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Logic talk

Abstract

Sentences about logic are often used to show that certain embedding expressions (attitude verbs, conditionals, etc.) are hyperintensional. Yet it is not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. In this paper, I develop a formal system called hyperlogic that is designed to do just that. I provide a hyperintensional semantics for hyperlogic that doesn’t appeal to logically impossible worlds, as traditionally understood, but instead uses a shiftable parameter that determines the interpretation of the logical connectives. I argue this semantics compares favorably to the more common impossible worlds semantics, which faces difficulties interpreting propositionally quantified logic talk.

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Notes

  1. 1.

    Indeed, this project is in many ways connected to work in relevant logic (Dunn and Restall , 2002; Mares , 2004; Mares , 2020). I view hyperlogic as a step towards developing what Routley (2019) calls a “universal logic”: a logic that’s “applicable in every situation whether realised or not, possible or not” (see Nolan (2018) for a critical overview of Routley’s program). The approach taken here will be different from Routley’s in that I aim to develop a universal logic within a classical framework, rather than revise the base logic. Other choices of background logic may be equally (or even more) fruitful as starting points (Girard and Weber , 2015; Weber , 2014; Weber et al. , 2016). It would be also worth investigating how the hybrid aspects of hyperlogic change in nonclassical settings; see Braüner (2006, 2011); Braüner and de Paiva (2006); Chadha et al. (2006) for work on intuitionistic hybrid logic, as well as Standefer (2020) for work on relevant logics for ‘actually’.

  2. 2.

    Of course, (14) and (16) may be invalid on a stricter notion of validity. For example, these arguments could be deemed invalid in the sense of relevant logic (Anderson and Belnap , 1975; Dunn and Restall , 2002; Mares , 2004; Mares , 2020). Again, my assumption of a classical background logic is just a starting point. Alternative choices of background logic might lead to different views on what exactly is wrong with these arguments (see Weber (2014); Girard and Weber (2015); Weber et al. (2016) for semantical analyses within nonclassical metatheory). My claim is just that these arguments seem bad in some sense that is not accounted for by saying they’re invalid on a classical conception of validity.

  3. 3.

    We could add other connectives to the language, including those that arise in nonclassical logics such as “intensional” conjunction and disjunction operators from relevant logic (Dunn and Restall , 2002) or exponentials from linear logic (Di Cosmo and Miller , 2019). But I will stick with the standard “boolean” connectives for ease of exposition.

  4. 4.

    We could make \(\mathbin {\rhd }\) right-multigrade also to account for multiple-conclusion logics, but we’ll set this complication aside.

  5. 5.

    For more on propositional quantifiers, see Bull (1969); Fine (1969, 1970); Kaplan (1970); Kaminski and Tiomkin (1996); Holliday (2017); Holliday and Litak (2018); Fritz (2020).

  6. 6.

    See Williamson (2013); Bacon (2020) for related approaches.

  7. 7.

    See Areces and ten Cate (2006); Braüner (2017) for an overview.

  8. 8.

    The idea to introduce hybrid operators of this sort comes from Kocurek and Jerzak (2021), though they introduce it for different reasons, viz., to distinguish between convention-shifting readings of counterlogicals from non-shifty readings. I’ll discuss Kocurek and Jerzak’s (2021) motivation in Sect. 5.1.

  9. 9.

    One complication with this regimentation of (4): it’s not entirely clear how we should regiment a law’s failure according to some logic. There are at least three ways a law \(\lambda \) can “fail” for a logic l. There’s an “external” notion: \(\lambda \) externally fails for l if it’s not the case that according to l, \(\lambda \) holds (\(\mathop {\lnot }\nolimits \mathop {@}\nolimits _l\lambda \)). There’s also an “internal” notion: \(\lambda \) internally fails for l if its negation holds according to l (\(\mathop {@}\nolimits _l\mathop {\lnot }\nolimits \lambda \)). Finally, there is a third, intermediate “classical” notion: \(\lambda \) classically fails for l if according to l, \(\lambda \) does not hold, where the ‘not’ here is interpreted as classical negation. This is the notion of failure we get by holding fixed what we actually mean by ‘fails’ or ‘does not hold’ (given our background logic is classical) within the scope of according-to operators. Thus, we can regiment this notion using the \(\mathop {\downarrow }\nolimits \) binder (\(\mathop {@}\nolimits _l\mathop {\downarrow }\nolimits i.\mathop {@}\nolimits _{cl}\mathop {\lnot }\nolimits \mathop {@}\nolimits _i\lambda \)). For concreteness, I have regimented (4) using the internal notion, but nothing in what follows hinges on this choice.

