4. ●¬X & ¬◊¬X APN2, 3, &-intro
APN5. ●X & ¬◊X APN4, ¬X/X-subst
APN6. ●X APN5, &-elim
APN7. ¬◊X APN5, &-elim
CP maj. prem. ●X → ◊X CP
APN8. ◊X APN6, CP maj. prem., →-elim
APN9. APN7, APN8, ⊥-intro
APN10. ¬(●X → ◊X) APN9, ⊥-elim QED

CP’s proponents may respond to act-theorists relying on the Argument from APN, by contending that most examples of CP argumentation do not involve APN, and it is not easy to see how considerations of APN block inferences from X’s conceivability to X’s possibility in non-APN cases. For example, suppose that as a matter of fact the number of stars in α is 1023.  Then act gives that it is impossible that there are 1024 stars. But we can surely conceive of there being 1024 stars. Thence, application of CP gives that it is possible that there are 1024 stars.

There is something right about this complaint. The Argument from APN shows only that CP argumentation is not deductively valid – in particular, that it is invalidated in cases of APN’s. Nevertheless, even if CP argumentation is not deductively valid, there is still good reason for thinking that in a great many non-APN cases conceivability serves as an useful (if occasionally fallible) guide to possibility. As a matter of fact, I think that the act-theoristcan accept that CP is applicable in the ‘stars’ example above, without needing to resort to an error theory. He may endorse the associated possibility, viz. of there being 1024 stars, by renewing his appeal to either of:-
i. The C-method, whereby the property of having a power of cardinality n is treated as the counterpart of having a power of cardinality n+1. Thus, the property of having 1023 stars is C-related to the property of having 1024 stars.
ii. Physical Actualism, whereby some island universe, part of α, actually contains 1024 stars. 
I would contend, hopefully not too controversially, that analogous considerations apply in many other non-APN cases. Finally, the more robustly error-theoretical sort of act-theorist may contend that the failure of CP argumentation in the case of APN indicates scope for CP failure in those, hopefully relatively few, non-APN cases in which neither the C-method nor Physical Actualism avails.


(5.7) A Posteriori Necessity and Two-Dimensionalism

Certain proponents of Two-Dimensionalist semantics, most famously.Jackson and Chalmers have attacked the notion of APN itself. The implication of their arguments for the act-theorist is that he cannot prosecute Kripke-esque error-theories with respect to the denials of APN’s. Now it is open to the act-theorist to go along with their prognoses. To anticipate proceedings below, he may endorse e.g. the Jackson-Chalmer-ish prognosis that water might not have been H2O by recourse, again, to either of
i. The C-method – there’s all this other stuff with properties rather like the properties of H2O.
ii. Physical Actualism – XYZ plays the H2O role in some other island universe. 
Equally, the act-theorist may, as I consider likely, find the APN cases presented in (§5.5.1-4) more persuasive than the Jackson-Chalmer argument against APN, to the presentation of which I now turn.

Two-Dimensional (2D) semantics is based on a term e.g. “water” having two intensions:-

A primary or epistemic intension, corresponding to the term’s descriptive or Fregean sense. To this purpose, suppose as before that the descriptive sense of “water” is “the watery stuff”. Hence, at Twin-Earth, the extension of “water” as determined by its primary intension is XYZ. In this sense, the primary intension can also be seen as the indexical intension – “water” means at w what it refers to at w, e.g. it means XYZ at Twin-Earth.

A secondary or metaphysical intension, corresponding to the term’s reference at α. For example, here in (the anglophone regions, at any rate) of α “water” refers to H2O. Hence, at Twin-Earth, the extension of “water” as determined by its secondary intension is H2O. In this sense, the secondary intension can also be seen as the rigid intension – “water” means at w what it refers to at α, viz. H2O.

In a 2D semantics, a sentence is associated with both primary and secondary propositions. For instance, the sentence
11. Water is H2O
is associated with the following primary and secondary propositions, respectively:-
12. <The watery stuff is H2O>
13. <H2O is H2O>

On the face of things 2D semantics affords an elegant accommodation of APN. That is to say, APN comes about when a sentence is associated with a contingent primary proposition and a necessary secondary proposition. For instance, 11 is associated with a primary proposition – 12 – which is contingent, and with a secondary proposition – 13 – which is necessary. For Kaplan [1979b, 1989], 2D semantics represents a natural way to cash out his character/content conception of language – character corresponds to the primary intension, content to the secondary intension. For Davies & Humberstone [1979], 2D semantics represents a natural way of cashing out their notion of the fixedly actual – a sentence is fixedly actually true when it is associated with a primary proposition which is necessarily true.

It might be suggested that the foregoing amounts to some kind of argument against APN; after all, we have an instance of what is ostensibly APN i.e. 11, being separated into a proposition which is contingent and a posteriori i.e. 12, and a proposition which is necessary and a priori i.e. 13. But this would be to miss the point, and somewhat egregiously at that. The notion of APN is not refuted by showing that a sentence with an APN phenomenology such as 11 is in some more or less nebulous way associated with a couple of propositions which happen to align the a priori with the necessary and the a posteriori with the contingent. No argument has been given to the effect that, for a sentence with the form of ⎕Fa and an APN phenomenology, ◊¬Fa holds. In the case of water, 2D semantics merely associating 11 with 12 and 13 gives us no reason for thinking that 11 is possibly false rather than necessarily true.

Nevertheless, since the 1990’s Jackson and Chalmers have developed a 2D-based argument against APN. In Chalmers [2002] the argument turns on associating what he calls primary possibility with the primary intension, treating this primary possibility as being on a par with metaphysical possibility, and arguing that X being conceivable in the right way entails X’s being primarily possible – the right way being ideal primary positive conceivability, in the sense of his three-way taxonomy of conceivability rehearsed in §7.1 below. This bears on APN’s such as those detailed in §6 insofar as, for any statement ⎕Fa with an APN phenomenology, Chalmers finds ¬Fa ideally primarily positively conceivable, ergo primarily possible, ergo possible. I contend that Chalmers’s own argument fails, because X’s being primarily possible does not correspond to X’s possibility tout court/metaphysical possibility; and I produce a simple proof which shows this.

(5.7.1) Chalmers’s Taxonomy of Conceivability
Chalmers identifies three axes along which notions of conceivability can differ from one another [ibid. pp147-159]:-
i. Prima Facie Conceivability vs. Ideal Conceivability
X is prima facie conceivable iff X is conceivable on first appearances, after some consideration.
X is ideally conceivable iff X is conceivable on ideal rational reflection.

ii. Positive Conceivability vs. Negative Conceivability
X is positively conceivable iff s X is imaginable, i.e. a situation can be imagined which verifies X.
X is negatively conceivable iff X is not ruled out a priori.

iii. Primary vs. Secondary Conceivability
X is primarily conceivable X iff X is conceivably actually the case.
X is secondarily conceivable iff X conceivably might have been the case.

Much of the prose comprising Chalmers’s taxonomy is concerned with clarifying the various notions of conceivability, and with investigating whether the various notions at best entail, or at least function as good guides to, possibility. The long and the short of his prognostications is that the most plausible CP thesis is:-
Thesis I. Ideal primary positive conceivability entails primary possibility.
He also evinces sympathy for three more candidates, although he does not pursue these, and so neither will I. For the record, these are the following, viz:-
Thesis II. Ideal primary negative conceivability entails primary possibility
Thesis III. Secunda facie primary positive conceivability is an extremely good guide to primary possibility.
Thesis IV. Ideal secondary (positive/negative) conceivability entails secondary possibility.


Returning to a familiar theme, consider
14. Water is XYZ
Chalmers argues that 14 is ideally primarily positively conceivable. For instance, the Twin-Earth situation is an imaginable situation which would verify 14, which we can conceive of as actually being the case, and which remains conceivable on ideal rational reflection.

The next three questions are:-
* What is primary possibility?
* Does being conceivable in the prescribed manner really entail primary possibility?
* Is X’s primary possibility really a metaphysical possibility of X?

Positive answers to both of the second and third questions will jeopardize the defence of APN against CP, and more widely speaking the defence of act against the Completeness Objection.  In the event, as I will argue, the third question should be answered negatively. This is the root explanation for the failure of his argument from 2D against APN: on Chalmers’s own terms, the primary possibility which water has of being XYZ is really the anodyne metaphysical possibility of “water” designating XYZ.

(5.7.2) Primary Possibility
In Chalmers’s own words,
“S is primarily possible if its primary intension is true in some possible world” [p164].
Going by this characterization, 14 is primarily possible, because it is true that the watery stuff on Twin Earth is XYZ.

(5.7.3) Does Ideal Primary Positive Conceivability Really Entail Primary Possibility?
To quote Chalmers again,
“When we find it conceivable that water is not H2O ... there is a possible world that satisfies ‘water is not H2O’ when the world is conceived as actual. Put differently ... the primary intension is true in some ... worlds.” [p165]
This looks correct. At any rate, we may grant it for the sake of advancing the dialectic

(5.7.4) Is X’s Primary Possibility Really a Metaphysical Possibility of X?
Chalmers seems to think that there is some a-fortiori-type of principle whereby we may argue from primary possibility to possibility tout court, understanding the latter as metaphysical possibility underwritten by the pw system in which both types of possibility – primary and tout court/metaphysical – are conveniently articulated. No doubt there is such a principle, and a perfectly sound one at that. However it is not X’s primary possibility which is to be identified as X’s possibility tout court. Chalmers goes badly wrong in supposing otherwise. Reverting yet again to the case of water and accepting Chalmers’s terms, it is indeed primarily possible that water is XYZ; indeed, we have already worked out that 14’s primary intension is true on Twin Earth. But this is just to say, in pw terminology,
15. ∃w(at w, “water” refers to XYZ)
which is the translation of the modal sentence
16. ◊ “water” refers to XYZ.
In short, Chalmers’s argument does not show, as he would wish, that
17. ●(water is XYZ) → ◊(water is XYZ)
but rather that
18. ●(water is XYZ) → ◊(“water” refers to XYZ).
Chalmers has not established that X’s conceivability implies X’s possibility, but rather that X’s conceivability implies the possibility of something, Y perhaps, distinct from X; which is entirely anodyne and acceptable to partisans of APN.


Conclusion

In this chapter, I have discussed the obvious objection against act, namely that in rescinding from pw theory it repudiates the resources it needs in order to deliver a complete account of modal truth – there will not be enough regions and individuals in α to truth-make all the analyses of all the true modal sentences.

I suggested two ways of reducing the range of modal truths in respect of which the problem arises; appealing to the C-method or to inflationary cosmology and Physical Actualism. Once again, more liberal inflationary regimes mean that there are fewer recalcitrant possibility claims for the act-theorist to deal with.

The problem range, thus reduced, is much more plausibly susceptible of disposal by means of an error theory (we would be wrong to consider deeply alien/merely metaphysical possibilities as genuine possibilities, but so what?).

One class of possibility claims that appears particularly amenable to error-theoretical disposal is the class of those involving APN-denial, since the literature on the subject boasts a number of pro-APN arguments. I have sounded a note of caution in relation to this procedure insofar as the prosecution of an error-theory in relation to APN-denial involves refuting the CP mode of argumentation, whereas CP argumentation seems to remain applicable in cases not involving APN. However, I consider that it is open to the act-theorist also to dispose of many of these claims by recourse to the C-method or Physical Actualism. Finally, I have defended error-theory in relation to APN-denial against the 2D-based attack mounted against it by Jackson and especially Chalmers.

Chapter 6. Soundness:- An Actuality Objection

The present chapter addresses a problem to do with soundness. The original insight is Hazen’s [1979] (q.v. §6.2), with more recent elaborations by M.Fara & Williamson (MFW) [2005] (§§6.3-6) and D.G.Fara (DGF) [2010] (§6.7). The problem bears on counterpart theories in general, and a fortiori on act being as act is a counterpart theory. Call the problem the Actuality Objection, insofar as it concerns the treatment of modal actuality sentences, i.e. sentences about what might have actually been the case.

The Actuality Objection is a problem of soundness insofar as its central complaint is that certain natural language sentences concerning modal actuality, although revealed as inconsistent when formalized in QMLA, translate to satisfiable sentences of counterpart theory. 

