Abstract
This paper investigates set-theoretical semantics for logics that contain unary connectives, which can be viewed as modalities. Indeed, some of the logics we consider are closely related to linear logic. We use insights from the relational semantics of relevance logics together with a new version of the squeeze lemma in our semantics for logics with disjunction (but no conjunction). The ideal-based semantics, which takes co-theories to be situations, dualizes the theory-based semantics for logics with conjunction (but no disjunction).
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Notes
- 1.
Semantical considerations may suggest that \(\,!\,\) and \(\lozenge \), on one hand, plus \(\,?\,\) and \(\square \), on the other hand, are better pairings (cf. Bimbó 2007; Bimbó and Dunn 2008). However, both the proof-theoretical and the semantic analogies are mere analogies, which we underline by not using a notation (like \(\square \) or \(\lozenge \)) that have deeply ingrained connotations in modal logic.
- 2.
The notation used here is the same notation that we used in Bimbó (2017).
- 3.
- 4.
For example,
is a pair that may be included or omitted. The cut theorem for the calculuses here is Theorem 15 in Bimbó (2017). The proof of the cut theorem in [Bimbó 2015a, §3] is triple-inductive proof for the intensional part of our calculuses with modalized structural rules. See also Chap. 7 in Bimbó (2015b). - 5.
The duality between filters and ideals, which is rooted in the duality of \(\wedge \) and \(\vee \), has a storied past in logic (see Halmos 1962, p. 22 and Dunn 2019, p. 28). Our semantics are also motivated by semantics for relevance logics in a broad sense of relevance (see Avron 1984, 2014; Bimbó and Dunn 2009).
- 6.
Lowercase Greek letters range over elements of U, and we will use logical symbols in the metalanguage which may be thought—for the sake of simplicity—to be those in two-valued logic.
- 7.
As a rule, we only indicate the situation that makes a formula true in a model in the \(\vDash \) notation, and we omit mentioning \(\mathfrak {F}\) or v—to enhance readability.
- 8.
is a pair that may be included or omitted. The cut theorem for the calculuses here is Theorem 15 in Bimbó (References
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Acknowledgements
I am grateful to Anna Zamansky and Ofer Arieli, the editors of the volume, for inviting me to contribute a paper. I would like to thank the organizers of and the audiences at IsraLog 2017 (The Third Israeli Workshop on Non-Classical Logics and their Applications) and AAL 2018 (2018 Annual Meeting of the Australasian Association for Logic) where I presented talks related to this paper. I am also grateful to the IU Logic Group at Indiana University–Bloomington for letting me talk about the results in this paper in November 2019. The research reported in this paper was funded by an Insight Grant (SSHRC IG #435–2014–0127) awarded by the Social Sciences and Humanities Research Council of Canada.
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Bimbó, K. (2021). Interpretations of Weak Positive Modal Logics. In: Arieli, O., Zamansky, A. (eds) Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Outstanding Contributions to Logic, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-71258-7_2
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