  10. 10.

    See Nolan (1997); Vander Laan (2004); Krakauer (2012); Brogaard and Salerno (2013); Jago (2014); Kment (2014); Berto et al. (2018); Berto and Jago (2019); French et al. (2020). There are other approaches to hyperintensionality that I do not have space to consider here. See Fine (2012); Schaffer (2016); Wilson (2018); Leitgeb (2019) for some prominent examples. For an overview of different approaches to hyperintensionality, see Berto and Nolan (2021).

  11. 11.

    My implementation of the impossible worlds approach to counterfactuals is largely in line with that of French et al. (2020). See also Mares (1997); Nolan (1997); Brogaard and Salerno (2013); Kment (2014); Berto et al. (2018) for other examples. This semantics is closely related to the Routley-Meyer semantics for relevant logics; see Mares (2004) for an overview.

  12. 12.

    Following Tanaka (2018), this clause could be generalized so that formulas can take multiple (or no) truth values at an impossible world. This additional complication does not affect the main arguments below, however, so I will set this aside.

  13. 13.

    As Tanaka (2018) and Sandgren and Tanaka (2020) note, there are two senses in which a world can be logically impossible: it can be logically different, in that the laws of logic differ from the actual laws, or it can contain a violation of a logical law (Kocurek and Jerzak, 2021: pp. 22–23 on actual vs. counterfactual logical impossibility). Here, by “logically impossible”, I mean to neutrally refer to either notion of logical impossibility. The semantics provided in Sect. 5 can represent either kind of logical impossibility.

  14. 14.

    Nolan (1997: p. 563) argues that not every set of impossible worlds counts as a proposition. Thus, we may want to include in our models a domain of sets of worlds the quantifiers can range over. (So the semantics would be closer to \(\mathbf{S5 }\pi \).) This won’t affect the substantive points in what follows, however.

  15. 15.

    Proof: suppose where \(\mathcal {I}= \left\langle W,P,f,V \right\rangle \) and \(w \in P\). Let \(X \mathrel {\subseteq }W\). Since possible worlds are classically consistent, . But since \(V^p_X\) doesn’t reassign the truth of \(p \mathbin {\wedge }\mathop {\lnot }\nolimits p\), \(V^p_X(p \mathbin {\wedge }\mathop {\lnot }\nolimits p,v) = V(p \mathbin {\wedge }\mathop {\lnot }\nolimits p,v)\) for all \(v \in \overline{P}\). Hence, , and so . Moreover, since \(q \ne p\), \(V^p_X(q,w) = V(q,w)\), and so . Therefore, if for any \(X \mathrel {\subseteq }W\), then it holds for all X. In other words, entails .

  16. 16.

    This view can be seen as a species of logical pluralism (Beall and Restall , 2006), though the latter is a broader category. Kocurek and Jerzak (2021: p. 17) argue that logics are effectively semantic conventions governing logical vocabulary. According to logical expressivism, adopting a logic is akin to adopting a language: just as there is no such thing as “the one true” semantic convention, so too, there is no such thing as “the one true” logic.

  17. 17.

    Since Kocurek and Jerzak’s (2021) semantics does not include propositional quantifiers or an entailment operator, we need to generalize their notion of a hyperconvention.

  18. 18.