The Objection arises in one of the following scenarios, when an individual does not have a unique actual counterpart:- the individual is possible and has no actual counterparts (NC) [MFW 2005], the individual is possible and has multiple actual counterparts (MC) [Hazen 1979, MFW ibid.], and the individual is actual and has at least one distinct actual counterpart (AC) [DGF 2010]. It is MFW’s particular claim, that such scenarios generate satisfiable sentences of counterpart theory as the translations of inconsistent QMLA sentences, no matter which QMLA-counterpart theory translation scheme is chosen.

MFW identify three possible responses to the Objection, all presumptive of its success:- Two of these involve replacing counterpart theory altogether – either with modalism, or with a Kripkean pw theory i.e. with transworld identity instead of the counterpart relation [2005 p26]. Both theories having been comprehensively rebutted in the present work, I do not address them again in ch6. The third response involves retaining counterpart theory, but stipulating that an individual has a unique counterpart at every world. MFW reject this response because it is ad hoc, and I agree with them.

Two more responses, which MFW overlook, aim to defend counterpart theory by rebutting the Objection. I am more sympathetic to the second of these:-

I. In §6.8, I take up Lewis’s postulation of multiple counterpart relations [Lewis 1979, 1986 p230]. In this way, it is hoped, some unique individual is always the sole actual counterpart of a possible individual relative to any given context. DGF has pointed out [ibid.] that such an arrangement requires A not to e.g. distribute over &. As she argues, it is very unlikely that A does behave in this way; such behaviour on the part of A certainly doesn’t answer to the phenomenology.

II. In §6.9, I advert to a forthcoming paper [Meyer 2012], in which Ulrich Meyer argues that A is eliminable in SQML with set-theory, and that SQML regimentations of natural language modal actuality sentences avoid the Actuality Objection – there appear to be no cases of inconsistent SQML sentences translating to satisfiable sentences of counterpart theory. In §6.10 I show how the Objection bears upon act, and then how to apply Meyer’s solution in the case of act. Finally, a potential drawback of Meyer’s solution is that SQML generates notorious issues in connection with its validation of the Barcan Formula (BF) and the Thesis of Necessary Existence (NE). Nevertheless, I argue that act palliates the consequences of these validations much more convincingly than rival SQML-based proposals suggested by Plantinga, Linsky&Zalta, and Williamson (q.v. ch2).


(6.1) Setting Up the Actuality Problem

Recall from ch1 the motivation for enriching QML with an actuality operator A: doing so at least looks like the only way for a formalized modal language to express modal actuality sentences – sentences about what might have actually been the case. In §6.1.1 below, I discuss how
(a) conventional Kripkean pw theory with identity across worlds, and
(b) counterpart theory
are expected to account for modal actuality. In the latter case this is a question of how many actual counterparts of a being F it takes for AFa to hold, i.e. which of the following is true:-
A∃. a is Aly F ≡ at least one of a’s actual counterparts is F
A∀. a is Aly F ≡ all a’s actual counterparts are F
AQ. a is Aly F ≡ a quantity Q of a’s actual counterparts are F
In §6.1.2, I explain the notion of non-standard scenarios in counterpart theory, where an individual has something other than an unique counterpart at every world, and how these scenarios are what generate different instances of inconsistent QMLA sentences translating to satisfiable sentences of counterpart theory, depending on how A is characterized. In §6.1.3, I introduce five schemes which have been proposed for QMLA-counterpart theory translation. Finally, §6.1.4 features some remarks about the logic of actuality, and the extent to which respecting or disrespecting the logic of actuality bears on the ability or otherwise of the respective schemes to avoid the respective versions of the Actuality Objection. This forms the basis for §§6.2-7, in which I provide detailed specifications of the cases built by Hazen, MFW and DGF against individual variants of counterpart theory – and, in the case of MFW, against counterpart theory as a whole.

(6.1.1) Actuality, PW Theory and Counterpart Theory
Suppose we accept Hazen’s argument that A is ineliminable from QML – that we can only formalize the actuality fragment of modal language by adding A to QML to make QMLA. The question that most naturally arises now is: how are actuality and modal actuality to be characterized in the pw-theoretical language into which QMLA is to be translated? 

Intuitively, pw theory analyses modal actuality in the propositional case by means of:-
Apw. Aφ is true at a world w ≡ φ is true at α.
What about modal actuality de re? Such a claim, that an individual a might have actually been F, is formalized by means of AFa. Given Apw, we will expect a Kripkean pw theory, in which identity is permitted across worlds, to analyse de re modal actuality by means of
Apwk. AFa is true at w ≡ Fa is true at α.
Unfortunately, as we know from ch2, there are good reasons for objecting to Kripkean pw theory:- the problems of transworld identity and accidental intrinsics. For present purposes, regard the associated Objections as decisive against the Kripkean system.

And so in short order we turn our attention to counterpart theory. Intuitively, this latter analyses de re modal actuality in terms of what happens to actual counterparts of a, i.e. by means of
Act. AFa is true at w ≡ X actual counterparts of a are F.
This characterization raises an obvious question over what to substitute in the place of ‘X’. How many actual counterparts of a being F does it take to make AFa come out true? One, all, or some quantity in between? That is to say, do we quantify over actual counterparts existentially, universally, or by means of some other quantifier? In the vernacular of this chapter, ∃ ∀ or Q? To repeat the three candidate analyses from earlier:-
A∃. a is Aly F ≡ at least one of a’s actual counterparts is F
A∀. a is Aly F ≡ all a’s actual counterparts are F
AQ. a is Aly F ≡ a quantity Q of a’s actual counterparts are F

(6.1.2) Counterpart Theory, Non-Standard Scenarios, and the Actuality Objection
In standard scenarios of counterpart theory, an individual has a unique counterpart at every world. Typically, given Lewis’s classical similarity-based conception of counterparthood [1968, 1986], your counterpart at a world w is the unique individual there most closely resembling you. The ‘how many is X?’ question has a straightforward answer if a standard scenario is under consideration, namely “one”. AFa holds iff a’s unique actual counterpart is F.

In non-standard scenarios, a has something other than a unique counterpart at some world. And, as Lewis famously concedes [1968 pp115 et passim], worlds satisfying non-standard arrangements of counterparts are not plausibly postulated away, so non-standard scenarios have to be taken into consideration. (Moreover, to anticipate proceedings, this is also true of act, q.v. ch4 in the present work.)

 Now modal actuality generates the following non-standard scenarios for counterpart theory:-
* a has multiple mutually distinct actual counterparts – the MC scenario.
* a has no actual counterparts – the NC scenario.
* a is actual and has an actual counterpart distinct from itself – the AC scenario.
Note that, since in most versions of counterpart theory an individual is its own counterpart in its own world, AC is in fact a special case of MC – because the individual then has multiple actual counterparts, viz, itself and also its distinct actual counterpart.

These non-standard scenarios are, finally, what generate the Actuality Objection in its various guises. The basic idea underlying the Objection is that allegedly inconsistent QMLA sentences – specifically certain A-involving sentences invoking MC, NC or AC – translate by means of the chosen translation scheme ts into satisfiable sentences of counterpart theory. To anticipate later sections of this chapter, consider an MC scenario, in which a has both F and ¬F actual counterparts. If the ‘how many is X’ question is answered ‘some’, that is if 
A∃. a is Aly F ≡ at least one of a’s actual counterparts is F
 then the corresponding MC scenario is formalized in QMLA by means of (AFa & A¬Fa), which is inconsistent on the further assumption that A commutes with ¬, which it does given the logic of actuality. However, the MC scenario forms a satisfiable sentence in a counterpart-theoretical language, as is attested by the very fact that a is spoken of as having both F and ¬F actual counterparts. In Joe Melia’s useful phrase, inconsistent QMLA sentences map onto satisfiable sentences of counterpart theory.

(6.1.3) An Overview of Five Translation Schemes
Counterpart theory has been subjected to attacks based on the Actuality Objection by three parties: Hazen, MFW and DGF:-

* Hazen’s concern (q.v. §6.2)  is to argue that two extensions of Lewis’s translation scheme tsl generate bad mappings. The schemes in question are tsA∀ and tsA∃, obtained by adding respectively A∀ and A to tsl. Thus Hazen may be construed as arguing that there is no systematic translation from QMLA into Lewis’s cct; leaving open the possibility of there being a systematic translation from QMLA into some other variant of counterpart theory.

* MFW build on Hazen’s insight. According to them, bad mappings affect not just tslA∀ and tslA∃, but also
(i) a third tsl extension tslAQ, obtained by adding AQ to tsl (§6.3) and 
(ii) the ts of Forbes (tsf) (§6.4) and Ramachandran (tsr) (§6.5).
The prognoses in respect of these five ts invite the inference that no ts avoids bad mappings. In short, the Objection bears on counterpart theory as a whole, and not just particular variants corresponding to particular ts. The final nail in the coffin comes in the form of a Symmetry version of the MC scenario (§6.6). According to MFW, this last is fatal to all ts.

* Recently, DGF [2010] has proposed an extension to the case built by MFW. She considers a scenario – a world of eternal recurrence to be exact – where a is actual and has a distinct actual counterpart, which is F to a’s ¬F. Yet another bad mapping ensues, in that counterpart theory appears to satisfy (¬Fa & AFa), which is inconsistent assuming that AFa≡Fa holds, as it does given the logic of actuality. Thus §6.7.

(6.1.4) The Logic of Actuality
For present purposes, the salient components of the logic of actuality are the principles that AFa≡Fa holds, and that A commutes with ¬ and distributes over the connectives.

The logic of actuality will be a recurrent theme in any thorough exposition of the Objection:-
* The problematic QMLA sentences, the QMLA components of individual bad mappings, are described as inconsistent assuming one or other of the A-logic principles above.
* The three tsl extensions are described as failing to respect the logic of actuality. In particular, their failure to respect the fact that AFa≡Fa holds is ascribed to their inheriting the original tsl translation of Faw as Fa, whilst AFa receives its respective translation (A∀, A∃, or AQ depending on which tsl extension) from the additional clauses supplied by Hazen and MFW. Moreover, the bad mappings generated by the three tsl extensions are ascribed to their failures to respect the logic of actuality.
* Unlike the three tsl extensions, Forbes’s tsf and Ramachandran’s tsr respect the logic of actuality. In particular, that they both respect AFa≡Fa is ascribed to Fa being defined in each case by means of a clause specifying how many of a’s actual counterparts are F. In the case of tsf, Fa is defined in terms of some of a’s actual counterparts being F. In the case of tsr, Fa is defined in terms of some and all of a’s actual counterparts being F. This leaves Forbes and Ramachandran free to stipulate in their respective clauses for A precisely that AFa≡Fa, since the “how many is X?” question is settled simply by attending to the clause for Fa.

It is perhaps unrealistic to expect ts which fail to respect the logic of actuality to effect unproblematic translation from QMLA into counterpart theory. So the failings of the tsl extensions perhaps do not come as too much of a surprise. Correspondingly, we might expect tsf and tsr to effect unproblematic translation, since they do respect the logic of actuality.  However, tsf and tsr in fact face bad-mapping difficulties of their own; although these difficulties tend to relate to satisfiable translations of QMLA sentences which are inconsistent in a first-order logic with identity, rather than inconsistent in propositional logic as is the case with the tsl extensions. Finally, the Symmetry argument, presented as a form of MC scenario, is alleged by MFW to be fatal to all counterpart theories, whether or not they respect the logic of actuality.

Over §6.2-7 now following, I provide detailed specifications of the permutations of the Objection, as they bear on the various ts and associated counterpart theories.


(6.2) Hazen vs. Lewis

A prototype of the Actuality Objection is first aired in Hazen’s [1979]. As described above, Hazen’s attack focuses on two extensions of tsl for translating sentences of QMLA into sentences of counterpart theory. Hazen has to extend tsl because its clauses as set out in Lewis [1968] make no provision for modal actuality. The two extensions are the result of enriching tsl with a clause characterizing A either in ∃ mode, which gives us A∃ and therefore tslA∃ ; or in ∀ mode, which gives us A∀ and therefore tslA∀. To recall:-
A∃. a is Aly F ≡ at least one of a’s actual counterparts is F
A∀. a is Aly F ≡ all a’s actual counterparts are F.

