Abstract
I review Ninan’s Hundred Tickets case pertaining to quantification into epistemic modal contexts, and his counterpart theoretic way to address it (Ninan, Philos Rev, 2018. https://doi.org/10.1215/00318108-6973010). Ninan’s solution employs a ‘counterpart relation’ parameter intended to reflect how the domain of quantification is thought of in a context. This approach theoretically rules out the possibility of contexts where different ways of thinking about the domain can be deployed through different quantificational noun phrases. I bring out the case of the multiply de re modal statement Any ticket in photo #2 might be any ticket in photo #1 to challenge Ninan’s approach. I propose a different approach adapting a more complex ‘counterpart relation’ parameter due to Rabern (Inquiry, 2021. https://doi.org/10.1080/0020174X.2018.1470568). I attempt to flesh it out by relating it to a finer grained notion of epistemic possibility involving assignments to discourse referents. My approach can account for the aforementioned multiply de re statement, as well as address the Hundred Tickets case.
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Notes
Of these, I will address only HT, but my proposal can deal with TT as well.
For the present paper, I assume the following outlook. A statement is a locutionary act considered as a one-off event. Each statement is essentially associated with a context, a logical form (LF), a semantic value and a sentential semantic content. The LF determines the semantic value via the grammar of the language the statement belongs to. The semantic value is a function which yields a semantic content after the post-semantic feeding of the relevant contextual parameters. Semantic contents are at the ground of the truth or falsity of statements and they will be assumed to be sets of possible worlds. Kindred outlooks are variously described in Lewis (1980), Ninan (2010), Rabern (2012), Santorio (2012), Yalcin (2014). So I assume LFs, and the formal sentences are primarily meant to represent LFs. But I will not be able to give full LF descriptions, which would have required me to address the semantics of the natural language directly.
I take the notion of entailment that applies to English sentences (strictly speaking LFs, see the previous note) to be such that, \( \left\{{\phi }_{1}, \ldots ,{\phi }_{n}\right\}\models \psi \) iff there is no context \( c \) such that \( {\phi }_{1}, \ldots ,{\phi }_{n} \) are all true and \( \psi \) is false at \( c \). The notion of entailment that applies to QML formulas is the standard one: \( \left\{{\phi }_{1}, \ldots ,{\phi }_{n}\right\}\models \psi \) iff there is no model \( \mathcal{M} \) no evaluation point \( \mathfrak{I} \) such that \( {\phi }_{1}, \ldots ,{\phi }_{n} \) are all true at \( \mathcal{M},\mathfrak{I} \) and \( \psi \) is false at \( \mathcal{M},\mathfrak{I} \). I use entail or \( \models \) to express both, as the context will make clear which one is meant.
Strictly speaking, in relation with the HT puzzle Ninan focuses only on the issue of the context sensitivity of (1) (his (4)). But he also claims that “the counterpart story [his proposal regarding the TT puzzle] extends straightforwardly to the one hundred-ticket case” (2018b, p. 470). So I here perform that straightforward transfer.
In Ninan’s paper and in this paper formal languages are employed to address a puzzle about the way a set of sentences are intuitively evaluated and ascribed a certain semantic relation. How do formal representations in QML pertain to English (NL) sentences? How can such representations provide a solution to a puzzle about an aspect of NL? Here is my picture. The theoretician singles out the elements of the input which she claims determines the intuited truth conditions relative to context, and she customs a formal language that can form sentences which can be paired with targeted NL sentences in such a way that they yield the same truth conditions relative to the same input and bear among themselves analogous semantic relations. What is the benefit of this? The theoretician thereby obtains a preliminary proof of concept for ideas regarding which factors are relevant for the evaluation of the targeted NL sentences and how these contrive to yield the intuited outcomes. The ideas used in the formal medium can then guide a direct treatment of NL.
The values of what I call the ‘counterpart relation’ parameter (for which I will use the letter \( K \)) need not always be individual level counterpart relations—hence the quotation marks. I am just using it as a common label for parameters that play a comparable role in different versions of QML.
\( \fancyscript{w} \) is interpreted by a concept \(\lambda w. \{the\, winner\, at \, w\}\).
Strictly speaking between the contexts, accessibility relations will be different too. But the point of the HT puzzle is that this difference is not sufficient to capture the difference in the intuited truth conditions. I will ignore this difference.