    One reason to allow \(\mathbin {\rhd }\) to be noncontingent is to model logics such as \(\mathbf{K3 }\) on which there are no logical validities. It would be natural to capture this fact about \(\mathbf{K3 }\) with the formula . But if entailment facts are necessary, then this formula will be falsifiable.

  19. 19.

    Proof: let \(w \in W\) be such that there is a \(w' > w\) in W. Let . Then:

    figuread

    But , since \(w' \ge w\) but \(w \ngeq w'\). Thus, .

  20. 20.

    Proof: just pick an index proposition whose restriction to some \(c \in C_{\mathcal {K}}\) is .

  21. 21.

    While Kocurek and Jerzak (2021) model logics (which are the referents of interpretation terms) as hyperconventions, this proposal is too stringent for our purposes, as it would require logics to provide interpretations of every propositional variable. To avoid this concern, we can let the interpretation terms denote conventions, i.e., sets of hyperconventions (Kocurek and Jerzak, 2021, fn. 21).

  22. 22.

    Note, however, that V(p) need not be \(P_p\); this is essential, since quantifiers need to be capable of reinterpreting atomics.

  23. 23.

    Kocurek and Jerzak argue elsewhere, following the work of Einheuser (2006), that even conditionals not about logic can shift the convention used to interpret material in their scope (Kocurek et al. , 2020).

  24. 24.

    Note, this does not mean that counterfactuals are “metalinguistic” in the sense of being about language or conventions. Rather, the semantics models hyperintensionality by allowing counterfactuals to shift the conventions used to interpret material in their scope. Compare: quantifiers shift variable assignments, but that doesn’t mean they’re “about” variable assignments.

  25. 25.

    We could remove relativization to c and say \(R_{\mathop {\textsf {B}}\nolimits }\) relates worlds to indices. Either approach would be fine for our purposes.

  26. 26.

    If an agent accepts a logic that is not factive (so a validity might be false) or is contingent (so a validity need not be necessarily valid), we should not expect their beliefs to be closed under that logic anyway.

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Correspondence to Alexander W. Kocurek.

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Thanks to Harold Hodes, Jens Kipper, Carlotta Pavese, Rachel Rudolph, Zeynep Soysal, Will Starr, James Walsh, and two anonymous reviewers, as well as the participants of the 11th Semantics and Philosophy in Europe conference, the 22nd Amsterdam Colloquium, and the Fall 2019 Cornell Semantics Reading Group for helpful comments and discussion. An early (albeit unrefined) version of the system I develop here can be found in Kocurek (2019).

Appendix

Appendix

A representability theorem

In this appendix, I establish that any finitary single-conclusion logic can be represented in the hyperconvention semantics. In fact, we can represent any logic using just a single hyperconvention. In addition, I show that this improves upon the semantics in Kocurek and Jerzak 2021, which can only represent a smaller set of logics.

First, we need to clarify what we mean by ‘logic’ and ‘represent’.

Definition 11

(Logic) A logic over \({\mathcal {L}}\) is a set \(\mathbf{L }\mathrel {\subseteq }{\mathcal {L}}^{<\omega } \mathrel {\times }{\mathcal {L}}\) of pairs of the form . (We allow the lefthand side to be the empty tuple \(\diamondsuit \).) We may write in place of . We use \(\vec {\phi }\) as shorthand for , where the length is implicit (possibly zero).

Definition 12

(Representability) A logic \(\mathbf{L }\) is representable in \(\mathcal {M}= \left\langle W,D_{\mathbb {C}},D_{\mathbb {P}},V \right\rangle \) if there’s a \(c \in \mathbb {H}_{W}\) such that for all :

figurean

Theorem 13

(Representation) Every logic is representable in an infinite hypermodel (i.e., \(\left| {W}\right| = \aleph _{0}\)).

Proof

Let be an injection. Let . Define c as follows.

figureao

Let \(\mathcal {M}\) be a hypermodel over W with c in its hyperconvention domain such that \(V(p)(c) = [p]\) (e.g., we can let \(V(p) = P_p\)). By a simple induction, for all \(\phi \in {\mathcal {L}}\).