Hazen argues that adopting tslA∀ leads to “a sort of failure of excluded middle” [1979 p330]. This comes about in the sense that the second conjunct of the following QMLA sentence
1. ◊∃x(A∃y x=y & ¬(AFx v A¬Fx))
violates the Law of Excluded Middle, assuming that A commutes with ¬. Nevertheless 1 maps onto a satisfiable sentence of counterpart theory. Hazen does not specify the sentence in question. However, it is reasonable to assume that he has in mind an MC case, sc. a possible object x having multiple actual counterparts, some of which are F and some of which are ¬F. For this scenario procures the required result, i.e. that a possible x is neither A-ly F nor A-ly ¬F

Alternatively, adopting tslA∃ leads to “something at least equally repellent” [ibid.], which Hazen might consistently have described as a (sort of) failure of non-contradiction. This comes about in the sense that the second and third conjuncts of the following QMLA sentence
2. ◊∃x(A∃y x=y & AFx & A¬Fx)
together negate the Law of Contradiction, assuming again that A commutes with ¬. Nevertheless, 2 maps onto a consistent sentence of counterpart theory. Hazen is again a little dilatory with his specifications. However, it is reasonable to assume that the same MC scenario is again involved, since this again procures the required result, in this case that a possible x is both A-ly F and A-ly ¬F. Current proceedings are summarised by means of Table 1, with ✗ signifying the occurrence of bad mapping.



Tab. 1. Hazen vs. tsl:- MC
Contradiction? Excluded Middle?
tslA
tslA


(6.3) MFW vs. Lewis

The early part of MFW’s [2005] is taken up with elaborating Hazen’s criticisms of tsl, by confronting tslA∃, tslA∀, and also tslAQ – the latter being the result of augmenting tsl with
AQ. a is Aly F ≡ a quantity Q of a’s actual counterparts are F
 – first with NC (§6.3.1), and then with MC (§6.3.2). MFW run the procedure in each case both for sentences featuring names for possible individuals and for sentences featuring bound variables, whereas Hazen only runs the procedure for sentences featuring bound variables.

(6.3.1) Lewis and the Case of No Actual Counterparts
MFW begin by investigating NC in relation to tslA∃. There are two cases of this, depending on whether it is permissible to name mere possibles, or whether we are to confine ourselves to sentences with bound variables.

tslA∃ with proper names
MFW show how the QMLA sentence
3. Fa & ¬AFa
which is inconsistent assuming AFa≡Fa, maps via A∃ and NC to the following satisfiable sentence of counterpart theory:-
4. Fa & ¬∃x(Ixα & Cxa & Fx).
4 is satisfied by a considered as non-actual, as F, and as lacking actual counterparts.

tslA∃ with bound variables
MFW show how the QMLA sentence
5. ◊∃x(AFx ≡ A¬Fx),
which again is inconsistent assuming A/¬ commutation, maps onto the following satisfiable sentence of counterpart theory:-
6. ∃w∃x(Ixw & (∃y(Iyα & Cyx & Fy) ≡ ∃y(Iyα & Cyx & ¬Fy))).
Again, 6 is satisfied as one would expect in an NC case, by a possible individual lacking actual counterparts, because both conjuncts of the embedded biconditional are false, so the biconditional is true.

This concludes the investigation of tslA∃ in relation to NC. MFW now confront tslA∀ with NC. Again, there are proper-name and bound-variable cases to be dealt with.

tslA∀ with proper names
MFW show how the QMLA sentence
7. Fa & A¬Fa
which is inconsistent assuming A/¬ commutation, maps via A∀ and NC to the following satisfiable sentence of counterpart theory:-
8. Fa & ∀x((Ixα & Cxa) → ¬Fx).
8 is satisfied by a non-actual individual which is F and lacks actual counterparts, because the conditional forming the second conjunct is vacuously true.
tslA∀ with proper names
MFW show how the QMLA sentence
7. Fa & A¬Fa
which is inconsistent assuming A/¬ commutation, maps via A∀ and NC to the following satisfiable sentence of counterpart theory:-
8. Fa & ∀x((Ixα & Cxa) → ¬Fx).
8 is satisfied by a non-actual individual which is F and lacks actual counterparts, because the conditional forming the second conjunct is vacuously true.

tslA∀ with bound variables
MFW show how the QMLA sentence
5. ◊∃x(AFx ≡ A¬Fx)
which as before is inconsistent assuming A commutation, maps onto the following satisfiable sentence of counterpart theory:-
9. ∃w∃x(Ixw & ((∀y(Iyα & Cyx) → Fy) ≡ (∀y(Iyα & Cyx) → ¬Fy))).
9 is satisfied by a non-actual individual lacking actual counterparts, because the conditionals forming the conjuncts of the embedded biconditional are both vacuously true, so the biconditional is true.

This concludes the investigation of tslA∀ in relation to NC. MFW now confront tslAQ with NC. As with tslA∃ and tslA∀, there are proper-name and bound-variable cases to be dealt with.

tslAQ with proper names
MFW show how the QMLA sentence
10. Fa & (A¬Fa v ¬AFa)
which is inconsistent assuming A commutation, maps onto the following satisfiable sentence of counterpart theory:-
11. Fa & ([Qx: Ixα & Cxa](¬Fx) v ¬[Qx: Ixα & Cxa](Fx))
11 is satisfied by a non-actual individual lacking actual counterparts. This is because the left-hand conjunct is true; and the right-hand disjunct of the disjunction forming the right hand conjunct is true if a lacks actual counterparts, which makes the right hand conjunct true also. 

tslAQ with bound variables
MFW show how the QMLA sentence from earlier
5. ◊∃x(AFx ≡ A¬Fx)
which as before is inconsistent assuming A/¬ commutation, maps onto the following satisfiable sentence of counterpart theory:-
12. ∃w∃x(Ixw & ([Qy: Iyα & Cyx](Fy) ≡ [Qy: Iyα & Cyx](¬Fy)))
which yet again is satisfied by a non-actual individual a lacking actual counterparts. Firstly, a satisfies Ixw; and secondly, both sides of the biconditional forming the right-hand of the main conjunction come out false. If a lacks actual counterparts, then it is false both that Q actual counterparts of a are F, and that Q actual counterparts of a are ¬F. Since the two sides of the biconditional share the same truth value – i.e. they’re both false, the biconditional itself comes out true, which makes the right-hand conjunct formed by the biconditional true also.

This concludes the investigation of tslAQ in relation to NC. MFW now turn their attention to the case of tsl and MC.

(6.3.2) Lewis and the Case of Multiple Actual Counterparts
Recall my earlier remark, that standard models of counterpart theory give an individual exactly one counterpart per world; and Lewis’s [1968] remark, to the effect that postulates ought not to be abused for the purpose of precluding non-standard cases. Both Lewis and MFW devise MC models; although MFW perhaps make heavy weather out of this, mainly because their MC model involves Allen Gibbard’s [1975] famous ‘Statue’ case, which crops up throughout their exposition of MC:-

* Lewis points out that, on the plausible supposition that there is an individual whose closest resemblants at a world are identical twins, it is reasonable to treat both twins as counterparts in their world of that individual [Lewis 1968 p116].

* MFW argue for multiple counterparts on the basis that they are the outcome of a natural counterpart-theoretical solution to a problem of material constitution [MFW ibid. p13]. Briefly, on the counterpart-theroretical view, a statue Goliath existing at some world w is identical with the lump LumpL of clay constituting it, avoiding the absurd possibility of distinct but spatiotemporally coincident objects. However, Goliath and LumpL are only contingently identical, because Goliath/LumpL has at a world w1 both a statue-like counterpart and a lump-like counterpart which are distinct from one another. Finally, if it is plausible that an individual has multiple counterparts at a world, then it is reasonable to suppose that α is one of the worlds at which an individual has multiple counterparts. Hence, we arrive at speaking of an individual a having multiple actual counterparts.

Having by this means established the reasonableness of MC-type scenarios, MFW next set about showing how MC scenarios generate bad mappings for tsl. Their procedure is much the same as it was with NC. They begin by investigating MC with tslA∃. As is to be expected, there are proper-name and bound variable cases of this.

tslA∃ with proper names
MFW show how the QMLA sentence
13. a=b & A¬a=b
which is inconsistent assuming A/¬ commutation, maps onto the following satisfiable sentence of counterpart theory:-
14. a=b & ∃x∃y(Ixα & Iyα & Cxa & Cyb & ¬x=y)
Continuing with the statue case, let a be Goliath and b be LumpL. Now let x and y range over, respectively, statue-like and lump-like counterparts of a/b. Thus 14 is satisfied by the other-worldly statue having distinct multiple actual counterparts – i.e. the statue-like and lump-like counterparts ranged over by x and y.

tslA∃ with bound variables
MFW show how the QMLA sentence
15. ◊∃x(AFx & A¬Fx)
which is inconsistent assuming A commutation, maps onto the following satisfiable sentence of counterpart theory:-
16. ∃w∃x(Ixw & ∃y(Iyα & Cyx & Fy) & ∃y(Iyα & Cyx & ¬Fy)).
Again, MFW illustrate 16’s satisfiability by means of the statue case, with F thought of as the property of being statue-like. 16 is satisfied by the other-worldly statue having distinct counterparts, one of which is statue-like and one, presumably the lump-like counterpart, not statue-like.

This concludes the investigation of tslA∃ in relation to MC. MFW now confront tslA∀ with MC. Again, there are proper-name and bound-variable cases to be dealt with.

tslA∀ with proper names
MFW show how the QMLA sentence
17. a=b & ¬Aa=b
which is inconsistent assuming AFa≡Fa, maps onto the following satisfiable sentence of counterpart theory:-
18. a=b & ¬∀x∀y((Ixα & Iyα & Cxa & Cyb) → x=y)
which is satisfied by an other-worldly statue with distinct actual counterparts, e.g. a statue-like counterpart and a lump-like counterpart.

tslA∀ with bound variables
MFW show how the QMLA sentence
19. ◊∃x(¬AFx & ¬A¬Fx)
which is inconsistent assuming A/¬ commutation, maps onto the following consistent sentence of counterpart theory:-
20. ∃w∃x(Ixw & (¬∀z((Izα & Czx) → Fz) & ¬∀z((Izα & Czx) → ¬Fz))).
With F considered as the property of being statue-like, 20 is satisfied by a possible statue with distinct actual counterparts, which are neither all statue-like (because some of them are lump-like instead), nor all non-statue-like (because some of them are statue-like).

This concludes the investigation of tslA∀ in relation to MC. MFW now confront tslAQ with MC. MFW only provide a bound-variable case of this.

tslAQ with bound variables
MFW show how the QMLA sentence
21. ◊∃x(ALx ≡ A¬Lx)
which is inconsistent assuming A/¬ commutation, maps onto the following satisfiable sentence of counterpart theory:-
22. ∃w∃x(Ixw & ([Qy: Iyα & Cyx](Ly) ≡ [Qy: Iyα & Cyx](¬Ly)))
Let Lx be characterized as a predicate true of the statue’s lump-like counterparts but false of its statue-like counterparts. Thus 22 is satisfied by a possible object with both Q actual counterparts which are L in virtue of being lump-like, and Q actual counterparts which are not L, in virtue of being statue-like.

To summarise the current section, if MFW’s arguments are correct, then every way of extending tsl in order to accommodate A maps some inconsistent QMLA sentence onto some satisfiable sentence of counterpart theory. Hence MFW’s interim conclusion, that there is no plausible way of extending tsl so as to effect a systematic translation from QMLA into counterpart theory.

This is scarcely to be wondered at. It should not surprise us that, when theoreticians translate logically inconsistent QMLA sentences by means of a ts in a metalanguage which does not itself respect the logic of actuality, bad mappings ensue. However, tsl’s failure does not by itself establish MFW’s terminating conclusion, sc. that there is no systematic translation whatsoever from QMLA into counterpart theory. In order to establish this, MFW have to show that bad mappings also ensue for competing translation schemes which do respect the logic of actuality.

Two such schemes are Forbes’s tsf [1990] and Ramachandran’s tsr [1989]. Both schemes address, inter alia, Hazen’s original version of the Objection. Recall this from §1. Assuming MC and A/¬ commution, the exposure of tslA∀ to
1. ◊∃x(A∃y x=y & ¬(AFx v A¬Fx))
generates a violation of the Law of Excluded Middle; and the exposure of tslA∃ to
2. ◊∃x(A∃y x=y & AFx & A¬Fx)
generates a violation of the Law of Contradiction. I take up the Objection in relation to tsf and tsr in the two sections now following.


