On the Function of Advanced Modalizing

Abstract

The Lewisian account of modality based on counterpart theory suffers from the problems of advanced modalizing, where claims about spatiotemporally disunified entities are modalized. In this paper, I first discuss a strategy to bypass the problem, one which treats cases of advanced modalizing as cases of equivocation lying outside the scope of the translation. I then argue that the strategy does not satisfactorily generalize to the case of advanced modal claims involving abstract entities. This failure not only reveals the limitations of counterpart theory in handling abstracta, but also weakens the abductive argument for modal realism that posits it as a robust account of nominalism. Furthermore, advanced modalizing is an indispensable part of the Lewisian framework when doing metaphysics, since advanced modal claims made by Lewis himself strengthen his abductive argument for modal realism. Hence advanced modalizing plays a crucial role in the Lewisian project, and an account of advanced modalizing is needed for the Lewisian reduction of modality to succeed.

Graduate Paper from the 2023 Joint Session

I

Advanced Modalizing: The Current State of Play. Counterpart Theory (ct) provides a systematic translation to convert every formula ϕ of quantified modal logic, which I call the modal language L_M, into some formula ψ of the non-modal first-order language of ct, L_C (Lewis 1968). ct follows the possible-worlds interpretation of de dicto modality, and identifies ϕ’s being possibly true with ϕ’s being true at some world w. However, since the theory postulates that every object exists in at most one world, it interprets the modal profile of an object a via its ‘counterparts’ which represent a at other worlds, and identifies a’s being possibly F with a’s having some counterpart which is F.¹

The counterpart relation is formally flexible and need not be one of equivalence. This flexibility allows the counterpart-theoretic frameworks (i) to be more widely available for implementation in logic (Schwarz 2012, Varzi 2020)—especially when compared to the classical Kripkean semantics based on the strict relation of identity—and (ii) to have a puzzle-solving strength in metaphysics (Lewis 1986). Most crucially, Lewis (1986) argues that ct provides the best semantic analysis of modality, and this is the main reason why one should endorse his modal realism, which posits that there are many worlds ontologically on a par with the universe we are part of. However, all these theoretical benefits depend upon whether ct’s translation is truth-preserving (Kripke 1979),² that is, whether the translation pairs an L_M-formula ϕ with an L_C-formula ψ if and only if they have the same truth-value, that is, if and only if ⌜ϕ if and only if ψ⌝ holds.

The problems of advanced modalizing suggest that the translation fails to be truth-preserving by its own lights when the modal statement to translate concerns inter-world matters because it pairs modal formulas that one accepts to be true with false counterpart-theoretic translations. More precisely, whilst basic modalizing is modalizing about spatiotemporally unified, ordinary entities that are wholly located in some world, advanced modalizing concerns extraordinary entities that are not spatiotemporally unified and that do not wholly exist in one world (Divers 1999, p. 220), and ct’s translation results in inadequacy in the latter case.³ As an example of advanced de dicto modalizing, consider the main postulate of modal realism (Divers 1999, pp. 221–3):

• (1) There are many worlds.

By the T axiom (that is, if ϕ, then possibly ϕ), the following modalized instance of (1) should be true:

• (2) Possibly, there are many worlds.

However, the counterpart-theoretic translation of (2) is false:

• (3) There is a world such that it contains many worlds.

The translation is false because it is an axiom of ct that no object is part of more than one world. Hence we have a failure of truth-preservation where the modal sentence which we expect to be true, namely (2), is paired with a counterpart-theoretic sentence which is false.

Divers (1999, p. 230) proposes to solve this problem by introducing an extension of ct, called the redundancy analysis, which holds that for each sentence ϕ involving extraordinary entities, the modalizing of ϕ (that is, ◇ ϕ and ☐ ϕ) is semantically equivalent to ϕ. So, given the redundancy analysis, (2) is equivalent to (1), and the translation is, once again, truth-preserving. The idea here is to distinguish between an intra-world truth holding restrictedly at a world and an inter-world truth holding unrestrictedly of the logical space. Hence, whilst ct’s translation deals with intra-world issues, the scope of the redundancy analysis concerns inter-world matters.

