Abstract
Reflexive transitive closure modalities represent a number of important notions, such as common knowledge in a group of agents or non-deterministic iteration of actions. Normal modal logics with such modalities are well-explored but weaker logics are not. We add a reflexive transitive closure box modality to the modal non-associative commutative full Lambek calculus with a simple negation. Decidability and weak completeness of the resulting system are established and extensions of the results to stronger substructural logics are discussed. As a special case, we obtain decidability and weak completeness for intuitionistic modal logic with the reflexive transitive closure box.
This work has been supported by the joint project of the German Science Foundation (DFG) and the Czech Science Foundation (GA ČR) number 16-07954J (SEGA: From shared evidence to group attitudes). The author would like to thank the anonymous reviewers for a number of suggestions, and to Adam Přenosil for reading a draft of the paper. A preliminary version of the paper was presented at the 8th International Workshop on Logic and Cognition in Guangzhou, China; the author is indebted to the audience for valuable feedback.
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Notes
- 1.
Due to space limitations, we do not provide an introduction to substructural logics and their relational semantics. See [9], for example.
- 2.
See [8], for example.
- 3.
Stated more precisely, counterexamples to the K-axiom can be constructed if the frame property \(Syx \implies Rxxx\) fails. Similarly, counterexamples to distributivity of \(\Box \) over \(\otimes \) can be found if we have Swx and Ryzx but also \(Ry'z'w\) and \(Sy'u\) with \(y \not \le u\) for some u. Counterexamples to the converse implication can be found if a similar frame condition holds.
- 4.
The reason is that if \(\Box ^{*}\varphi \in \varGamma \), then \(\varphi , \Box ^{n} \varphi \in \varGamma \) for all \(n \in \omega \) by \((*3 )\), but the converse implication cannot be established (our axiomatization is finitary).
- 5.
Note that \((\varphi \otimes \psi ) \otimes \chi \rightarrow \varphi \otimes (\psi \otimes \chi )\) is valid in M if the model satisfies this frame condition. On the other hand, (5) entails the validity of \(\varphi \otimes \psi \rightarrow \psi \otimes \varphi \).
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Sedlár, I. (2017). Substructural Logics with a Reflexive Transitive Closure Modality. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_25
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DOI: https://doi.org/10.1007/978-3-662-55386-2_25
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