(6.4) MFW vs. Forbes

MFW concede that tsf does not map 1 and 2 onto consistent sentences of counterpart theory. Indeed, they concede that tsf does not generate the bad mappings generated by the tsl extensions, because unlike the latter it respects the logic of actuality. However, they complain [ibid. pp17-20] that tsf still generates other bad mappings based on first order logic with identity, showing this with respect to sentences featuring i. relational and ii. monadic predicates. These two cases are quite beside the Symmetry version of the MC scenario, which MFW later identify as fatal to counterpart theory whichever translation scheme is adopted [q.v. §7].

(6.4.1) How tsf Works
tsf differs from tsl and its extensions in how atomic sentences and modal actuality are translated. We have seen how tsl and its extensions translate Faw as Fa, leaving the ‘how many is X?’ question to be settled by the respective tsl extensions with their respective clauses for A - tsl∃, tsl∀ or tslQ. In contrast, tsf settles the ‘how many is X’ question at the stage of defining the translation of Faw. In brief, the translation of a sentence Fa relativizes Fa to α – i.e. Fa is translated as Faα. For a sentence Fa and a world-term w, the relativization of Faw to α is defined as ∃x(Ixα & Cxa & Fx). Fa is true at w iff some actual counterpart of a is F. Settling the ‘how many is X’ question at this stage enables tsf to define the translation for AFa with a clause for A stipulating that AFa≡Fa, whence is derived
23. AFa ≡ Fa ≡ ∃x(Ixα & Cxa & Fx).

It is because tsf respects the logic of actuality, that it avoids the objections raised by MFW against the tsl extensions. To illuminate this MFW reconsider
10. Fa & (A¬Fa v ¬AFa).
Precisely because it respects the fact that A commutes with ¬ , tsf translates 10 into a sentence of counterpart theory with the logical form of φ&(¬φv¬φ) – to obtain 10’s translation, read φ as ‘∃x(Ixα & Cxa & Fx)’. Since a sentence with this logical form cannot be satisfied even in counterpart theory, it seems that Forbes’s version of counterpart theory, avoids mapping QMLA sentences which are inconsistent in propositional logic onto satisfiable sentences of counterpart theory.

Nevertheless, MFW show that tsf does not avoid mapping other inconsistent QMLA sentences onto satisfiable sentences of counterpart theory. They show this in two cases – in a case of relational predicates (§6.4.2), and in a case of monadic predicates (§6.4.5). Both cases utilize the same story, about a possible individual Yolanda, and her actual counterparts Alfred and Agnes.


(6.4.2) tsf and the Case of Relational Predicates
The case of relational predicates purports to show that tsf renders intransitive relations which are intuitively transitive. This case is susceptible of formalization as yet another instance of bad mapping. That is to say, where R is virtually any reflexive and intuitively transitive relation, tsf maps:-
24. ∃x∃z◊∃yA(Rxy & Ryz & ¬Rxz)
onto a satisfiable sentence of counterpart theory.

This counts as a bad mapping, in the sense that, because R is ex hypothesi transitive, 23 states a metaphysical inconsistency – transitive relations can’t be intransitive – and yet translates to a satisfiable sentence of counterpart theory. In MFW’s exposition of the case [ibid. pp18-19], the relation of identity stands in for R, the non-actual Yolanda stands in for y, and her actual counterparts Alfred and Agnes stand in for, respectively, x and z. Now the relation of Being-a-Counterpart-Of is traditionally understood [q.v. Lewis 1968] as replacing Transworld-Identity in counterpart theory. MFW’s contention is then that tsf maps the following instance of 24, which clearly violates first order logic with identity,
25. ∃x∃z◊∃yA(x=y & y=z & ¬x=z)
onto a satisfiable sentence of counterpart theory; Yolanda might have been identical with either of her counterparts, Alfred and Agnes, whereas Alfred and Agnes are distinct.

One slightly complicating feature of the foregoing is that MFW represent their exposition as showing that tsf renders identity non-transitive, whereas going by Forbes’s own explicit proposal, tsf has the resources to render transitive only identity out of pretty much all the reflexive and intuitively transitive relations. That tsf has the resources to render identity necessary, a fortiori transitive, is a consequence of rescinding the Falsehood Convention (FC) in the particular case of identity, as Forbes proposes in an earlier paper [1982 p34 clause (iii)]. In pw language, FC is the following claim:-
FC. Faw requires Iaw
i.e. for an individual to have a property at a world requires its presence there. Let F be the relation of identity with b. Rescinding FC for the case of identity has the effect of making a and b identical with respect to every world, including worlds not containing them, and therefore necessarily identical. Hence, contra MFW, tsf does not render identity intransitive unless Forbes’s proposal is discounted.

However, this does not damage MFW’s underlying point which, to repeat, is that almost any reflexive and intuitively transitive relation is rendered intransitive by tsf. This is easily seen by adverting to a model in which Yolanda is both a possible ancestor of Alfred and a possible descendant of Agnes, although Alfred and Agnes are unrelated. This modification establishes MFW’s point, because being an ancestor or descendant of someone is a transitive relation rendered intransitive by tsf.

(6.4.3) A Pertinent Digression: act and the Necessity of Identity
Consider FC’s act-theoretical analogue, r being as throughout an arbitrary region of α:-
FCact. Far requires Iar
i.e. for an individual to have a property in a region of α requires its presence there.

The suggestion is that one further codicil be added to the formal exposition of act in ch4. The codicil consists in the rescinding of FCact. There may be scruples over the prospect of rescinding FCact wholesale. If so, how about rescinding it for the case of identity? The suggestion is that, alone of all the predicates if need be, identity predications if true at all are true pervasively throughout α, even at regions from which the individuals named are absent. This Forbesian treatment of identity enables act to validate the Necessity of Identity (NiD). Validating NiD can otherwise prove problematic for counterpart theories in general, as witness Lewis’s attempt to parlay into a strength of cct the fact that it enables the expression of contingent identity claims. Moreover, the idea of treating identity distinctly from other notions both has a fairly venerable history in philosophy and logic, and is if anything better motivated in act. Intuitively, it is reasonable for an actual individual to postulate identities between other actual individuals even if these latter are located in distant regions of α. The postulation of identities between mere possibilia raises familiar Quinean issues about propriety, unless these latter are understood, as they are in cct, as concrete. However, the individuals in act which play the role analogous to that of mere possibilia in a pw theory such as cct are just those actual individuals not in the @ of act – and these are typically concrete.

A final point: why restrict the rescindment of FC just to the case of identity? It seems fairly mandatory to extend this treatment to mathematical and logical truths.

(6.4.4) A Less Pertinent Digression: the Falsehood Convention and Strong Necessity
This short and slightly tangential subsection addresses Forbes’s motivation for rescinding FC. Briefly, Forbes suggests rescinding FC as a way of maintaining the principle of Strong Necessity, without relinquishing the highly plausible claim that some contingently existing individuals have necessary properties.

Here is the principle of Strong Necessity:-
SN. ⎕A’s truth at a world requires A’s truth at every world.
SN might be considered beyond plausible. For example if S5 is assumed for simplification, then doesn’t SN just implement the original Leibnizian dictum that necessity is a matter of truth at all possible worlds? Surely to let go of SN is to lose one’s grip on necessity.

To illustrate the idea of contingently existing individuals with necessary properties, consider my own case. I modestly concede the contingent nature of my own existence, yet maintaining that I am necessarily human. If I am correct in this, then by SN “RC is human” must be true at every world. But just now I conceded my contingency, so in standard cct/act-sprach there are worlds/regions from which I am absent.
Diagnosis:- SN + contingent individuals with necessary properties = ¬FC.
Forbes’s solution:- rescind FC.
My point:- act is also motivated to rescind FC. By this means, “RC is human” can be necessarily true on act’s terms, i.e. true at every region r of α, whilst my existence itself is plainly contingent, since there are spatiotemporal regions from which I am absent.

(6.4.5) tsf and the Case of Monadic Predicates
The case of monadic predicates appeals to a notion of essence-specifying property, such as Being-Alfred, or Being-Agnes. The crucial point here is that distinct essence-specifying properties, like the Plantingan individual essences which they closely resemble – are mutually incompatible; i.e. the same individual cannot have both the property of Being-Alfred and that of Being-Agnes. MFW’s complaint is then that tsf renders compatible properties which are incompatible, i.e. renders distinct essence-specifying properties compatible. Continuing with the example of Yolanda, Alfred and Agnes, let F and G be respectively Being-Alfred and Being-Agnes. Then the following “manifestly inconsistent” [p19] QMLA sentence
26. ◊∃xA(∃y x=y & Fx & Gx & ∀y¬(Fy & Gy))
is satisfied by x ranging over Yolanda and y ranging over Alfred and Agnes.


(6.5) MFW vs. Ramachandran

MFW accept that, like tsf, tsr avoids all the bad mappings incurred by the tsl extensions. tsr avoids this fate for much the same reason that tsf does – because it respects the logic of actuality. MFW also accept that tsr avoids the problems in relation to sentences 24-26, which proved “fatal” [p21] to tsf, in virtue of a modified translation for atomic sentences. However, again like tsf, tsr suffers a number of problems of its own in relation to bad mappings. And again, these are quite beside the Symmetry argument awaiting us in §6.6.

(6.5.1) tsr Translates Fa in Terms of ∃&∀
In short, whereas tsf defines the translation of Faw in terms of existential quantification over actual counterparts, tsr defines it in terms of, so to speak, Russellian i.e. existential-plus-universal quantification over actual counterparts. Thus, whereas
23. AFa ≡ Fa ≡ ∃x(Ixα & Cxa & Fx).
is deriveable from tsf, the following is deriveable from tsr, viz:-
27. AFa ≡ Fa ≡ (∃x(Ixα & Cxa & Fx) & ∀x((Ixα & Cxa) → Fx))

As well as accepting that tsr matches tsf’s success in avoiding the bad mappings which bore upon tsl’s extensions, MFW also maintain that tsr avoids tsf’s problems with the relational and monadic predicate cases, although they do not argue for this. I supply this deficit in the two subsections immediately following.


(6.5.2) tsr and the Case of Relational Predicates
Recall what the case of Relational Predicates seemed to show with respect to tsf, sc. that tsf renders intransitive just about any reflexive and intuitively transitive relation. This was then recast as a bad mapping in the style of the Actuality Objection, in the sense that, where R is virtually any reflexive and intuitively transitive relation, tsf maps the metaphysically inconsistent:-
24. ∃x∃z◊∃yA(Rxy & Ryz & ¬Rxz)
onto a satisfiable sentence of counterpart theory, this being shown by means of the Yolanda/Alfred/Agnes example. To recall, 24 is satisfied by a possible individual Yolanda being both a possible ancestor of the actual individual Alfred and a possible descendant of the actual individual Agnes, although Alfred and Agnes are not actually related to one another.

We are now in a position to see why, unlike tsf, tsr does not map 24 onto a satisfiable sentence of counterpart theory. On the one hand, tsf maps 24 onto a satisfiable sentence of counterpart theory because, for virtually any R, some trio consisting of a possible individual y and its two actual counterparts x and z can be found, which satisfies the tsf translation of A(Rxy & Ryz & ¬Rxz). We saw, from the example of Yolanda, Alfred and Agnes, that such a trio is not particularly difficult to come by. In contrast, tsr maps 24 onto a satisfiable sentence of counterpart theory only if, for virtually any R, all <x,y,z> trios satisfy the tsr translation of A(Rxy & Ryz & ¬Rxz). Since many such trios do not satisfy the tsr translation of A(Rxy & Ryz & ¬Rxz), tsr clearly does not map 24 onto a satisfiable sentence of counterpart theory.

 (6.5.3) tsr and the Case of Monadic Predicates
Recall what the case of Monadic Predicates seemed to show with respect to tsf, sc. that tsf renders compatible any intuitively incompatible monadic properties. This was then recast as a ch6-style bad mapping, in the sense that, where F and G are virtually any two metaphysically incompatible properties – e.g. the essence-specifying properties of distinct individuals – tsf maps the “manifestly inconsistent”
26. ◊∃xA(∃y x=y & Fx & Gx & ∀y¬(Fy & Gy))
onto a satisfiable sentence of counterpart theory, this being illustrated again by means of the Yolanda/Alfred/Agnes story.