Hazen (1979) and Lewis (1983) discuss issues faced by a straightforward implementation of the ideas of Lewis (1968) in QML. The problem I will discuss here is related, but different from these. The problems discussed by Hazen and Lewis show that ‘other worldly’ representatives of individuals should be sought in conjunction -as if they constitute a single system- but the relevant manner of representation may be the same for each individual considered in conjunction. The problem case at focus here concerns epistemic modality and rather shows that sometimes individuals’ representatives should be sought not just in conjunction but also via different manners of representation. This idea too has a precursor in Lewis’ writings (1971, 1986), but as a component of Lewis’ defense of some of his metaphysical positions.
My (3) highlights essentially the same problem as Aloni’s case of the soccer game (2001, pp. 100, 113), which she reports to be a modification of an example due to Paul Dekker. Aloni’s system has the capacity to associate different covers with different variables. Thus the system can yield adequate predictions regarding some multiply de re statements. For example, it can in the said soccer game case where the relevant covers are based on looks and names—both presumably give singleton partitions of the domain consisting of players. In cases where the epistemically relevant attributes partition the domain rather coarsely—like (3) relative to \( {c}_{4} \) where relevant attributes are red/blue and circular/triangular—the system faces the ‘bad covers’ objection raised by Ninan (2018b, pp. 471–473).
Rabern (fn.10) explained his adopting a different system than Schwarz’ (2012), which is essentially the same as Ninan’s, as due to issues regarding multiply de re modality raised in Hazen (1979) and in Lewis (1983). These issues and what is at issue in my paper are related but different. Rabern did not highlight or put to use his version’s capacity to bring into play different counterpart relations simultaneously, although he was mainly motivated by that capacity (p.c.).
There are a number of proposed systems which manifest the capability of associating different counterpart relations with different terms. As these are miscellaneous systems addressing miscellaneous problems, here I cannot possibly introduce their definitions and base my assessment on these. I just cite my view of these systems in the context of the HT case, intended as hints which may be helpful for readers interested to assess the systems for themselves. In the epistemic front Santorio’s (2012) treatment of epistemic might enables to bring into play multiple counterpart relations simultaneously. Yet the approach cannot be brought to address cases that need to be represented by formulas where we quantify into the modal sub-formula. Again on the epistemic front there is the concept generators approach exemplified in Charlow and Sharvit (2014) which concerns attitude ascriptions. Among the conceptual underpinnings of the approach are acquaintance functions; as such the approach may appear unsuitable to apply to the HT cases. The HT scenario is purposefully designed to give cases where we are not acquainted with the individuals themselves. However, formally concept generators appear very much like Kocurek’s super-indexes and as much capable (see below). In the metaphysical front there is Fara (2012), Schwarz (2014) and Kocurek (2018). Only the latter two are fully worked out proposals. Schwarz (2014) addresses a puzzle about identity. The variable assignments of Schwarz’ formal system assign ‘sorts’ to variables, thus different variables can be related with different counterpart relations. But the system in the state it is presented cannot address the HT cases, due to the ‘sortal blindness’ of the quantifier. The sortal blindness may be overcome by moving sorts to syntax or/and with sortal quantifiers, both options are suggested by the author himself. Kocurek (2018) concerns the interpretation of counteridenticals (sentences like If I were you I would not bet at all). Clearly his principle of explicit identity which disallows terms with different counterpart indexes to flank identity constructions like ‘is the same ticket as’ has to be dropped. The proposed formal system which does not depend on the principle appears capable to deal with the HT cases discussed in this paper. For instance (3R \( Kocurek \)) could satisfactorily be \( \forall x\forall y\diamond {x}^{2}={y}^{1} \). But the quantifier which is not sensitive to super-indexes is a problem: \( \exists x\left(\diamond {x}^{2}={y}^{3}\wedge \neg \diamond {x}^{1}={y}^{3}\right) \) would be true if interpreted relative to the \( {c}_{4} \) as specified in the main text, so it would be hard to explain why Some ticket both might be the winner and cannot be the winner would be false. Thanks to anonymous reviewers for drawing my attention to Fara (2012), Schwarz (2014), Kocurek (2018) and Charlow and Sharvit (2014).
Rabern notes that for the aforementioned reason Ninan proceeds thus in his own system with his brand of \( K \) (fn. 20).
Taking \( \mathcal{G} \) as the set of assignments we effectively make QML-\( g \) a free logic. In the appendix I describe a restricted neutral free logic version, along a standard version with \( {\mathcal{D}}^{\mathbb{N}} \) as the set of assignments. Opting for assignments defined over subsets of \( {\mathbb{N}} \) does not lack other commendable aspects, but it is not essential for this paper. It just enables me to keep the treatment of variable indexes in better alignment with the significance I will ascribe to NP indexes (to be introduced below). If pressed we can simply revert to the standard version; in connection with a representation task we can assign an arbitrary element of the domain to all quantified NP indexes and be mindful of that stipulation as we deal with the task (the matter about the quantified NP indexes will be coming shortly).
