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Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) |
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We show that the lattices of the normal extensions of four well-known logics—propositional intuitionistic logic \, Grzegorczyk logic \, modalized Heyting calculus \ and \—can be joined in a commutative diagram. One connection of this diagram is an isomorphism between the lattices of the normal extensions of \ and \; we show some preservation properties of this isomorphism. Two other connections are join semilattice epimorphims of the lattice of the normal extensions of \ onto that of \ and of the (...) |
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We will outline the contributions of A.V. Kuznetsov to modal logic. In his research he focused mainly on semantic, i.e. algebraic, issues and lattices of extensions of particular modal logics, though his proof of the Full Conservativeness Theorem for the proof-intuitionistic logic KM (Theorem 17 below) is a gem of proof-theoretic art. |
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The fact that many modal operators are part of an adjunction is probably folklore since the discovery of adjunctions. On the other hand, the natural idea of a minimal propositional calculus extended with a pair of adjoint operators seems to have been formulated only very recently. This recent research, mainly motivated by applications in computer science, concentrates on technical issues related to the calculi and not on the significance of adjunctions in modal logic. It then seems a worthy enterprise (both (...) |
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Let |$\textbf{A}$| be a |$BCK$|-algebra and |$f:A^{k}\rightarrow A$| a function. The main goal of this article is to give a necessary and sufficient condition for |$f$| to be compatible with respect to every relative congruence of |$\textbf{A}$|. We extend this result in some related algebras, as e.g. in pocrims. |
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In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions. |
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Gödel logic is the extension of intuitionistic logic by the linearity axiom. Symmetric Gödel logic is a logical system, the language of which is an enrichment of the language of Gödel logic with their dual logical connectives. Symmetric Gödel logic is the extension of symmetric intuitionistic logic. The proof-intuitionistic calculus, the language of which is an enrichment of the language of intuitionistic logic by modal operator was investigated by Kuznetsov and Muravitsky. Bimodal symmetric Gödel logic is a logical system, the (...) |
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We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Gödel–Löb logic GL (provability logic) and Grzegorczyk’s logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically interesting, requiring (...) |
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We present some results on algebraic and modal analysis of polynomial distortions of the standard provability predicate in Peano Arithmetic PA, and investigate three provability-like modal systems related to the Gödel–Löb modal system GL. We also present a short review of relational and topological semantics for these systems, and describe the dual category of algebraic models of our main modal system. |
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We study some operations that may be defined using the minimum operator in the context of a Heyting algebra. Our motivation comes from the fact that 1) already known compatible operations, such as the successor by Kuznetsov, the minimum dense by Smetanich and the operation G by Gabbay may be defined in this way, though almost never explicitly noted in the literature; 2) defining operations in this way is equivalent, from a logical point of view, to two clauses, one corresponding (...) |
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We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider (...) |
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In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation $${\tau(a) \leq b \vee (b \rightarrow a)}$$, for all $${a, b \in A}$$. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia (...) |
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In this paper we shall prove that certain subvarieties of the variety of Salgebras (Heyting algebras with successor) has amalgamation. This result together with an appropriate version of Theorem 1 of [L. L. Maksimova, Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras, Algebra i Logika, 16(6):643-681, 1977] allows us to show interpolation in the calculus IPC S (n), associated with these varieties.We use that every algebra in any of the varieties of S-algebras studied in this work has (...) |
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This work extend to residuated lattices the results of [ 7 ]. It also provides a possible generalization to this context of frontal operators in the sense of [ 9 ]. Let L be a residuated lattice, and f : L k → L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L . We use this characterization of compatible functions in order to prove that the variety of (...) |
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This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) |
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