
Syntactic cutelimination and backward proofsearch for tense logic via linear nested sequents (Extended version)
We give a linear nested sequent calculus for the basic normal tense logi...
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The MultiplicativeAdditive Lambek Calculus with Subexponential and Bracket Modalities
We give a prooftheoretic and algorithmic complexity analysis for system...
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A sequent calculus for a semiassociative law
We introduce a sequent calculus with a simple restriction of Lambek's pr...
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Reasoning about Unreliable Actions
We analyse the philosopher Davidson's semantics of actions, using a stro...
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Proof Theory of Partially Normal Skew Monoidal Categories
The skew monoidal categories of Szlachányi are a weakening of monoidal c...
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The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them
In sequent calculi, cut elimination is a property that guarantees that a...
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A Light Modality for Recursion
We investigate the interplay between a modality for controlling the beha...
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Reconciling Lambek's restriction, cutelimination, and substitution in the presence of exponential modalities
The Lambek calculus can be considered as a version of noncommutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the socalled "Lambek's restriction," that is, the antecedent of any provable sequent should be nonempty. In this paper we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek's restriction. Interestingly enough, we show that for any system equipped with a reasonable exponential modality the following holds: if the system enjoys cut elimination and substitution to the full extent, then the system necessarily violates Lambek's restriction. Nevertheless, we show that two of the three conditions can be implemented. Namely, we design a system with Lambek's restriction and cut elimination and another system with Lambek's restriction and substitution. For both calculi we prove that they are undecidable, even if we take only one of the two divisions provided by the Lambek calculus. The system with cut elimination and substitution and without Lambek's restriction is folklore and known to be undecidable.
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