Ninan (2021) discusses essentially the same sentence (names repeated, no anaphoric pronouns) among a group of sentences illustrating the puzzling differential ways definite descriptions and proper names interact with epistemic and metaphysical modals. He cites his paper (2018) and Rabern (2021) as possible static approaches that can address the issues.
The notion of discourse referent originates in Karttunen's work (1968, 1976). Heim (1982) identifies them with file cards. Similar notions have variously been introduced for similar or related reasons. Groenendijk et al. (1996) postulate pegs, (Ninan, 2012) postulates tags. Recanati’s files would be a possible cognitive counterpart to the semantic-pragmatic notion at use here (2012). It is beyond the scope of this paper to further coordinate these precedents with the notion I will use.
Using variables instead of individual constants is arguably the better option. Names’ ‘reference’ should be variable across the epistemic space but rigid in the metaphysical space. This is exactly what you can get with variables. Epistemic modals will shift \( (g,w) \) thus the output will vary; metaphysical modals will shift only \( w \), the output will be constant. Treatment of names as variables has been defended in connection with epistemic modality in Santorio (2012), Ninan (2012). Schoubye (2020) adds consideration of the predicative and bound uses of names to motivate the move. Şişkolar (2023) notes, in connection with Stalnaker's (2014) framework, the treatment’s suitability to derive sensible assertoric contents from statements involving names.
The idea of directly relating referential indexes with the variables of formal representations was already present in Karttunen (1976) who attributed it to other authors (p. 25).
Why does this relativity need to be understood in terms of a disposition that obtains at \( w \) rather than simply in terms of ways of thinking actually deployed at \( w \)? Because in what follows I intend to employ the same notions in dealing with quantifications into modal contexts. In the evaluation thereof we will have to consider all the information states deriving from a collection of assignment-world pairs where the world component remains the same but assignment component varies relative to an index over the whole domain of quantification. The same agents cannot actually deploy the same way of thinking in several differently tuned ways in the same world (at the same time). That is, \( g \) and all \( g\left[2\to d\right] \) s for \( d\in \mathcal{D} \) cannot all obtain in \( w \) (at \( t \)). See also fn. 29 below.
What will happen when modal formulas are evaluated relative to points for which the set of accessible possibilities is empty? For example relative to a world \( w \) where the participants of the conversation do not exist? Under the standard QML-\( g \) version we will then have \( \left[\kern-0.15em\left[ {\diamond\,\phi } \right]\kern-0.15em\right]^{g,K,w} = 0 \) and \( \left[\kern-0.15em\left[ {\neg \diamond \;\phi } \right]\kern-0.15em\right]^{g,K,w} \)\( =1 \) for any formula \( \phi \). Thus we predict, for example, that the statements expressed by Alice can’t be infected and Alice must be infected will both be true relative to the worlds where the speaker and its audience do not exist. Authors using a version of QML to analyze epistemic modal statements seldom take note of their variant of the issue. I too will ignore it as far as the standard version is concerned. The issue does not arise in the free logic version; according to the definition I gave in the appendix, modal formulas are trivalent and evaluate as neutral (neither true nor false) at such points.
The qualification individual directed (id.) is important. When we consider quantified cases, it will be seen that such a set cannot reflect an important dimension of the information that is common ground tout court in a context. Namely the dimension expressed by quantified de re modal statements. An entity that perspicuously reflects that dimension as well can only be described by carving a more intricate chunk from \( {K}_{c} \) (or \( {\mathcal{S}}_{c} \)).
It is very convenient to use variables to represent definite noun phrases so long as we deal only with epistemic modality. Over the counter alternative options are conceptual constants which are interpreted by functions in \( {\mathcal{D}}^{\mathcal{W}} \) or Russellian translation. The former would complicate the semantics of the formal system. Furthermore such a system would not stand true to the generalization that NPs are associated with discourse referents with respect to definite phrases. This would be especially odd given that one major motive for introducing discourse referents was to account for the definiteness of definite noun phrases. Under the latter option, the Russellian translation, (1F) could be represented by the following:
$$ \exists {x}_{3}(\forall {x}_{4}\left(W{x}_{4}\supset {x}_{3}={x}_{4}\right)\wedge \forall {x}_{2}\diamond {x}_{2}={x}_{3}) $$Using this translation we could keep all the other points of the main discussion practically the same as they now stand, and we would not have to change anything in the semantics of QML-\( g \). But our representations would be more complex. Also not all definite phrases can be packed with a uniqueness claim, She hit the man and everybody was in shock. On the other side, the issue with using variables to represent definites is that, then, assuming a standard treatment of metaphysical modality, definites are predicted to be rigid designators. Which they cannot be as the possibility of acceptably using The CEO of Twitter might not have been the CEO of Twitter appears to suggest. Using over the counter options the best outcome can perhaps be obtained by postulating two types of definite noun phrases Russellian-quantificational and pronoun like. At the end of the day in this text we can simply leave the issue open, since we could have conducted practically the same discussion about MDP using Any ticket might have won.