We are now in a position to see why, unlike tsf, tsr does not map 26 onto a satisfiable sentence of counterpart theory. On the one hand, tsf maps 26 onto a satisfiable sentence of counterpart theory so long as some possible individual e.g. Yolanda has some mutually distinct actual counterparts e.g. Alfred and Agnes, each of which has its own essence-specifying properties, rendering the possible individual a possibly actual bearer of mutually incompatible essence-specifying properties, e.g. Being-Alfred and Being-Agnes. On the other hand, as MFW describe matters, tsr manages to avoid a bad-mapping in the case of 26
“by requiring that if there could be an object which is actually F and G, it is not enough that the object have some actual counterpart which is F and some actual counterpart is G, in addition, all of the object’s actual counterparts must be both F and G, which is ruled out by the last conjunct of [26].” [MFW ibid. p21]

(5.4) Three Actuality Problems for tsr
Unfortunately, as MFW find [pp21-22], tsr faces bad-mapping problems of its own. They consider three cases, 28, 29, 31 below, for the purposes of which let E be an atomic existence predicate, true at a world w of an individual with a counterpart at w:-

28. ◊∃xA(Ex & ¬(Fx v Gx) & ∀y(Fy v Gy))
The problem here is that 28 is inconsistent, but translates via tsr to a satisfiable sentence of counterpart theory. This comes about when a possible individual a has two actual counterparts, b which is F but not G, and c which is G but not F. But then not all of this possible object’s actual counterparts are F, and not all of them are G, so ¬(Fx v Gx) is satisfied on a tsr understanding. On the further supposition, true with G understood as ¬F, that everything is either F or G, the corresponding sentence of counterpart theory is satisfied.

29. A∃x(Fx & A¬Fx)
Likewise, the problem here is that 29 is inconsistent assuming A/¬ commutation, but translates via tsr to a satisfiable sentence of counterpart theory, namely
30. ∃x(Ixα & Fx & ¬∃y(Iyα & Cyx & ∀y((Iyα & Cyx) → Fx)))
According to MFW, 30 is true when an actual object is F, and has an actual counterpart which is not F. Actually, this is not quite right. In fact, 30 is true when an actual object is F, but does not have any actual counterparts which are F. As MFW accept, it is open to Ramachandran to follow Lewis and Forbes in stipulating ∀w∀x (at w, Cxx), and doing this would enable tsr to avoid the bad mapping just described. However, as they also point out, Ramachandran nowhere indicates a wish to make such a stipulation.

31. ◊∃x∃y(A(Ex & Ey) & ¬Rxy & ¬Ryx))
This sentence relates to a complaint made by MFW, that tsr makes intuitively connected binary relations unconnected. To explain: a binary relation R is said to be connected over a domain d iff all pairs <x, y> of distinct individuals in d are such that either Rxy or Ryx. To illustrate, let R be the relation of being-at-least-as-tall-as. Under this assignment, R is connected over the domain of people. The problem with this is that tsr only allows a binary relation Rxy to be satisfied if each of x’s actual counterparts bears R to each of y’s actual counterparts. However, it is perfectly possible for there to be some pair of possible people x and y such that an actual counterpart of x is taller than one but shorter than another of y’s actual counterparts. In this case, according to tsr, neither Rxy nor Ryx holds. So tsr has the consequence that the relation of being-as-tall-as is not connected. Since being-as-tall-as most certainly is a connected relation, we are driven to reject tsr. Finally, I reconstrue MFW’s argument about this case [ibid. pp21-22] in the terminology of bad mapping, with 31 above as the inconsistent QMLA sentence mapping onto a satisfiable sentence of counterpart theory, viz. a sentence about x having an actual counterpart taller than some and shorter than other of y’s counterparts.
This concludes the investigations into tsf and tsr individually. Taking stock of things, tsf and tsr both avoid the bad mappings which form the basis of the Actuality Objection against tsl. They achieve this by means of quantifying over actual counterparts at the stage of defining the translation of a de re atomic sentence Faw, stipulating that AFa≡Fa, and thereby enabling A to be characterized in such a way as to respect a standard logic of actuality. However, tsf proves vulnerable to the Relational and Monadic Predicate Objections, because of its existential characterization of actuality. In contrast, tsr avoids these two Objections, because of its existential-plus-universal characterization of actuality. But this success for tsr is bought at the cost of tsr generating further actuality-related problems of its own, the cases of 28-31 above. Moreover, this is to say nothing of the Symmetry Objection, the subject of the section immediately following, and which MFW regard as
“a quite general reason for thinking that there cannot be a plausible translation scheme for any version of counterpart theory that allows for possible objects with multiple actual counterparts.” [p22]


(6.6) The Symmetry Argument

The Symmetry Argument [p23] asks us to consider a special kind of MC scenario, in which a possible individual has two actual counterparts, neither of which is more of a counterpart to it than the other is. To illustrate this, consider anew Gibbard’s statue Goliath, which we now consider as symmetrically related to its actual counterparts Lumpl (l) and Statue (s), in the sense that neither of these latter is more of a counterpart to Goliath than the other is. Given the symmetry, argue MFW, Goliath should satisfy
32. A(l=x) iff A(s=x)
Since ((l=x) v (s=x)) is satisfied for either value of x, then no matter which of tsf and tsr is adopted, Goliath should satisfy
33. A((l=x) v (s=x))
Furthermore, since Lumpl and Statue are actually distinct, neither satisfies ((l=x) & (s=x)) as a value of x. So Goliath should satsify
34. ¬A((l=x) & (s=x)).
Next, if Goliath satisfies each of 32-34, then it should satisfy their conjunction. That is to say, the inconsistent QMLA sentence
35. ◊∃x((A(l=x) iff A(s=x)) & A((l=x) v (s=x)) & ¬A((l=x) & (s=x)))
maps onto a satisfiable sentence of counterpart theory.

That 35 is inconsistent follows, MFW allege, from the facts that, firstly A commutes with both & and v even within the scope of ◊, and secondly (p≡q) & (pvq) & ¬(p&q) is truth-functionally inconsistent. That is to say, from 35 and the commutation of A with the two connectives, we obtain
35a. ◊∃x((A(l=x) ≡ A(g=x)) & (A(l=x) v A(s=x)) & ¬(A(l=x) & A(s=x)))
Now let A(l=x) be p, and let A(g=x) be q. From 35a and the renamings, we obtain
35b. ◊∃x((p≡q) & (pvq) & ¬(p&q))
35b says that there might exist an individual such that an inconsistency is satisfied, which itself is surely inconsistent.

This concludes the exposition of MFW’s elaboration of the Actuality Problem. To take stock of matters, if MFW are right, then no translation scheme entirely avoids mapping inconsistent QMLA sentences onto satisfiable sentences of counterpart theory. tsl’s extensions generate bad mappings in the cases of 1-22, tsf generates bad mappings in the cases of 23-26, and tsr generates bad mappings in the case of 27-35. The differences between ts, concerning which ts generates which bad mappings, are to be explained in terms, firstly of whether the ts respects the logic of actuality; secondly, whether A is characterized in terms of ∃, ∀, or Q, or indeed in terms of both ∃ and ∀ as in the case of tsr. MFW also contend that, as well as facing these objections individually, ts quite generally are under severe pressure from the Symmetry Argument which, to repeat, is a special case of the MC scenario.


(6.7) DGF’s Eternal Recurrence Objection

DGF’s Eternal Recurrence Objection takes its source from Lewis’s attempt to devise a “cheap substitute for haecceitism”, i.e. an analysis of modality which allows for haecceitistic-looking possibilities whilst preserving the purely qualitative nature of the counterpart relation. DGF argues that Lewis is motivated to do this by the inability of cct  to accommodate haecceitistic possibilities without compromising the purely qualitative character of the counterpart relation as conventionally conceived. The haecceitistic-style effect is achieved by means of rescinding Lewis’s P5, i.e. admitting distinct same-world counterparts, and in particular the actual counterparts of actual individuals. (q.v. §6.7.1)

DGF’s main purpose is to argue that the qualitative-counterpart-theorist who admits haecceitistic possibilities by this means
“precludes himself from accepting very plausible and traditional, if not incontrovertible, claims about actuality. In particular, [he] cannot accept both (i) that if something’s the case in the actual world, then it’s actually the case; and (ii) that [α] is a possibility. If he does, then for him being the case in [α] does not always suffice for possibly being the case.” [DGF 2009 p286 my itals.]

DGF’s wider argument for this conclusion need not detain us. What is of present interest is that a part of her argument can be recast and elaborated as a special case of the MC version of the Actuality Objection. The special case of MC is the case whereby an individual with multiple actual counterparts is itself actual – is one in a plurality of its own actual counterparts, in fact. Predictably, I have called this case AC. Depending on choice of translation scheme, AC gives rise in the usual way to further A-involving bad mappings.

(6.7.1) The Qualitative Problem and its Cheap Haecceitist Solution
Consider DGF’s notion of a “qualitative description, along with related notions suitably defined mutatis mutandis” [ibid.]. A qualitative description is one involving no reference to individuals. Thus, for a pair of worlds w1 and w2, let =Q denote the relation of qualitative identity, whereby “w1 =Q w2” holds iff w1 and w2 share the same qualitative description.

Given the terminology of qualitative description along with related notions etc etc, haecceitism can be defined as the claim that there are distinct possible worlds that are qualitatively identical, but that which nevertheless differ as to how they represent some individual as being. That is, for some property F, individual a, and distinct worlds w and v:-
Haec. (w =Q v) & (Faw & ¬Fav)
Many haecceitistic claims look frankly preposterous, but some look “hard to reject” [ibid. p290]. For example, we may find hard to accept the idea that Lewis could have swapped his place in the grand scheme of things with a poached egg; whereas perhaps we are slightly less troubled by the idea of the Milliband twins swapping places. At any rate, we may follow DGF in assuming that at least some haecceitistic claims are to be accepted.

In which context, consider as factually true the supposition that α is a world of eternal recurrence, either an infinite chain or a cycle of qualitatively identical epoques whereof you and I and DGF inhabit the 17th, and have twins – our qualitative duplicates – inhabiting each of the other epoques and living lives qualitatively identical to our own. Now consider DGF’s 18th epoque twin DGF*. Surely, given their qualitative identity, DGF could have taken DGF*’s place in the 18th epoque. In short, we have a haecceitistic possibility; individuals change places, but nothing qualitative changes.

On the conventional cct picture, a de re possibility ◊Fa is analysed in terms of there being some world w featuring a counterpart of a which is F. However, consider a as Lewis and b as a particular poached egg, both being inhabitants of α. Moreover, consider F as the property of being a poached egg. Thus ◊Fa states a haecceitistic possibility, the possibility of Lewis being a poached egg. De re haecceitistic possibility is obtained by a’s counterpart at w occupying the role – of being F – which corresponds to the role actually occupied by b.

The problem with the conventional analysis of de re haecceitistic possibility is that it militates against the qualitative conception of the counterpart relation, which is comparably conventional. If a’s w-counterpart Schmay occupies the w-role corresponding to the α-role occupied by b, it is reasonable to expect Schmay to be more similar to b than to a. But the qualitative conception of the counterpart relation is one founded on similarity. Your counterpart at w is the individual at w most like you, surely! On a purely qualitative conception of the counterpart relation, Schmay would have to be b’s counterpart at w. a’s counterpart at w should be whichever individual filled the a-ish role at w.

In the present case, the problem is more subtle. The conventional way of analysing haecceitistic possibility invites us to posit a world w, qualitatively identical to α, but in which DGF’s counterpart at w DGF’ occupies the DGF*-ish role of inhabiting that world’s 18th epoque. The subtlety resides in the fact that ex hypothesi DGF and DGF’ are as much duplicates of one another as are DGF’ and DGF*. That there still is a problem resides in the fact that DGF’ is qualitatively identical to an individual sc. DGF* who is distinct from DGF.

The cheap-haecceitism solution involves rescinding Lewis’s P5 which, as will be familiar from earlier chapters, prohibits an individual from having distinct same-world counterparts. In the particular case of interest to us, P5 prohibits actual objects from having distinct actual counterparts. The rescinding of P5 is clearly critical to the act project, and has already been argued for extensively, particularly in ch3, and I do not reproduce those proceedings here. The relevant point for present purposes is that, with P5 rescinded, it becomes permissible to treat the actual individual DGF* as DGF’s 18th epoque actual counterpart. In this way, we obtain a haecceitistic-looking possibility – the possibility of DGF having DGF*’s 18th epoque life – without having to compromise on the qualitative nature of the counterpart relation – for DGF* really is the individual most suited for playing the role of DGF’s 18th epoque counterpart.