However, even if the redundancy analysis might be helpful to deal with the cases of advanced de dicto modalizing, it does not function well in de re cases. Consider, for instance, Anna and Bill, who are not worldmates, and assume that (Jago 2016, pp. 630–1):⁴

• (4) Anna is taller than Bill.

Now, given that Anna and Bill are not worldmates, we have an inter-world content that should receive a translation by the redundancy analysis. Then it follows that from (4) one can infer:

• (5) Necessarily, Anna is taller than Bill.

But (5) is strongly counterintuitive. For, since Anna and Bill are non-worldmates but otherwise perfectly standard human beings, it conflicts with our intuitions of contingency to suggest that spatiotemporal relations between them are governed by some kind of necessity.

These de re problems of advanced modalizing are often treated by introducing further extensions to ct, and in particular, by revising the redundancy analysis so that it also covers cases of advanced de re modalizing. However, such proposals suffer from serious problems. For instance, one such account (Noonan 2014) eliminates the counterpart-theoretic translation scheme (for de dicto modality) in favour of the redundancy analysis and results in an unattractive account of modality, namely, a certain de dicto necessitism, where all and only contingencies are de re in nature. Another (Divers and Parry 2018), which uses both the counterpart-theoretic translation and the redundancy analysis, results in an ambiguous semantics for modal languages, where it is not clear which sense of possibility is to be assigned to a given modal sentence. An alternative solution to the advanced modalizing problems, which initially looks promising, bypasses all these problems, and passes through reclarifying the issue. I will now discuss this reply.

II

The Equivocation Reply. The equivocation reply provides an initially promising diagnosis of and solution to the advanced modalizing problems. This reply is based on the simple observation that the cases of advanced modalizing are (explicit or implicit) cases of mixing the two languages, that is, the target language L_M and the object language L_C (Parsons 2012, pp. 151–2; Steinberg 2018, pp. 552–5). The result is the emergence of seemingly translatable sentences of mixed vocabulary, which are in effect not within the domain of the target language. Hence advanced modalizing is not a phenomenon of the failure of truth-preservation, but rather one of equivocation, where the translation cannot be properly applied.

More precisely, Lewis (1968) clarifies the range of the translation, that is, the target language, by saying that ct’s translation is an algorithm that converts L_M into L_C. And L_M is a first-order language enriched with the modal operators ‘◇’ (‘possibly’) and ‘☐’ (‘necessarily’), whilst L_C is a first-order language which designates four non-logical predicates (Lewis, 1968, p. 113):

• Wx: x is a world

• Ax: x is actual

• Ixy: x is in y

• Cxy: x is a counterpart of y

Now, formalizing the problematic line in the first example, we obtain:

• (6) ◇

  ∃w∃v(Ww ⋀ Wv ⋀ w ≠ v)

(6) is not a sentence of the target language L_M because it contains vocabulary (namely, ‘W’), which belongs to the syntax of L_C. And it is not a sentence of the object language L_C because it contains L_M-vocabulary, ‘◇’.⁵ Hence (6) is an explicit case of mixed vocabulary, where the translation cannot be properly applied.

Moving on to the second case, given the assumption that Anna and Bill are not worldmates, (4) is stated in a language which quantifies over worlds and objects in them. So, even though there is no explicitly detectable use of L_C- or L_M-vocabulary in the sentence, (4) is a sentence of L_C (cf. Steinberg 2018, p. 558) and cannot be translated using ct’s translation. In effect, if the assumption that Anna and Bill are not worldmates (that is, that they are not in the same world, which we cannot formalize in L_M but only in L_C, as ‘Iaw ⋀ Ibv ⋀ w ≠ v’) is removed from the example, then the sentence can be unproblematically translated by ct’s original translation as:

• (7) In the actual world, Anna is taller than Bill.

Crucially, one cannot infer the necessity of Anna being taller than Bill from (7), since (7) does not entail that every counterpart of Anna is taller than every counterpart of Bill.