Let \( i,j\in {\mathbb{N}} \), \( d\in \mathcal{D} \), \( g\in \mathcal{G} \). If \( i\ne j \) and \( g \) is defined for \( j \), \( g\left[i\to d\right]\left(j\right)=g(j) \). If \( i\ne j \) and \( g \) is not defined for \( j \), \( g\left[i\to d\right] \) is not defined for \( j \). Else, \( g\left[i\to d\right]\left(j\right)=d \).
Kindred ideas have been expressed by others. In connection with examples that involve quantification into epistemic modal contexts (Smith believes that every ticket probably lost the lottery) Moss suggests that quantifier bound pronouns might have assignment relative senses along assignment relative denotations (2018, pp. 161–162). Recanati takes file-like special vehicles to play a discourse-referential role in connection with quantificational statements (2012, pp.172–176).
Index 3, standing for a way of thinking about the winning ticket, is also deployed, and these ranges of possibilities pertain to it too. There is added the condition ‘\( \hbox{if }g\left(2\right)\hbox{ is defined} \)’ in the description of the set constraining \( {K}_{{c}_{2}} \). This is needed in case along (1R\( g \)) we had to evaluate a formula like \( \diamond \,F{x}_{3} \). Without the added condition to the constraint, for no \( w \) can \( \left({g}_{{c}_{2}}, w\right) \) be the first relatum of \( {K}_{{c}_{2}} \) constrained thus. For, \( {g}_{{c}_{2}} \)(2) is not defined. This will cause \( \diamond F{x}_{3} \) to evaluate neutral (false, under the standard version) for all \( w \) relative to \( {g}_{{c}_{2}} \).
Is it possible for them to think about a particular ticket \( d \), given that according to the scenario they should not have singled out any particular ticket? Suppose some outsider says I picked a ticket from photo #2 and has indeed singled out a ticket \( d \). If the agents think about \( d \) through the way corresponding to index 2 then \( d \)’s being the winning ticket will be an open possibility. Our framework is static but it ascribes to universal any-phrases a dynamic significance. How can the same way of thinking be potentially used to think about different individuals? In this paper all I can say is that it should be a special way of thinking, of different type than those associated with referential noun phrases like Musk. The specialty is not perspicuously reflected in the present formalism. There is nothing special about \( {x}_{2} \), except that it is bound by a universal quantifier in (1R\( g \)). This much suffices given the exploratory purposes of this paper. A more elaborate formalism and perhaps a truly dynamic framework may enable us to capture the difference and to model an actually obtaining cognitive structure that grounds the dispositions that we had to allude to while expounding the truth conditions. This is a topic left for future work.
The intuition which may lead one to use Pick a ticket, any ticket, it might be the winner to explicate what any ticket might be the winner means.
If \( {\mathcal{D}}_{{c}_{4}} \) just consists of tickets \( \forall {x}_{2}\forall {x}_{1}\diamond {x}_{2}={x}_{1} \) is also a suitable representation. As already noted (5), Any ticket might be any ticket, can have the same effect as (3) in \( {c}_{4} \).
Under QML-\( g \), \( \forall {x}_{2}\diamond {x}_{2}={x}_{3} \) entails \( \forall {x}_{2}\left(R{x}_{2}\supset \diamond {x}_{2}={x}_{3}\right) \).
\( \forall {x}_{2}\diamond {x}_{2}={x}_{3} \) does not entail \( \forall {x}_{1}\left(R{x}_{1}\supset \diamond {x}_{1}={x}_{3}\right) \)
In this characterization way of thinking and connotation can be understood in the intuitive, pre-theoretical manner. For a theoretical definition of connotation (of \( n \) relative to \( K \), \( {C}_{K}(n)) \) see fn. 36 below.
Ninan raises a similar question in connection with the TT puzzle (2018b, pp. 446–447).
