The system of intuitionistic modal logic $\textbf{IEL}^{-}$ was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic (S. Artemov and T. Protopopescu. Intuitionistic epistemic logic. The Review of Symbolic Logic, 9, 266–298, 2016). We construct the modal lambda calculus, which is Curry–Howard isomorphic to $\textbf{IEL}^{-}$ as the type-theoretical representation of applicative computation widely known in functional programming.We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a Cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study complete Kripke–Joyal-style semantics for predicate extensions of $\textbf{IEL}^{-}$ and related logics using Dedekind–MacNeille completions and modal cover systems introduced by Goldblatt (R. Goldblatt. Cover semantics for quantified lax logic. Journal of Logic and Computation, 21, 1035–1063, 2011). The paper extends the conference paper published in the LFCS’20 volume (D. Rogozin. Modal type theory based on the intuitionistic modal logic IEL. In International Symposium on Logical Foundations of Computer Science, pp. 236–248. Springer, 2020).
In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Hirsch and Hodkinson (2002, Relation Algebras by Games). The finite representation property for residuated semigroups also implies that the Lambek calculus has the finite model property with respect to relational models, the so-called $R$-models. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatization. For this purpose, we introduce representability games for join semilattice-ordered semigroups.
In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov [10] and its quantale semantics. In our approach, subexponential modalities have an interpretation in terms of quantic conuclei. We show that this extension of Lambek calculus is complete w.r.t quantales with quantic conuclei. Also, we prove a representation theorem for quantales with quantic conuclei and show that Lambek calculus with subexponentials is relationally complete. Finally, we extend this representation theorem to the category of quantales with quantic conuclei. Some of these results were presented here [20].Here we assign the special syntactic categories to the words of this sentence. np denotes "noun phrase", n -noun, ad -adjective, p -phrase, s -sentence. This sequent denotes that this Oscar Wilde's quote is a well-formed sentence. The verb "changed" has type "np under (s over p)". In other words, one needs to apply some noun phrase ("The Thames nocturne of blue and gold") from the left, apply some phrase from the right ("Changed to Harmony in grey") and obtain sentence after that. The other syntactic categories might be considered similarly.The general case of such derivations in categorial grammars is axiomatised via Lambek calculus [13], non-commutative linear logic: Definition 1. Lambek calculus (the system L) ax
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