Hence the problems of advanced modalizing are not problems of truth-preservation, that is, they do not show that ct’s translation (and the Lewisian analysis of modality resulting therefrom) fails to preserve truth by its own lights. These cases merely mix the two languages and expect the translation to function normally, even though Lewis (1968) clarifies that the target of the translation is not such a mixed language. However, this reply suffers from crucial limitations, which I will now discuss.

III

The Question of Completeness and the Function of Advanced Modalizing. One crucial worry is that the equivocation reply, which distinguishes the two languages L_C and L_M, and underlines that the translation cannot be properly applied in cases of advanced modalizing, is incomplete: although ct’s translation is not targeted to render cases of mixed vocabulary, sentences of advanced modalizing make sense in a natural language, and are part of our ordinary usage, hence the translation should be well-equipped to deal with them (Parsons 2012, p. 152; Steinberg 2018, p. 555).

Then, as the objection goes, sentences such as ‘Possibly, there are many worlds’ or ‘Necessarily, there is a world with blue swans in it’ are part of our modalizing, and unless ct develops a way to make sense of them, it is seriously incomplete. So, though not via a failure of truth-preservation, the advanced modalizing problems still undermine the success of the counterpart-theoretic translation via an issue of incompleteness. More precisely, the objection attributes some kind of artificiality to the equivocation reply by suggesting that sentences of advanced modalizing are part of our ordinary usage.

This objection could have been renounced if the Lewisian were to convince us that those modal claims which lie outside the scope of her translation had no interesting use or indispensable function in our theorizing, and played no role in the Lewisian metaphysics.⁶ For if advanced modalizing does not play any role at all, why should we give a philosophically serious account of it?

One potential argument is that extending ct to translate a mixed language containing both L_C- and L_M-vocabularies lacks independent motivation. ct’s translation is functional because it converts a language into a stronger one, allowing us to use the inferential resources of the latter to evaluate our reasonings in the former. So this conversion is purposeful, familiarizing our modal judgements in a purely extensional language. But what is the function of re-modalizing this extensional language? Surely, it is not to make sense of metaphysical modality, which is already accounted for by the counterpart-theoretic translation. And unless some serious motivation is provided for the modalizing of L_C, the Lewisian is entitled to refrain from making pronouncements about the modal status of her own language, and treat the sentences expressing her own theoretical commitments as amodal, that is, devoid of a modal status (Cowling 2011).

This defence, however, does not get us far. First, even granting that the above defence is solid enough for advanced modalizing claims involving transworld material objects (such as Anna and Bill), there is a family of advanced modalizing claims (about abstracta) that appear to be utterly indispensable in our theorizing such that ct clearly falls short for failing to cover. Secondly, advanced modalizing is an indispensable part of the abductive argument for modal realism.⁷ So advanced modalizing plays a crucial role in the Lewisian framework.

IV

The Question of Generalisability: Abstract Entities. Abstract entities such as sets, propositions and properties are spatiotemporally disunified, and any modalizing involving them is advanced (Divers 1999). But the equivocation reply does not successfully generalize to the case of advanced modalizing concerning these entities. What’s worse, modalizing about these entities is an absolutely indispensable part of our theorizing, and should be accounted for.

More precisely, modal realism posits two fundamental kinds of entity: concrete individuals (worlds amongst them) and sets (Divers 2002, p. 45). And along nominalist lines, Lewis (1986) defines abstract entities by using these individuals and sets. For instance, numbers are identified with pure sets, propositions with sets of possible worlds, and properties with sets of possibilia, that is, possible objects (across worlds). Furthermore, Lewis (1983) claims that these abstract entities exist, not in a world, but from the standpoint of (potentially many) worlds,⁸ whereas ct’s translation only applies to ordinary individuals which are spatiotemporally unified and wholly exist in a world.