 (6.7.2) Recasting DGF’s Complaint Against Cheap Haecceitism as an AC Actuality Objection
In short, cheap haecceitism is a way of obtaining haecceitistic-looking possibilities, without compromising on the purely qualitative character of the counterpart relation. For the most part, the specifics of DGF’s subsequent complaint against cheap haecceitism are tangential to present concerns. However, in the course of stating her complaint, DGF highlights two sentences, 37 and 38 below, to which cheap haecceitism is committed, and which form the basis for my recasting her complaint as an AC case of the Actuality Objection.

The cheap haecceitist treats DGF’s 18th epoque twin DGF* as DGF’s actual counterpart. Since DGF* actually lives in the 18th epoque, actually living in the 18th epoque is a possibility for DGF. Moreover, we have it ex hypothesi that DGF lives in the 17th epoque. Hence the truth of
36. ADGF lives in the 17th epoque & ◊ADGF lives in the 18th epoque.
From 36 and the assumption that the possibly actual is actual, i.e. ◊Aφ→Aφ, we derive
37. ADGF lives in the 17th epoque & ADGF lives in the 18th epoque.

We now have the material for an AC version of the Actuality Objection. To echo the signature refrain of the present chapter, 37 is inconsistent on the assumption – this time argued for by DGF – that A commutes with &. And yet 37 translates to a satisfiable sentence of counterpart theory, being satisfied by the supposition that DGF lives in a universe of eternal recurrence and has twins living in other epoques who, with Lewis’s P5 rescinded in accordance with the promptings of cheap haecceitism, function as her actual counterparts.

DGF’s argument is susceptible of the following elaboration. Implicitly, her characterization of A is in terms of ∃, and this is what gives 37 the flavour it has, of failure of the Law of Contradiction. However, a version of AC can also be devised, in which A is characterized in terms of ∀, and doing things this way delivers a kind of – recalling Hazen – failure of Excluded Middle. In what follows, let F be the property of living in the 18th epoque:-
* On DGF’s implicitly existential characterization of A, DGF is A-ly F if at least one of her actual counterparts is F. Thus, with A characterized existentially, DGF does not herself live in the 18th epoque, and yet does A-ly live in the 18th epoque. Hence, A’s existential characterization leads to a sort of failure of the Law of Contradiction. This is how we obtained 37.
* With A characterized universally, DGF is A-ly F only if all of her actual counterparts are F. But being F means living in the 18th epoque, whereas almost all of DGF’s actual counterparts live in other epoques. For instance, DGF is one of her own actual counterparts, and she lives in the 17th epoque. Furthermore, DGF A¬-ly lives in the 18th epoque iff all her actual counterparts live in other epoques. But the hypothesis tells us that one of her actual inhabitants does live in the 18th epoque. On this basis neither AFa nor A¬Fa. In short order, we obtain yet another bad mapping in that
38. ¬Fa & ¬(AFa v A¬Fa)
which is inconsistent assuming that A commutes with ¬, but is satisfied by DGF’s eternal recurrence scenario – which amounts to a “sort of failure of excluded middle” similar to what we have already seen being denounced by Hazen.

This completes the exposition of the Actuality Objection in its various permutations. The fundamental contention, to reiterate, is that no QMLA-counterpart theory translation scheme avoids translating inconsistent sentences of QMLA into satisfiable sentences of counterpart theory – although precisely which QMLA sentences map onto which sentences of counterpart theory depends on the choice of ts. Generally speaking, ts which do not respect the propositional logic of actuality, i.e. the three tsl extensions, generate narrowly logical inconsistencies; whereas ts which do respect the propositional logic of actuality, e.g. tsf and tsr, tend to generate what I loosely describe as metaphysical inconsistencies, based on problems accommodating first-order logic with identity – intuitively transitive or connected relations come out as intransitive or unconnected etc. Moreover, the Symmetry Argument version of MC is fatal to all ts.

In what remains of the present chapter, I canvass the possible responses to the Actuality Objection, address the Actuality Objection in the matter of its applicability to act, and show how what I regard as the most promising response to the Objection – Ulrich Meyer’s SQML-based proposal – can be applied to act,  as well as disposing of certain problems associated with SQML, concerning its validations of the Barcan Formula and the Thesis of Necessary Existence.

MFW suggest two responses motivated by their view of counterpart theory as a failed theory of modality which needs to be replaced with an alternative. MFW’s solutions:-
* Modalism
* Kripkean PW Theory
MFW also suggest a third response, motivated by the recognition that the Objection only arises in what were referred to earlier as non-standard cases, i.e. when an individual has something other than a unique counterpart at every world:-
* Stipulate just exactly that an individual has a unique counterpart at every world. Call this view Standard Counterpart Theory – or sct.

The problems of the first two views have been well exposed in the early chapters of the present work, and so both are discounted for present purposes. sct seems inescapably ad hoc, and in addition MFW tax it with a further problem, relating to the following two sentences [MFW ibid. p24]
41. ◊∃x∃y(x=y & A¬x=y)
42. ∃x∃y(x=y & ◊¬x=y)
As MFW point out, we would expect 41 and 42 either both to translate to satisfiable sentences of sct, or both to translate to unsatisfiable sentences of sct. But 41 translates to an unsatisfiable sentence, and 42 to a satisfiable sentence, of sct. Whereas
“Lumpl and Goliath are identical and forever spatially coincident, but could have been different; but Lumpl and Goliath could not have been identical and forever spatially coincident while being actually different. This seems an odd combination of views.” [ibid. MFW’s itals.]

But there are also responses – or better, perhaps: response strategies – which defend counterpart theory and attack the Objection. Three of these are as follows:-
i. Produce a new ts which avoids bad mappings.
ii. Note that the Objection depends on the inconsistency of the QMLA sentences involved in the bad mappings described in §§6.2-7. Deny that the QMLA sentences in question really are inconsistent.
iii. Note that the Objection depends on A being ineliminable from the formalized modal language. Discover a means of eliminating A from the formalized modal language.

I am somewhat pessimistic about the prospects for responses of type i, as I think that the Hazen/Fara/Williamson camp has come very close to establishing that, modulo ii, there really is no systematic translation from QMLA into counterpart theory. Their success may induce some to flirt with a fourth response, viz:-
iv. Refrain from formalizing modal language by means of QMLA or any other system. Instead, translate modal sentences directly into counterpart theory on a case-by-case basis.
However, as MFW rightly remark, this response amounts to accepting that the semantics of natural language is “radically non-compositional”, which is
“an extremely strong commitment, one which seems insufficiently motivated by present considerations.” [2005 p27].

Setting aside i and iv leaves us with ii and iii. The ii-strategy has some pedigree. As will be explained in §6.8 presently following, the standard implementation of ii has been by means of an appeal to multiple counterpart relations, the early proponent of this approach being Lewis [1979, 1986 p230]. Very recently, Ulrich Meyer [2012] has proposed a way of implementing iii. Meyer’s proposal, which involves the antecedent elimination of A, an SQML logic [Linsky & Zalta 1994], and some set-theoretical apparatus, is discussed in §6.9.


(6.8) First Response:- Multiple Counterpart Relations

Table 3 on the page following is a schedule of all the QMLA sentences which are the QMLA elements of all the bad mappings diagnosed by Hazen, MFW and DGF. A striking feature of these sentences is that, although the Hazen/Fara/Williamson argument proceeds on the basis that they are inconsistent, in the preponderance of cases it is not obviously the case that they really are inconsistent, or so it may be argued. For instance, consider
5. ◊∃x(AFx ≡ A¬Fx).
In the case of 5, the alleged inconsistency consists in AFx being inconsistent with A¬Fx. But is this right? For there undoubtedly are operators ● such that (●φ & ●¬φ) is consistent. For example, ◊ is one of these operators, since (◊φ & ◊¬φ) is consistent; it holds when φ is contingent, because ◊ does not commute with ¬; although ¬◊φ implies ◊¬φ, the converse does not hold. On such a basis, compare the allegedly inconsistent
37. (ADGF lives in the 17th epoque & ADGF lives in the 18th epoque)
with the consistent
39. (◊DGF lives in the 17th epoque & ◊DGF lives in the 18th epoque)
39 is consistent, because ◊ does not distribute across &, i.e. 39 does not have as a derivation
40 ◊(DGF lives in the 17th epoque & DGF lives in the 18th epoque)
which is certainly inconsistent.If some justification can be found for arguing that A behaves in the same way as ◊, in not commuting with ¬ and not distributing across &, then it may be argued that 5 and 37 and the other QMLA sentences on Table 2 below are consistent. In which case, mappings from consistent QMLA sentences to satisfiable sentences of counterpart theory are not bad mappings. Such a result would show that MFW had failed to establish their claim that there is no systematic translation from QMLA into counterpart theory. 

Prima facie, this is a hopeless undertaking. Of course A commutes with ¬ and distributes across &. Surely, if the dog actually did not eat my PhD thesis, then the dog did not actually eat my PhD thesis. Surely, if my PhD thesis is actually anathematized and is also actually excoriated, then it is actually both anathematized and excoriated.



Tab. 2. Disputed sentences of QMLA
Hazen:- tslA 1. ◊∃x(A∃y x=y & ¬(AFx v A¬Fx))
tslA 2. ◊∃x(A∃y x=y & AFx & A¬Fx)
MFW NC:- tslA names 3. Fa & ¬AFa
variables 5. ◊∃x(AFx ≡ A¬Fx)
tslA names 7. Fa & A¬Fa
variables 5. ◊∃x(AFx ≡ A¬Fx)
tslAQ names 10. Fa & (A¬Fa v ¬AFa)
variables 5. ◊∃x(AFx ≡ A¬Fx)
MFW MC:- tslA names 13. a=b & A¬a=b
variables 15. ◊∃x(AFx & A¬Fx)
tslA names 17. a=b & ¬Aa=b
variables 19. ◊∃x(¬AFx & ¬A¬Fx)
tslAQ variables 21. ◊∃x(ALx ≡ A¬Lx)
MFW tsf relat. preds. 24. ∃x∃z◊∃yA(Rxy & Ryz & ¬Rxz)
monad. preds. 26. ◊∃xA(∃y x=y & Fx & Gx & ∀y¬(Fy & Gy))
tsr 28. ◊∃xA(Ex & ¬(Fx v Gx) & ∀y(Fy v Gy))
29. A∃x(Fx & A¬Fx)
31. ◊∃x∃y(A(Ex & Ey) & ¬Rxy & ¬Ryx))
symmetry 35. ◊∃x((A(l = x) ≡ A(g = x)) & A((l = x) v (g = x)) & 
¬A((l = x) & (g = x)))
DGF:- AC:- 37. (ADGF lives in the 17th epoque & ADGF lives in the 
18th epoque).
Craven:- AC:- 38. ¬Fa & ¬(AFa v A¬Fa)
(6.8.1) Lewis’s  Current Problem with His Own Body
Lewis [1979] addresses what is essentially an Actuality Objection in slightly different terminology. His proposed solution, which involves appealing to multiple counterpart relations, is adaptable to the Table 2 cases. The problem which he identifies, again a problem for counterpart theory, arises when the modal language in which he conducts his business is enriched with the term ‘today’. ‘Today’ serves a function in the Lewis [1979] language analogous to that of A in QMLA: a function from worlds back to the locus of evaluation, this latter being the actual date of the utterance, rather than the world/region of the utterance as in the case of cct/act.

Lewis invites us to regard persons and bodies as composed of stages. As a materialist, Lewis believes that persons are identical with their bodies. However, personal identity is derivative of stage identity at a time t. That is to say the person Lewis (l) is identical with his body (b) at t iff the l-stage at t and the b-stage at t are identical simpliciter. Thus l is identical with b simpliciter only if every l-stage is identical with some b-stage and vice-versa. Where it appears, the term ‘today’ occupies the extra argument place in the three place derivative notion of identity – as it might be, some x and y are identical today.