Thus, if , then \(([\phi _1],\dots ,[\phi _n] \mathbin {\rhd }_c [\psi ]) = W\), so . Conversely, if , then \(([\phi _1],\dots ,[\phi _n] \mathbin {\rhd }_c [\psi ]) = [p_1]\), so .

Theorem 13 contrasts with the situation for the semantics developed in Kocurek and Jerzak 2021. There, they do not introduce an entailment operator \(\mathbin {\rhd }\) into the language, so there is no obvious way to represent logics in the usual sense. In a footnote, they suggest the following (rephrased in our notation): c represents \(\mathbf{L }\) in \(\mathcal {M}\) iff iff for all w, if , then \(\mathcal {M},w,c \Vdash \psi \). I will now show that on this understanding of representability, a strictly smaller class of logics are representable by a hyperconvention.

Definition 14

(Properties of logics) A logic \(\mathbf{L }\) is intensional if it satisfies the following properties:

  • Reflexive: \(\phi \mathbin {\Rightarrow }_\mathbf{L }\phi \)

  • Transitive: if \(\vec {\psi } \mathbin {\Rightarrow }_\mathbf{L }\chi \) and \(\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\psi _i\) for each \(\psi _i\), then \(\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\chi \)

  • Commutative: if \(\vec {\phi },\vec {\psi } \mathbin {\Rightarrow }_\mathbf{L }\chi \), then \(\vec {\psi },\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\chi \)

  • Monotonic: if \(\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\psi \), then \(\vec {\phi },\chi \mathbin {\Rightarrow }_\mathbf{L }\psi \)

  • Congruential: it obeys replacement of logical equivalents:

    • for : if \(\phi \mathbin {\Leftrightarrow }_\mathbf{L }\phi '\), then \(\mathop {\bullet }\nolimits \phi \mathbin {\Leftrightarrow }_\mathbf{L }\mathop {\bullet }\nolimits \phi '\)

    • for : if \(\phi \mathbin {\Leftrightarrow }_\mathbf{L }\phi '\) and \(\psi \mathbin {\Leftrightarrow }_\mathbf{L }\psi '\), then \(\phi \mathbin {\circ }\psi \mathbin {\Leftrightarrow }_\mathbf{L }\phi ' \mathbin {\circ }\psi '\).

Observe that reflexivity, commutativity, and monotonicity entail (in fact, given transitivity, are equivalent to) the following property:

  • Reiterative: \(\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\phi _i\) for each \(\phi _i\).

Definition 15

(Intensional representability) A logic \(\mathbf{L }\) is intensionally representable in \(\mathcal {M}= \left\langle W,D_{\mathbb {C}},D_{\mathbb {P}},V \right\rangle \) if there’s a \(c \in \mathbb {H}_{W}\) such that for all :

figureap

Theorem 16

(Intensional Representation). A logic \(\mathbf{L }\) is intensionally representable in an infinite hypermodel \(\mathcal {M}\) iff \(\mathbf{L }\) is intensional. Regardless of \(\mathcal {M}\)’s size, \(\mathbf{L }\) is intensionally representable in \(\mathcal {M}\) iff \(\mathbf{L }\) is intensional and the number of nonequivalent finite sequences of formulas is no greater than \(\left| {W}\right| \).

Proof

The left-to-right direction is straightforward: one simply checks that necessary implication has all the features of an intensional logic.

For the right-to-left direction, we begin by defining the finite theory space of \(\mathbf{L }\), which is a generalization of the Lindenbaum-Tarski algebra. Where , define:

figureaq

The finite theory space of \(\mathbf{L }\) is the order \(\mathbb {F}_\mathbf{L }= \left\langle F_\mathbf{L },\le _\mathbf{L } \right\rangle \) where:

  • \([\vec {\phi }]_\mathbf{L }\le _\mathbf{L }[\vec {\psi }]_\mathbf{L }\) iff \(\vec {\phi } \mathbin {\Rightarrow }_\mathbf{L }\psi _i\) for each \(\psi _i \in \vec {\psi }\).

