Here, we provide a detailed description of the mutual relation of formulas with finite propositional variables p1, …, pm in modal logic S4. Our description contains more information on S4 than those given in Shehtman (1978) and Moss (2007); however, Shehtman (1978) also treated Grzegorczyk logic and Moss (2007) treated many other normal modal logics. Specifically, we construct normal forms, which behave like the principal conjunctive normal forms in the classical propositional logic. The results include finite and effective methods to (...) find a normal form equivalent to a given formula A by clarifying the behavior of connectives and giving a finite method to list all exact models. (shrink)

We construct the finitely generated free algebras and determine the free spectra of varieties of linear equivalential algebras and linear equivalential algebras of finite height corresponding. respectively, to the equivalential fragments of intermediate Gödel-Dummett logic and intermediate finite-valued logics of Gödel. Thus we compute the number of purely equivalential propositional formulas in these logics in n variables for an arbitrary n ∈ N.

A deductive system (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas.

A deductive system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{S}$$ \end{document} (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{ \left\langle {\alpha,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \},$$ \end{document}is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the (...) deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective. (shrink)

Sahlqvist theory is extended to the fragments of the intuitionistic propositional calculus that include the conjunction connective. This allows us to introduce a Sahlqvist theory of intuitionistic character amenable to arbitrary protoalgebraic deductive systems. As an application, we obtain a Sahlqvist theorem for the fragments of the intuitionistic propositional calculus that include the implication connective and for the extensions of the intuitionistic linear logic.

In the paper we consider complexity of intuitionistic propositional logic and its natural fragments such as implicative fragment, finite-variable fragments, and some others. Most facts we mention here are known and obtained by logicians from different countries and in different time since 1920s; we present these results together to see the whole picture.

In this paper we develop a theory of Bosbach and Riečan states on non-trivial Hilbert algebras.

In her well-known book, Rasiowa states without proof that in implicative algebras there is a one-to-one correspondence between kernels of epimorphisms and the so-called special implicative filters, and that in the logic whose algebraic counterpart is the class of implicative algebras the deductive filters coincide with the special implicative filters. We show that neither claim is true, and how to repair the situation by redefining some of the notions involved. We answer other questions concerning special implicative filters, taking the theory (...) of algebraizable logics of Blok and Pigozzi as a framework to approach the question in a systematic way. (shrink)

In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been (...) called ‘Hilbert algebras with infimum’. (shrink)

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧, →) homomorphisms, (∧, →, 0) homomorphisms, and (∧, →, ∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev’s (...) theorem that each intermediate logic can be axiomatized by canonical formulas. (shrink)