For a given \( K \), for \( n\in {\mathbb{N}} \), \( n \) ’s connotation relative to \( K \) is the relation \( {C}_{K}(n) \):
$$ {C}_{K}\left(n\right)=\left\{\left(\left(d,w\right),\left({d}^{\prime},{w}^{\prime}\right)\right)|\exists g,{g}^{\prime}\in \mathcal{G}:\matrix{g\left(n\right)=d\cr {g}^{\prime}\left(n\right)={d}^{\prime}}<!end array> \wedge K\left(g,w\right)\left({g}^{\prime},{w}^{\prime}\right)\right\} $$\( \phi \left[ {x_{j} /x_{i} } \right] \) is the formula that results from substituting all occurrences of \( {x}_{i} \) which are not in the scope of a \( {x}_{i} \)-quantifier with occurrences of \( {x}_{j} \).
The gist of HT puzzle is that we seem to be able to derive a falsity from a truth using a pattern of reasoning which we deem valid. From SDP’s perspective the surprise element does not lie in the reasoning but in the shift of context brought about by the use of (2). From MDP’s perspective the source of the surprise may well be where it prima facie appears to be, in the reasoning pattern.
Attributions made to QML-\( g \) in this section apply both to the standard version and the free logic version. But note that attributions of entailment do not have exactly the same import in both cases, although nominally the definition remains the same. In the case of the standard version, entailment is truth preserving. In the free logic version it is just non-falsity preserving.
As expected by Lewis (1971) as well.
As (8) obtains in QML-\( d \), it also follows that \( x_{1} = x_{2} { \nvDash }\neg \diamond{x_{1}} \ne x_{2} \) in it. This is an advantage. Even if unbeknownst to us this computer is the same computer as that one we may be uncertain as to whether they are the same.
The issue here is that when we evaluate \( \square (F{x}_{1}\wedge G{x}_{2}) \) and \( \square F{x}_{1} \) relative to the same evaluation point under QML-\( d \), the evaluation of the former need not draw on the same accessible worlds base as the evaluation of the latter. For example, if \( \left(F{x}_{1}\wedge G{x}_{2}\right) \) is true at all accessible worlds in which both \( g\left(1\right) \) and \( g\left(2\right) \) have counterparts, but \( g\left(2\right) \) does not have counterparts at some additional accessible worlds in which \( g \)(1) does and in those worlds \( F{x}_{1} \) is false then we have \( \square (F{x}_{1}\wedge G{x}_{2}) \) true and \( \square F{x}_{1} \) false. By the design of the semantics such evaluation points are not possible in QML-\( g \). Modals’ evaluation depends on accessible possibilities but possibility bases relative to a given evaluation point cannot vary from formula to formula. Because possibilities themselves are systems of objects in worlds, we do not have to seek independently specified counterparts, separately for each formula, in accessible possibilities. Let \( \left(g,K,w\right) \) be any evaluation point where \( \square (F{x}_{1}\wedge G{x}_{2}) \) is true. Then, \( (F{x}_{1}\wedge G{x}_{2}) \) will be true at all \( \left({g}^{\prime},K,{w}^{\prime}\right) \) such that \( K\left(g,w\right)\left({g}^{\prime},{w}^{\prime}\right) \); and so will the conjunct \( F{x}_{1} \) as well. Now note that \( \square F{x}_{1} \) will be true at \( \left(g,K,w\right) \) if \( F{x}_{1} \) is true at all \( \left({g}^{\prime},K,{w}^{\prime}\right) \) such that \( K\left(g,w\right)\left({g}^{\prime},{w}^{\prime}\right) \). In the free logic version, if \( \left(g,K,w\right) \) is an evaluation point where \( \square (F{x}_{1}\wedge G{x}_{2}) \) is true, then \( (F{x}_{1}\wedge G{x}_{2}) \) will be either true or neutral at all \( \left({g}^{\prime},K,{w}^{\prime}\right) \) such that \( K\left(g,w\right)\left({g}^{\prime},{w}^{\prime}\right) \); again, so will the conjunct \( F{x}_{1} \); thus, \( \square F{x}_{1} \) will be true at \( \left(g,K,w\right) \). A propos the last step, note that if \( \left(g,K,w\right) \) is an evaluation point where \( \square (F{x}_{1}\wedge G{x}_{2}) \) is true, then there must be at least one \( \left({g}^{\prime},K,{w}^{\prime}\right) \) such that \( K\left(g,w\right)\left({g}^{\prime},{w}^{\prime}\right) \) where \( F{x}_{1} \) is true; so, \( \square F{x}_{1} \) will definitely be true at \( \left(g,K,w\right) \), not possibly neutral.
\( \exists t\phi , \left(\phi \vee \psi \right), \left(\phi \supset \psi \right), \left(\phi \equiv \psi \right), \square \phi \) stand for the same formulas as in the standard version.