The equivocation reply generalizes to the case of modal claims about propositions and properties to a certain extent. For instance, ‘certain propositions necessarily follow from one another’ is just ‘certain sets of worlds necessarily follow from one another’, and thus implicitly involves a certain mixture of the two languages L_M and L_C (‘necessarily’ and ‘(sets of) worlds’).⁹ However, the theorizing one obtains from such a mixture is ultimately indispensable when, for instance, we are making sense of our logical inferences, and that ct is unable to translate it points to a serious limit in its scope.

Another, even more problematic, example is ‘necessarily, 2 + 2 = 4’. Au fond, this is a modal claim about pure sets, which involves neither equivocation nor language-mixing, and most crucially, it is a fundamental piece of mathematical theorizing that is usually taken as a paradigmatic example of metaphysical necessity. However, ‘necessarily, 2 + 2 = 4’ lies outside the scope of ct. But if even the paradigmatic examples of necessity cannot be accounted for via ct, then the theory is seriously limited, and the equivocation reply is not in a position to dismiss this limitation.

Furthermore, this limitation seriously undermines the abductive argument for modal realism. For the abductive argument for modal realism posits it as a robust account of nominalism, that is, the success of the nominalist account of properties and propositions is a reason why one should be a modal realist. But if modal claims about these entities lie outside the scope of ct, then the Lewisian metaphysical account (of properties and propositions) is not even analysable by using the Lewisian (counterpart-theoretic) semantics, suggesting that either (i) the metaphysical account is not as promising as it initially appears, or (ii) the semantic account is deeply limited. In either case, the argument for the systematic Lewisian philosophy is substantially weakened.

V

The Question of Indispensability: The Abductive Argument for Modal Realism. I’ve just argued in §iv that the equivocation reply does not satisfactorily generalize to the case of modal claims about abstract entities, and that ct’s limitation in translating these claims weakens Lewis’s overall abductive argument for modal realism. Before concluding, I want briefly to defend the claim that the phenomenon of advanced modalizing is an indispensable part of the Lewisian metaphysics, especially in formulating the abductive argument itself for modal realism.

More precisely, Lewis (1986) asserts that the totality of worlds has a necessary character, and the necessity of logical space is then used repeatedly to lessen certain objections against the view. Let us take the use of advanced modalizing in responding to the epistemological objection as an example.

The epistemological objection tells us that since other possible worlds are spatiotemporally isolated and not open to direct inspection, there is no causal connection between the knower (of modal realism) and the subject matter of her knowledge. Hence, knowledge of modal realism is unattainable.

However, Lewis, using a recurring analogy between modal realism and mathematics, claims that just as knowledge of mathematics does not require causal acquaintance with mathematical objects, knowledge of modal realism is attainable without acquaintance with possible worlds and their inhabitants. For, just as mathematical objects are necessary existents, the objects of modal realism—that is, the totality of possible worlds, or what possibilities there are—exist as a matter of necessity. Hence, that there exists a world containing blue swans, for instance, is necessary, making ‘Necessarily, there exists a world with blue swans in it’ not only syntactically well-formed, but also true.

Parsons (2012, p. 143) takes this definitive pronouncement about the modal status of modal realism to be ‘a moment of weakness on Lewis’s part’, which is only put forward to dismiss the epistemological objection and not repeated elsewhere. However, §§2.1 and 2.5 from the same chapter of Lewis (1986, p. 101 n.1; p. 126) also attribute a necessary character to the logical space to defend the Lewisian understanding of actuality and to lessen certain moral concerns (Roca-Reyes 2020). So, regardless of whether these are some moments of weakness, Lewis engages with advanced modalizing in the most crucial manoeuvres of his defence of modal realism. Advanced modalizing therefore plays a major role in the Lewisian metaphysics.

Can a Lewisian thus do without advanced modalizing? It seems not. Both modalizing about abstracta and the use of advanced modalizing in metaphysics suggest that advanced modalizing plays an indispensable role in the Lewisian account of modality. An adequate account of advanced modalizing is therefore needed, and the equivocation reply is incomplete: the Lewisian cannot simply reject cases of advanced modalizing as relying on equivocation or lying outside the scope of her translation.¹⁰

References

Footnotes