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Hyperintensional logics for everyone

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Abstract

We introduce a general representation of unary hyperintensional modalities and study various hyperintensional modal logics based on the representation. It is shown that the major approaches to hyperintensionality known from the literature, that is state-based, syntactic and structuralist approaches, all correspond to special cases of the general framework. Completeness results pertaining to our hyperintensional modal logics are established.

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Notes

  1. For a brief historical overview, see Chap. 1.1 of Duží et al. (2010), for example.

  2. It is natural in a hyperintensional setting to consider all the usual Boolean connectives as primitive. For the sake of simplicity, we do this already when discussing MS models.

  3. The definition of logical consequence over a class of models is standard: \(\varGamma \) entails F iff, for all models in the class, \(\bigcap _{G \in \varGamma } \llbracket G\rrbracket \subseteq \llbracket F\rrbracket \). Using \({\mathscr {L}}\), consequence relations over all classes of MS models have the property that \(\varGamma \) entails F iff there is a finite \(\varGamma ^{\prime } \subseteq \varGamma \) such that \(\left( \bigwedge _{G \in \varGamma ^{\prime }} G\right) \rightarrow F\) is valid in the given class. For such classes, (RR) entails that N(w) is closed under consequence. (RM) says that every N(w) is closed under single-premise consequence and (RN) entails closure under zero-premise consequence, i.e. validity.

  4. Stronger normal modal logics correspond to adding more assumptions. For instance, the logic \({\mathbf {T}}\) is determined by models where \(w \in N(w)\) for all w. Obviously, \(\Box F \rightarrow F\) is valid in any such model.

  5. For instance, assume that different arithmetical laws hold in these states.

  6. An account of hyperintensionality using impossible worlds has been put forward recently by Jago (2014); see also (Ripley 2012). (Berto 2007, 2013) gives a general overview of the most important topics related to impossible worlds.

  7. (A3) is retained in the sense that structuralist frameworks typically do not include other kinds of states in addition to possible worlds.

  8. We note that even though neither compositionality principle excludes the possibility that, say \(O(\lnot p) = O(p)\), there is no hyperintensional model where \(O(\lnot p) = O(p) = c\). If this were the case, then \(I(c) = W \setminus I(c)\), but this is not possible if W is non-empty.

  9. Consider, for example, the sentences ‘John is not a bachelor’ and ‘John is not an unmarried man’. One might be inclined to say that they have the same meaning because ‘John is a bachelor’ and ‘John is an unmarried man’ have the same meaning.

  10. Note that a related schema \(\Box F \leftrightarrow \lnot \Diamond \lnot F\) is not valid. The reason is that we may have models where \(O(F) \ne O(\lnot \lnot F)\).

  11. Note that it is possible to have \(\llbracket \Box F\rrbracket = \llbracket \Box G\rrbracket \) while \(O(F) \ne O(G)\).

  12. The completeness argument utilises a canonical model similar to the one presented in the proof of Theorem 1. The only difference is that the universe of the model is the set of maximal consistent theories with respect to the extended axiomatisation.

  13. Wansing defines these models over a language where only \(\lnot , \wedge , \Box \) are primitive, a detail we will pass over.

  14. Observe that \(M^{*}\) is not necessarily weakly compositional. The reason is that \(V(F) \setminus W\) in Rantala models does not depend on the structure of F. This may be a reason for proponents of weak compositionality to refute Rantala models as an adequate representation of hyperintensional modalities.

  15. In particular, Wansing proves this for Levesque’s logic of implicit and explicit belief (Levesque 1984), Fagin and Halpern’s logic of awareness, logic of general awareness and logic of local reasoning (Fagin and Halpern 1988) and van der Hoek and Meyer’s logic of awareness and principles (van der Hoek and Meyer 1989).

  16. A full discussion of these structuralist approaches is not provided mainly because of space limitations and proportionality considerations. Nevertheless, a more detailed account of how (fragments of) these structuralist accounts of sentential content can be formalised using hyperintensional models is an interesting topic of future research.

  17. We need to be careful here as, for instance, if \(\textit{mod}\) would be identical to the intension of the modal operator, i.e. to \(N_C\), then the present definition of neo-Russelian models would be circular.

  18. We reiterate that it is not our intention here to provide a faithful account of (a fragment of) TIL. The special case discussed in the text aims at providing only a hint of how TIL might be represented within our framework. Developing this hint to a complete account is a task that needs to be carried out in a separate article.

  19. Bloom and Suszko add to classical propositional logic a new equivalence connective ‘\(\equiv \)’ representing identity of content as opposed to mere identity of truth value represented by classical equivalence. For details of their approach, the reader is referred to Bloom and Suszko (1971, 1972).

  20. Since \(\dfrac{F \rightarrow G}{\Box _U F \rightarrow \Box _U G}\) is an admissible rule, as can be shown using (U4) and (U5).

  21. As an example of such a relation, consider various conversion relations between the denotations of \(\lambda \)-terms in Transparent Intensional Logic (Duží et al. 2010). The relation of procedural isomorphism on the set of constructions was defined to express a similar notion, see (Duží 2017). Another example, if \(\Box \) is assumed to correspond to epistemic attitudes of an agent, is intensional equivalence ‘recognized’ by the agent.

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Acknowledgements

This work was supported by the long-term strategic development financing of the Institute of Computer Science (RVO:67985807). The author is grateful to Pavel Cmorej, Marie Duží, Daniela Glavaničová, Miloš Kosterec, Pavel Materna, Jaroslav Peregrin, Ivo Pezlar and Jiří Raclavský for comments on drafts of this article.

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  1. Institute of Computer Science, The Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07, Prague 8, Czech Republic

    Igor Sedlár

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Sedlár, I. Hyperintensional logics for everyone. Synthese 198, 933–956 (2021). https://doi.org/10.1007/s11229-018-02076-7

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