From now on, we’ll leave the \(\mathbf{L }\)-subscripts implicit. Recall that a meet-semilattice is a partial order where the meet (i.e., greatest lower bound) of any two elements exists.

Claim

\(\mathbb {F}\) is a meet-semilattice with a top element.

Proof

Since \(\mathbf{L }\) is reiterative, for \(1 \le i \le n\). So \(\le \) is reflexive. The transitivity and antisymmetry of \(\le \) follow from the transitivity of \(\mathbf{L }\). The top element is simply \([\left\langle \right\rangle ]\). As for the existence of meets, define the meet \(\sqcap \) operation as follows:

figurear

That \([\vec {\phi },\vec {\psi }] \le [\vec {\phi }]\) and \([\vec {\phi },\vec {\psi }] \le [\vec {\psi }]\) follows from reiterativity. Moreover, if \([\vec {\chi }] \le [\vec {\phi }]\) and \([\vec {\chi }] \le [\vec {\psi }]\), then \([\vec {\chi }] \le [\vec {\phi },\vec {\psi }]\) by definition of \(\le \). So \(\sqcap \) really is a meet operation.

It is a well-known fact of order-theory that any partial order can be order-embedded into its powerset algebra in a meet-preserving manner. Since \(F_\mathbf{L }\) is at most the size of W, that means there’s an injective map such that:

  1. (i)

    \([\vec {\phi }] \le [\vec {\psi }]\) iff \(f([\vec {\phi }]) \mathrel {\subseteq }f([\vec {\psi }])\)

  2. (ii)

    \(f([\vec {\phi },\vec {\psi }]) = f([\vec {\phi }]) \cap f([\vec {\psi }])\).

What’s more, we can take \(f([\left\langle \right\rangle ]) = W\). For if \(f([\left\langle \right\rangle ]) \ne W\), define \(f^*\) to be exactly like f except \(f^*([\left\langle \right\rangle ]) = W\). It is easy to then verify that \(f^*\) is still an order-embedding that preserves finite meets.

Define c as follows (where and ):

figureas

This is well-defined. For if \(f([\phi ]) = f([\psi ])\), then \([\phi ] = [\psi ]\) by injectivity. Thus, \(\phi \mathbin {\Leftrightarrow }_\mathbf{L }\psi \). So by congruentiality, \(\mathop {\bullet }\nolimits \phi \mathbin {\Leftrightarrow }_\mathbf{L }\mathop {\bullet }\nolimits \psi \), which means \([\mathop {\bullet }\nolimits \phi ] = [\mathop {\bullet }\nolimits \psi ]\). Hence, \(f([\mathop {\bullet }\nolimits \phi ]) = f([\mathop {\bullet }\nolimits \psi ])\). (And similarly for binary connectives.)

Throughout, let \(\mathcal {M}\) be a hypermodel of the relevant size with c in its hyperconvention domain where \(V(p)(c) = c(p)\). By induction:

Claim

For any \(\phi \in {\mathcal {L}}\), .

Finally, we show that c intensionally represents \(\mathbf{L }\) in \(\mathcal {M}\).

\(\mathbin {\Leftarrow }\):

Suppose . Thus, . Since f is order-reflecting, . So there is a . Now observe:

figureat

Hence, but \(\mathcal {M},w,c \nVdash \psi \).

\(\mathbin {\Rightarrow }\):

Suppose . Then . Since f is order-embedding:

figureau
Therefore, c intensionally represents \(\mathbf{L }\) in \(\mathcal {M}\). \(\square \)

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Kocurek, A.W. Logic talk. Synthese (2021). https://doi.org/10.1007/s11229-021-03394-z

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Keywords

  • Metalogical claims
  • Hyperconvention semantics
  • Hyperintensionality
  • Propositional quantifiers
  • Hybrid logic
  • Impossible worlds
  • Counterfactuals