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Appendices
Appendix
QML-\( {{\varvec g}} \)
I describe first the standard version which has its variable assignments defined over \( {\mathbb{N}} \).
The basic vocabulary and syntax are as follows:
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\( \mathrm{variables}\!\!:{x}_{0}, {x}_{1},\ldots \)
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\( \mathrm{logical \,symbols}:\, \neg, \,\wedge,\, \forall,\,=,\,\diamond \)
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\( \mathrm{predicates}:\,{A}^{n},\ldots , {Z}^{n} \;\left(n\in {\mathbb{N}}^{+},\,{\hbox{I}}\,{\hbox{will}}\,{\hbox{omit}}\, {\hbox{the}} \,{\hbox{superscripts}}\,{\hbox{indicating}}\,{\hbox{the}}\, {\hbox{arity}}\right) \)
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\( {\hbox{Let}}\;{\Pi }^{n}\,{\hbox{be}} \,{\hbox{a}} \,{\hbox{predicate}},{t}_{1},\ldots,\,{t}_{n}\;{\hbox{a}}\;{\hbox{sequence}}\,{\hbox{of}}\,{\hbox{variables}}, \) \( t,{t}^{\prime}\,{\hbox{variables}}, \) \( \phi {\mkern 1mu} \;{\hbox{and}}\;{\mkern 1mu} \psi \; {\hbox{formulas}},\;{\mkern 1mu} {\hbox{then}}\;{\mkern 1mu} {\hbox{the}}\;{\mkern 1mu} {\hbox{following}}\;{\mkern 1mu} {\hbox{are}}\;{\mkern 1mu} {\hbox{formulas}}\;{\mkern 1mu} {\hbox{too}}{:} \)
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\( \Pi ^{n} t_{1} \ldots t_{n} ,{\mkern 1mu} \neg \phi ,{\mkern 1mu} \;\left( {\phi \wedge \psi } \right),\;{\mkern 1mu} \forall t\phi ,\;{\mkern 1mu} t = t^{\prime } ,\;{\mkern 1mu} \diamond {\mkern 1mu} \phi. \)
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\( \exists t\phi , \left(\phi \vee \psi \right), \left(\phi \supset \psi \right), \left(\phi \equiv \psi \right), \square \phi \,{\hbox{respectively}}\,{\hbox{stand}}\,{\hbox{for}} \)
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\( \neg \forall t\neg \phi,\,\neg \left(\neg \phi \wedge \neg \psi \right),\, \neg \left(\phi \wedge \neg \psi \right),\,\neg \left(\phi \wedge \neg \psi \right)\wedge \neg \left(\neg \phi \wedge \psi \right),\,\neg \diamond \neg \phi. \)
The models are triples \( \langle \mathcal{W},\mathcal{D},\mathcal{I}\rangle \). \( \mathcal{I} \) is the interpretation function such that for predicate \( {\Pi }^{n} \), \( \mathcal{I}\left({\Pi }^{n}\right)\in {\left(\wp \left({\mathcal{D}}^{n}\right)\right)}^{\mathcal{W}} \).
For \( g\in {\mathcal{D}}^{\mathbb{N}} \), for \( i,j\in {\mathbb{N}} \), \( d\in \mathcal{D} \), let \( g\left[i\to d\right]\in {\mathcal{D}}^{\mathbb{N}} \) be defined as follows:
Let \( \Phi \) be the set of all formulas. If \( t \) is a variable, let \( \#(t) \) be the index \( i\in {\mathbb{N}} \) of \( t \). The evaluation function \( \left[\kern-0.15em\left[ {\,} \right]\kern-0.15em\right]_{{\mathcal{M}}} \) for model \( \mathcal{M} \) = \( \langle \mathcal{W},\mathcal{D},\mathcal{I}\rangle \) is the function from
into \( \left\{\mathrm{0,1}\right\} \) which satisfies the following:
\( \left[\kern-0.15em\left[ {\Pi^{n} t_{1} ,...,t_{n} } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1\;{\hbox{iff}}\;\left( {g\left( {\# \left( {t_{1} } \right)} \right),{ } \ldots ,{ }g\left( {\# \left( {t_{n} } \right) } \right)} \right) \in {\mathcal{I}}\left( {{\Pi }^{n} } \right)\left( w \right) \)
\( \left[\kern-0.15em\left[ {\neg \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1\;{\hbox{iff}}\;\left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \)
\( \left[\kern-0.15em\left[ {\left( {\phi \wedge \psi } \right)} \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1\;{\hbox{iff}}\;\left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1\;{\hbox{and}}\;\left[\kern-0.15em\left[ \psi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \)
\( \left[\kern-0.15em\left[ {\forall x_{i} \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1\;{\hbox{iff for}}\;{\hbox{all}}\;d \in {\mathcal{D}},\;\left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g\left[ {i \to d} \right],K,w}} = 1 \)
\( \left[\kern-0.15em\left[ {t = t^{\prime } } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1\;{\hbox{iff}}\;g\left( {\# \left( t \right)} \right) = g\left( {\# \left( {t^{\prime } } \right)} \right) \)