Against this background, Lewis wants to defend the materialist thesis
(T). Necessarily ((l occupies b at t) ≡ (l is identical to b at t))
Against (T), Lewis presents a ‘body-switching’ argument [1979 pp204 et passim]. Body-switching is logically possible, which is to say that l might yesterday have quit occupying b – although actually he didn’t. Since persons are never identical with bodies unless they occupy them, we obtain:-
bs1. l and b are such that they might not have been identical today.
Now suppose as is no doubt true that l is identical with b simpliciter, i.e. no l-stage exists distinct from some b-stage and vice versa. From bs1, l=b, and the indiscernibility of identicals, we obtain
bs2. b and b are such that they might not have been identical today,
which is inconsistent, inviting the rejection of (T).

Of present interest is that, despite its inconsistency, bs2 translates to a satisfiable sentence of counterpart theory, which Lewis gives:-
bs3. There is a world w, a counterpart x in w of b, and a counterpart y in w of b, such that x and y are not identical today. [p206]
bs3 is satisfied – the difficulty arises – because, unlike the identity relation which it replaces, the counterpart relation is not transitive, in virtue of the possibility of an individual having twin counterparts at a world.

Yet another bad mapping! It is now quite plain that Lewis has identified an instance of the Actuality Objection. Indeed, it looks very much like an instance of the Symmetry Problem. For if an individual a has counterparts b and c which are twins of one another, is not b as much and as little of a counterpart to a as c is?

 (6.8.2) Lewis’s Solution
To approach Lewis’s proposed solution, he asks us to exchange bs3 for
bs4. There is a world w, a unique counterpart x in w of l, and a unique counterpart y of b, such that x and y are not identical today.
If bs4 is true, then an instance of the Actuality Objection still arises, because we still have an individual i.e. l/b being possibly identical with two mutually distinct individuals i.e. x and y. However, Lewis now argues [p207] that bs4 is false,
“despite the fact that I might have switched bodies yesterday. What is true because I might have switched bodies is not [bs4] but ... ”
bs5. There is a world w, a unique personal counterpart x in w of l, and a unique bodily counterpart y of b, such that x and y are not identical today.

The main advantage of this approach is that things can be arranged so that just one individual serves as l/b’s counterpart in any given context. Thus l/b has a unique personal actual counterpart, the actual counterpart which is most like l, and a unique bodily actual counterpart, the actual counterpart which is most like b. But there is no longer a counterpart relation Cκ such that some distinct x and y are both Cκ-ly related to l, and so the Symmetry problem is avoided.

(6.8.3) Two Methods of Implementing Lewis’s Solution
There are two ways of realizing this approach:-
The first way involves appealing to a set of indexicalized actuality operators Aa, Ab  etc. Thus, where formerly we might have described a as A-ly F and A-ly G, and would have encountered a problem in cases where F and G denoted incompatible properties, the A-indexicalization method envisages us describing a as Af-ly F and Ag-ly G. The point is that there is no operator Ak such that a is Ak-ly related to two actual counterparts individually holding properties which are incompatible in a single individual. The nice result of this is that we get both of the following:-
*a is A-ly F and A-ly G without a being A-ly both F and G; for the reasons just explained.
*The logic of actuality is respected because it remains the case for any Ak that ((Akφ & Akψ)≡Ak(φ & ψ)).
However, the obvious disadvantage of A-indexicalization is that it really does not answer to the phenomenology of actuality discourse. We would be taken aback to learn that, very often when we say “... actually ...”, we are using either a homonym with a far, far wider range of application than e.g. “bank”, or a very large range of distinct words with a collectively homonymous appearance deriving from users’ unconscious suppression of the indices which are all that distinguish these very many words from one another. 

The second way involves locating indexicalization at the level of the multiple counterpart relations, and then arguing that A does not e.g. distribute over & when differently indexed counterpart relations are involved with the conjuncts. For example, by this C-indexicalization method, DGF might be envisaged as C18-ly related to her 18th epoque counterpart, and C17-ly related to her 17th epoque counterpart, i.e. herself. The thought is then that, in these special circumstances – the C-relations in the conjuncts being differently indexed – A does not commute; although it remains the case that in normal circumstances, when the C-relations in the conjuncts have the same indices, A does still commute.

 (6.8.4) DGF’s Criticisms
DGF criticizes the C-indexicalization approach. She begins by conceding that certain
 operators, which normally commute with other operators and distribute over binary connectives, sometimes do not do so in particular circumstances. She illustrates this claim with the case of ⎕, which as she describes fails to distribute over & when multiple counterpart relations are involved. She then claims to have two reasons for denying that the involvement of multiple counterpart relations will prevent A from distributing over &:-

DGF’s first reason for denying that multiple counterpart relations prevent A distributing over & is that
 “only very few contexts allow for the needed sort of counterpart-relation ... [in particular,] ... the two occurrences of [‘DGF’] in
[37. (ADGF lives in the 17th epoque & ADGF lives in the 18th epoque)]
are not in the argument positions of different sorts of things.” [DGF ibid. p295]
Here, she seems to be saying (a) that C-indexicalization requires the different conjuncts to relate to different sortals, which seems reasonable; and (b) that sortal difference supervenes on qualitative difference; qualitatively identical sortals are identical simpliciter. In particular, living in the 17th epoque and living in the 18th epoque, are things of the same sort. In contrast to (a), (b) is a relatively bold claim. In comparison, Lewis is noticeably reticent about how finely or coarsely sortals are to be graded [q.v. Lewis 1979 pp210-11]; probably rightly, given the likelihood that our intuitions will fail us when the subject matter becomes this obscure. Moreover, even if DGF is correct in denying the distinctness of the sortals in 37, and is therefore correct in denying that the two occurrences of A in 37 are susceptible of indexicalisation and multiple counterpart-ization, she is surely just wrong  in claiming that only very few contexts are susceptible of this treatment. Surely very many are! One would suppose the conjoining of different sortal predicates to be commonplace.

DGF’s second reason for denying that multiple counterpart relations prevent A distributing over & is that
“whereas [] quantifies over a multitude of possibilities, [A] quantifies over just one. The range of predicates that an object could satisfy is something that varies with different ways of conceiving it (on the view in question). Qua statue, a possibly satisfies ‘is made entirely of bronze’; qua lump of clay, he does not. But the range of predicates that the object does satisfy does not so vary. The object actually satisfies the predicates it satisfies in [α] – no matter how we conceive of it.” [DGF ibid. p295]
Again, this does not feel exactly right. Granted, Lewis is famous for defining possibilities as sets of possible worlds. However, in [1986 p230]  - the ‘poor Fred’ case, again – he concedes that possibilities can be more finely-grained. Modal operators can pick out intra-world possibilities: they can pick out me not being poor Fred, because I actually am not poor Fred. And they can pick out me being poor Fred, because he is my actual counterpart.

However, whether A is one of the modal operators which can pick out intra-world possibilities is another matter. What is inescapable, again, is that the notion of e.g. A not distributing over & really does not answer to the phenomenology; it really seems to us as though A just does distribute over &, even when the conjuncts relate to different sortal predicates. And when a proposed solution to an objection relies on assuming something which runs counter to the phenomenology, the proposal is likely to be objected to as ad hoc. The prognosis:- the possibility of appealing to multiple counterpart relations shows that, strictly speaking, MFW are wrong. Strictly speaking, there is a way of translating systematically between QMLA and counterpart theory. But it relies on assuming that A does not distribute over & when the conjuncts relate to different sortal predicates. And every indication is that A always distributes over &.




(6.9) Can A be Eliminated?

We return now to an issue raised quite early in ch1 of the present work. This is the claim that modalism cannot rely on the resources of QML alone if it is to entertain a realistic aspiration to expressive completeness. In particular, following Hazen, it has been widely accepted that modalism needs to add A to its armoury, specifically in order to enable the expression of claims about modal actuality, the ch1 illustration of this being
41. There could have been things which do not actually exist.
It will be recalled that 41, taken as one of our paradigm actuality claims, appears to escape capture by means of A-less QML. In brief, the widespread prognosis – echoed by MFW [ibid. p5] is that 41 is only captured by means of
42. ◊∃xA¬Ex
with E understood as in ch1 as a predicate abbreviating ‘∃y y=x’. Similar considerations apply to another paradigm actuality claim, namely
43. It might have been that everyone who is actually rich was poor.
which is said only to be captured by
44. ◊∀x(ARx → Px).

The crucial point now is that the entire Actuality problematic of the present chapter rests upon this very assumption, sc. that modal actuality claims are indeed only captured by recourse to QMLA. If there turned to be a way for QML to formalize modal actuality claims without recourse to A, then it would not matter that QMLA-counterpart theory translations generate bad mappings. What would matter would be whether the new method of formalizing modal actuality generated bad mappings.

Very recently, Ulrich Meyer has proposed an antecedent elimination strategy. He argues that modal actuality claims can be formulated in a QML in which A is redundant – specifically, SQML [Linsky & Zalta 1994] – without generating bad mappings. In this way, the Actuality problematic is cut off at its root.

(6.9.1) Meyer’s Antecedent Elimination Strategy – the Generalities
Meyer notes that A
“allows us to make claims about [α] inside the scope of other modal operators ... But describing [α] is something we could do already, by using unmodalized sentences outside the scope of other modal operators.” [p4]
Thus, “offhand”, we should expect any claim expressed by means of A within the scope of a modal operator, to be expressible outside any modal operator by means of a locution in which A is redundant. Meyer then proposes his antecedent elimination strategy, whereby
i. Occurrences of A are eliminated by translating the modal claims in which they occur into a QML in which A is redundant; this turns out to be SQML.
ii.A modification of tsl is required in order to accommodate the A-free sentences resulting from i.
iii.Systematic translation into counterpart theory ensues – Meyer produces SQML regimentations of some of the Table 2 cases, which are shown not to generate bad mappings.

 (6.9.2) The Eliminability of A in Propositional Modal Logic
The hunt is on for a stronger QML in which A can be eliminated. However, Meyer first sets about proving the eliminability of A in propositional modal logic (PML).

Firstly, Meyer notes that the occurrence of A in a PML subformula of the form Aφ is redundant, if the subformula is not within the scope of another modal operator – this gives us Aφ≡φ. I call this the bare case. Secondly, Meyer considers a subformula of a type, which I call purely modal, consisting of an uninterrupted string of ⎕’s and/or ◊’s terminating in a subformula of the form Aφ, e.g. ⎕◊◊⎕◊⎕⎕⎕◊⎕◊Aφ. Meyer notes that in such special cases the entire string preceding φ is redundant, again so long as the subformula is not within the scope of another operator.

This suggests a general strategy of either eliminating A immediately, thanks to its position at the ‘front of the queue’ of operators, as in the bare case; or trying to get it into a position directly behind an uninterrupted string of boxes and diamonds, thereby generating a special case.

A potential difficulty arises for this general strategy, when A occurs within the scope of ◊/⎕, but is separated from ◊/⎕ by another logical constant, as in e.g.
45. ◊(φ v Aψ)
However, as Meyer argues, various relations between modal operators and truth functions always allow A to escape past the obstacle, attach itself to the string of boxes and diamonds, and so generate a special case. To continue the illustration by means of 45, one of the relations between modal operators and truth functions is that ◊ distributes across v. This relation enables the derivation from 45 of
46. ◊φ v ◊Aψ
the right hand disjunct of which forms a special case, allowing A to be eliminated, QED. This and like cases suffice to prove that A is redundant in PML.

 (6.9.3) The Eliminability of A in Quantified Modal Logic
However, when we consider the elimination of A in QML, a second potential difficulty with generating special cases reveals itself: A might get stuck behind a quantifier in its journey to the back of the uninterrupted string of boxes and diamonds. There are ∃ and ∀ cases of this problem:-
47. ◊∃xAFx
48. ◊∀xAFx

It is in the resolution of this issue that SQML becomes relevant. The salient difference between SQML, and QML’s in which A proves not to be eliminable, consists in the ranges of the quantifiers in the respective logics:-
* In the vulnerable QML’s, ∃ and ∀ are world-relative. Their range at a world w is restricted to the objects existing in w.
* In SQML, ∃ and ∀ range over all possible objects.