\( \left[\kern-0.15em\left[ {\diamond\,\phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \) iff there is \( g^{\prime} \in {\mathcal{D}}^{{\mathbb{N}}} ,w^{\prime} \in {\mathcal{W}} \) such that \( K\left( {g,w} \right)\left( {g^{\prime},w^{\prime}} \right) \) and \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g^{\prime } ,K,w^{\prime } }} = 1 \)
\( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} \ne 1 \) iff \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \)
The free logic version has the same syntax, the same models, the same interpretation functions. But its variable assignments are defined over subsets of \( {\mathbb{N}} \). So, given a model \( \mathcal{M}=\langle \mathcal{D},\mathcal{W},\mathcal{I}\rangle \) the set of assignments will be \( \mathcal{G}=\left\{g\in {\mathcal{D}}^{X}|X\subseteq {\mathbb{N}}\right\} \).
For \( d\in \mathcal{D} \), for \( i\in {\mathbb{N}} \), \( g\left[i\to d\right]\in \mathcal{G} \) will be defined thus:
The evaluation function \( \left[\kern-0.15em\left[ {\,} \right]\kern-0.15em\right]_{{\mathcal{M}}} \) for model \( \mathcal{M} \)=\( \langle \mathcal{W},\mathcal{D},\mathcal{I}\rangle \) is the function from
into \( \left\{ {0,1,\emptyset } \right\} \) which satisfies the following:
If \( g\left(\#\left({t}_{i}\right)\right) \) is defined for all \( 1\le i\le n \),
\( \left[\kern-0.15em\left[ {\Pi^{n} t_{1} ,...t_{n} } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \) iff \( \left( {g\left( {\# \left( {t_{1} } \right)} \right),{ } \ldots ,{ }g\left( {\# \left( {t_{n} } \right)} \right)} \right) \in {\mathcal{I}}\left( {{\Pi }^{n} } \right)\left( w \right) \),
\( \left[\kern-0.15em\left[ {\Pi^{n} t_{1} ,...t_{n} } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \) iff \( \left( {g\left( {\# \left( {t_{1} } \right)} \right),{ } \ldots ,{ }g\left( {\# \left( {t_{n} } \right)} \right)} \right) \notin {\mathcal{I}}\left( {{\Pi }^{n} } \right)\left( w \right) \).
Otherwise, \( \left[\kern-0.15em\left[ {\Pi^{n} t_{1} ,...t_{n} } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \).
If both \( g\left(\#\left(t\right)\right) \hbox{and }g\left(\#\left({t}^{\prime}\right)\right) \) are defined,
\( \left[\kern-0.15em\left[ {t = t^{\prime } } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \) iff \( g\left( {\# \left( t \right)} \right) = g\left( {\# \left( {t^{\prime } } \right)} \right) \),
\( \left[\kern-0.15em\left[ {t = t^{\prime } } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \) iff \( g\left( {\# \left( t \right)} \right) \ne g\left( {\# \left( {t^{\prime } } \right)} \right) \).
Otherwise, \( \left[\kern-0.15em\left[ {t = t^{\prime } } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \).
\( \left[\kern-0.15em\left[ {\neg \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \) iff \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \),
\( \left[\kern-0.15em\left[ {\neg \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \) iff \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \),
\( \left[\kern-0.15em\left[ {\neg \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \) iff \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \).
\( \left[\kern-0.15em\left[ {\left( {\phi \wedge \psi } \right)} \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \) iff \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \) and \( \left[\kern-0.15em\left[ \psi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \),
\( \left[\kern-0.15em\left[ {\left( {\phi \wedge \psi } \right)} \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \) iff \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \) or \( \left[\kern-0.15em\left[ \psi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \),
\( \left[\kern-0.15em\left[ {\left( {\phi \wedge \psi } \right)} \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \) iff (\( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \) or \( \left[\kern-0.15em\left[ \psi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \)) and (\( \left[\kern-0.15em\left[ \psi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} \ne 0 \) and \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} \ne 0 \)).
If for all \( d \in {\mathcal{D}} \), \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g\left[ {i \to d} \right],K,w}} \in \left\{ {0,1} \right\} \),
\( \left[\kern-0.15em\left[ {\forall x_{i} \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \) iff for all \( d \in {\mathcal{D}} \), \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g\left[ {i \to d} \right],K,w}} = 1 \),
\( \left[\kern-0.15em\left[ {\forall x_{i} \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \) iff there is \( d \in {\mathcal{D}} \), \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g\left[ {i \to d} \right],K,w}} = 0 \).
Otherwise, \( \left[\kern-0.15em\left[ {\forall x_{i} \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \).
If there is \( g^{\prime } \in {\mathcal{G}},w^{\prime } \in {\mathcal{W}} \) such that \( K\left( {g,w} \right)\left( {g^{\prime } ,w^{\prime } } \right) \) and \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g^{\prime } ,K,w^{\prime } }} \in \left\{ {0,1} \right\} \),
\( \left[\kern-0.15em\left[ {\diamond\phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 1 \) iff there is \( g^{\prime } \in {\mathcal{G}},w^{\prime } \in {\mathcal{W}} \) such that \( K\left( {g,w} \right)\left( {g^{\prime } ,w^{\prime } } \right) \) and \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g^{\prime } ,K,w^{\prime } }} = 1 \),
\( \left[\kern-0.15em\left[ {\diamond\phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = 0 \) iff for no \( g^{\prime } \in {\mathcal{G}},w^{\prime } \in {\mathcal{W}} \) such that \( K\left( {g,w} \right)\left( {g^{\prime } ,w^{\prime } } \right) \), \( \left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g^{\prime } ,\;K,w^{\prime } }} = 1 \).
Otherwise, \( \left[\kern-0.15em\left[ {\diamond\phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,K,w} = \emptyset \).Footnote 43
According to this, the free logic version of QML-\( g \) amounts to a restricted neutral free logic. I could have proceeded more conservatively using weak Kleene connectives, and keeping the modal formulas bivalent as in the standard version. The advantage of the trivalent definition for modal formulas is that a modal formula \( \phi \) is then evaluated neutral (\( \emptyset \)) relative to any point \( \left(g,K,w\right) \) for which there is no \( \left({g}^{\prime},{w}^{\prime}\right) \) such that \( K\left(g,w\right)\left({g}^{\prime},{w}^{\prime}\right) \). Strong Kleene connectives are then needed to get things right when formulas of type \( \forall {x}_{i}(\alpha \supset \diamond \beta ) \) are used to represent (Any A)i might be B when the domain of quantification includes non-As.
QML-\( {{\varvec d}} \)
QML-\( d \) is the QML version employed by Ninan (2018b). Its models are the same as QML-\( g \). Variable assignments are functions in \( {\mathcal{D}}^{\mathbb{N}} \). The values of parameter \( K \) are binary relations over \( \mathcal{D}\times \mathcal{W} \). Evaluation depends also on an accessibility relation parameter \( R \), its values are binary relations over \( \mathcal{W} \). The clause for \( \diamond \) is,
\( \left[\kern-0.15em\left[ {\diamond \phi } \right]\kern-0.15em\right]_{{\mathcal{M}}}^{g,R,K,w} = 1\,\,{\hbox{iff}}\,\,{\hbox{there}}\,\,{\hbox{is}}\,g^{\prime } \in {\mathcal{D}}^{\mathbb{N}},\,w^{\prime } \in {\mathcal{W}}\,{\hbox{such}}\,\,{\hbox{that}}\,\,Rww^{\prime},\, {\hbox{for}}\,{\hbox{each}}\;{\hbox{free}}\,\,\,\,\,\,\,\,{\hbox{variable}}\,x_{i} \,{\hbox{in}}\,\phi \) \( K\left( {g\left( i \right),w} \right)\left( {g^{\prime } \left( i \right),w^{\prime } } \right), \,{\hbox{and}}\,\left[\kern-0.15em\left[ \phi \right]\kern-0.15em\right]_{{\mathcal{M}}}^{{g^{\prime } ,R,K,w^{\prime } }} = 1. \)
Another difference are concept constants \( {\fancyscript{a}}, ...\,,{\fancyscript{z}} \). These are syntactically treated as terms and interpreted by functions \( f\in {\mathcal{D}}^{\mathcal{W}} \). Ninan also introduces a \(\lambda\)-operator, but in our paper we did not need to use it.
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Şişkolar, C. Representing multiply de re epistemic modal statements. Linguist and Philos 47, 211–237 (2024). https://doi.org/10.1007/s10988-023-09394-1
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DOI: https://doi.org/10.1007/s10988-023-09394-1