To see the relevance of this, consider again
41. There could have been things which do not actually exist.
A QML with world-relative quantifiers will presumably try to formalize 41 by means of
49. ◊∃x¬∃yy=x
This is satisfied by a world in which there are things which don’t exist there, which if it makes any sense at all does so only on a Meinongian distinction between ‘is’ and ‘exists’ – and this is usually taken to be a commitment worth avoiding. Now, as throughout the present work, it is customary to abbreviate ∃yy=x by means of an existence predicate E. This fact in itself shows that Ex is redundant in QML. In contrast, E is not redundant in SQML, but is needed so as to allow existence to be predicated of x at the particular world under consideration. When Ex does not lie within the scope of a modal operator, it is evaluated at α. Thus 41 receives the following analysis in SQML
50. ∃x(◊Ex & ¬Ex)
Here, the ∃ ranges over all possible objects, the ◊ makes the first E hold for the objects at a world w, but the ¬E is not thus restricted, and therefore holds for the objects in α. This seems, as Meyer contends, to capture the 41-ish idea of the possible existence of things which don’t actually exist.

In any case, as Meyer points out, there really is no quantifer problem for SQML, simply because in that logic the Barcan Formula holds:-
BF. ∃x◊φ→◊∃xφ
That is to say, ∃ commutes with ◊. And this, finally, allows ◊ to be swapped into the purely modal string at the end of which A awaits.

Putting his finishing touch to the ∃ case, Meyer lists four of the problematic QMLA sentences – i.e. Table 2 sentences above, with the corresponding SQML renditions [p8]. Here are the QMLA sentences in question, together with the corresponding SQML renditions, and brief explanations of provenance:-

3. Fa & ¬AFa
It will be recalled that 3 is inconsistent assuming AFa≡Fa, but translates to a satisfiable sentence of counterpart theory under tslA∃ and under the assumption of NC, engendering an instance of the Actuality Objection. The corresponding SQML rendition per Meyer:-
51. Fa & ¬Fa
Like 3 which it replaces, 51 is inconsistent. However, the crucial point is that 51 does not translate to a satisfiable sentence of NC, thanks to A’s elimination, and so the Actuality Objection is avoided.

5. ◊∃x(AFx ≡ A¬Fx)
Again, it will be recalled that 5 is inconsistent assuming A/¬ commutation, but translates to a satisfiable sentence of counterpart theory under tsl∃, tsl∀, and tslQ, and under the assumption of NC. As with 3, the Actuality Objection looms. The corresponding SQML rendition per Meyer:-
52. x(Fx ≡ ¬Fx)
Like 5 which it replaces, 52 is inconsistent. Again, the crucial point is that 52 does not translate to a satisfiable sentence of counterpart theory, thanks to A’s elimination, and so as with 51, the Actuality Objection looms no more.

10. Fa & (A¬Fa v ¬AFa).
Again, 10 is inconsistent assuming A/¬ commutation, but translates to a satisfiable sentence of counterpart theory under tsl∃ and under the assumption of NC. Yet again, the Actuality Objection looms. The corresponding SQML rendition per Meyer:-
53. Fa & (¬Fa v ¬Fa)
Like 10 which it replaces, 53 is inconsistent. Again, the crucial point is that 53 does not translate to a satisfiable sentence of counterpart theory, thanks to A’s elimination. The Actuality Objection looms no more.

15. ◊∃x(AFx & A¬Fx)
Again, 15 is inconsistent assuming A/¬ commutation, but translates to a satisfiable sentence of counterpart theory under tslA∃ and under the assumption of MC, engendering yet another instance of the Actuality Objection. The corresponding SQML rendition per Meyer:-
54. ∃x(Fx & ¬Fx)
Like 15 which it replaces, 54 is inconsistent. And to labour the crucial point, 54 does not translate to a satisfiable sentence of counterpart theory, thanks to A’s elimination. And so once again the Actuality Objection is avoided.

So much for the ∃ case. There remains the ∀ case:-
48. ◊∀xAFx
The ∀ case is more difficult because, as Meyer concedes, A does not commute with ◊ even in SQML. However, Meyer goes on to propose that A can be eliminated in an SQML with quantification over sets.

This latter logic comes about by adding to SQML the set membership relation ∈, and some set variables X, Y, Z ... . A heirarchy of impure sets is formed, with the objects in the regular domain as the ur-elements. Set membership is necessary, insofar as both of the following hold in all models of the expanded logic:-
55. ∀x∀X(x∈X → ⎕x∈X)
56. ∀x∀X(¬x∈X → ⎕¬x∈X)

The ensemble put together by Meyer calls for a modified ts (tsm) [ibid. pp8-9]. The basic problem is that, in dispensing with QML’s world-relative quantifiers, SQML needs some means of ensuring that unmodalized sentences get evaluated at the α of cct. This is brought about by imposing a restriction on atomic formulae, achieved by means of the existence predicate E. The essentials of tsm are reproduced here:-
Exw is Ixw
(φx1 ... xn)w is Ix1w & ... & Ixnw & φx1 ... xn if φ is an atomic formula not 
containing E.
(¬φ)w is ¬φw
(φ→ψ)w is φw → ψw
(∀xφ)w is ∀xφw
⎕(φx1 ... xn)w is ∀v∀y1 ... ∀yn((Wv & Iy1v & Cy1x1 & ... & Iynv & Cynxn) →
(φy1 ... yn)v)
In SQML, quantification over actual objects is expressed by means of ∀x(Ex →φ), which translates via tsm to the sentence ∀x(Ixα→φx) of counterpart theory. Meyer notes that this is exactly how tsl handles quantification over actual objects, and also that the Barcan Formula translates via tsm to a theorem of cct.


In its essentials, A-elimination now proceeds as follows. Consider a QML sentence of the form
57. ◊( ... ∀x ... Aφ ... )
containing no operators or quantifiers other than those depicted. The first step is to define outside the scope of the ◊ in 57 a set X, the members of which are the objects actually satisfying φ. We then substitute the occurrence of Aφ with the formula x∈X. This generates the following A-free paraphrase:-
58. ∃X(∀x((x∈X) ≡ φ) & ◊( ... ∀x ... x∈X)
Consider the paradigm ∀ case
43. It might have been that everyone who is actually rich was poor.
which QMLA translates by means of
44. ◊∀x(ARx → Px).
The A-free SQML regimentation of 43 is
59. ∃X(∀x((x∈X) ≡ Rx) & ◊∀x(x∈X → Px)).

The remainder of Meyer’s paper concerns the procedures required to effect translation in more complicated cases, e.g. when a mixture of universally and existentially bound variables occurs between the embedded A and the rest of the purely modal string; together with sundry suggestions, such as that the set-theoretical conjuncts of the counterpart-theoretical elements of mappings, as in e.g. the left-hand conjuncts of 58 and 59, should always be suppressed (Meyer: “ignored”) in the final translation. These considerations need not detain us, as the Table 2 sentences which serve as the focus of the Actuality Objection are for the most part not complicated in this way.

To all appearances, Meyer’s strategy succeeds in delivering A-free regimentations of the Table 2 QMLA sentences, which translate systematically via tsm to sentences of counterpart theory, moreover without generating bad mappings. Thus MFW’s argument, that no ts achieves systematic QMLA-counterpart theory translation without generating bad mappings, is refuted.


(6.10) Actuality, Meyer and act

Up to the present point in the current chapter, the focus has been on the impact of the Actuality Objection on counterpart theory tout court. For historical and understandable reasons, counterpart theory has tended in the literature on the subject to be conflated with cctact is the new kid on the theoretical block – and I have colluded in the conflation. My project in the section now underway is to refocus on act. Doing so paves the way for the ensuing discussion of the applicability of Meyer’s solution to the case of act. The section concludes with an appraisal of the treatment within act of the Barcan Formula and the Thesis of Necessary Existence, to both of which Meyer’s solution in view of its SQML basis commits act.

(6.10.1) Actuality and act
To initiate the project, I take it that the act-theorist shares in the common presumption that the QML – on which, after all, the provisional ch4 formalization of act is based – only captures modal actuality by the addition of A, as per 43 and 44 in the preceding section.

How then is modal actuality translated into act? That is to say, what are the translations from QMLA into act? Analogously to the pw/cct case, this is a question of how many @ actual counterparts of a being F it takes for AFa to hold, i.e. which of the following is true:-
A∃. a is Aly F ≡ at least one of a’s @ counterparts is F
A∀. a is Aly F ≡ all a’s @ counterparts are F
AQ. a is Aly F ≡ a quantity Q of a’s @ counterparts are F
The act  translation scheme tsα given in ch4 made no provision for modal actuality. For this reason, we call the relevant ts , respectively, tsα∃, tsα∀ and tsαQ.

The next step is to introduce analogues of the non-standard scenarios NC, MC and AC:-
* a has multiple mutually distinct @ counterparts – the MC analogue.
* a has no @ counterparts – the NCanalogue.
* a is actual and has an @ counterpart distinct from itself – the AC analogue.

The final step consists in applying the various combinations of tsα∃, tsα∀, tsαQ, and the NC, MC and AC analogues, in order to derive the bad mappings which constitute the instances of the Actuality Objection. For ease of exposition in the case of existentially quantified sentences, I confine myself to the sentences treated by Meyer, namely 3, 5, 10 and 15:-

3. Fa & ¬AFa
As before 3 is inconsistent assuming AFa≡Fa. However, 3 translates to a satisfiable sentence of act under tsαA∃ and under the assumption of NC, namely
60. Fa & ¬∃x(Ix@ & Cxa & Fx).
which is satisfied by a considered as non-@, as F, and as lacking actual counterparts.

5. ◊∃x(AFx ≡ A¬Fx)
As before, 5 is inconsistent assuming A/¬ commutation. However, 5 translates to satisfiable sentences of act under tsα∃, tsα∀, and tsαQ, and under the assumption of NC, respectively:-
61. ∃r∃x(Ixr & (∃y(Iy@ & Cyx & Fy) ≡ ∃y(Iy@ & Cyx & ¬Fy))).
which is satisfied as one would expect in an NC case, by a non-@ individual lacking @ counterparts, because both conjuncts of the embedded biconditional are false, so the biconditional is true;
62. ∃r∃x(Ixr & ((∀y(Iy@ & Cyx) → Fy) ≡ (∀y(Iy@ & Cyx) → ¬Fy))).
which is satisfied by a non-@ individual lacking @ counterparts, because the conditionals forming the conjuncts of the embedded biconditional are both vacuously true, so the biconditional is true;
63. ∃r∃x(Ixr & ([Qy: Iy@ & Cyx](Fy) ≡ [Qy: Iy@ & Cyx](¬Fy)))
which is satisfied by a non-@ individual a lacking @ counterparts. Firstly, a satisfies Ixr; and secondly, both sides of the biconditional forming the right-hand of the main conjunction come out false. If a lacks @ counterparts, then it is false both that Q @ counterparts of a are F, and that Q @ counterparts of a are ¬F. Since the two sides of the biconditional share the same truth value – i.e. they’re both false, the biconditional itself comes out true, which makes the right-hand conjunct formed by the biconditional true also.

10. Fa & (A¬Fa v ¬AFa).
As before, 10 is inconsistent assuming A commutation. However, 10 maps onto the following satisfiable sentence of act under tsαQ and under the assumption of NC:-
64. Fa & ([Qx: Ix@ & Cxa](¬Fx) v ¬[Qx: Ix@ & Cxa](Fx))
which is satisfied by a non-@ individual lacking @ counterparts. This is because the left-hand conjunct is true; and the right-hand disjunct of the disjunction forming the right hand conjunct is true if a lacks @ counterparts, which makes the right hand conjunct true also. 

15. ◊∃x(AFx & A¬Fx)
As before, 15 is inconsistent assuming A/¬ commutation. However, 15 translates to the following satisfiable sentence of act under tsα∃ and under the assumption of MC, namely
65. ∃r∃x(Ixr & ∃y(Iy@ & Cyx & Fy) & ∃y(Iy@ & Cyx & ¬Fy)).
I follow the cue given by MFW in illustrating 65’s satisfiability by means of the statue case, with F thought of as the property of being statue-like. 65 is satisfied by a non-@ statue having distinct @ counterparts, one of which is statue-like and one, presumably the lump-like counterpart, not statue-like.

I take it that the foregoing establishes that the Actuality Objection bears upon act much in the manner of its bearing on cct.

(6.10.2) Meyer and act
The task of this subsection is to establish that Meyer’s solution avails in the case of act. In the cases of existentially-quantified sentences like 3, 5, 10 and 15, this is easy. We just repeat the corresponding SQML renditions/results from §6.9.3 above. These are, respectively:-

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