In this paper we study the variety \(\textsf{WL}\) of bounded distributive lattices endowed with an implication, called weak Lewis distributive lattices. This variety corresponds to the algebraic semantics of the \(\{\vee,\wedge,\Rightarrow,\bot,\top \}\) -fragment of the arithmetical base preservativity logic \(\mathsf {iP^{-}}\). The variety \(\textsf{WL}\) properly contains the variety of bounded distributive lattices with strict implication, also known as weak Heyting algebras. We introduce the notion of WL-frame and we prove a representation theorem for WL-lattices by means of WL-frames. We extended (...) this representation to a topological duality by means of Priestley spaces endowed with a special neighbourhood relation between points and closed upsets of the space. These results are applied in order to give a representation and a topological duality for the variety of weak Heyting–Lewis algebras, i.e., for the algebraic semantics of the arithmetical base preservativity logic \(\textsf{iP}^{-}\). (shrink)

This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.

Standard unary modal logic and binary modal logic, i.e. modal logic with one binary operator, are shown to be definitional extensions of one another when an additional axiom |$U$| is added to the basic axiomatization of the binary side. This is a strengthening of our previous results. It follows that all unary modal logics extending Classical Modal Logic, in other words all unary modal logics with a neighborhood semantics, can equivalently be seen as binary modal logics. This in particular applies (...) to standard modal logics, which can be given simple natural axiomatizations in binary form. We illustrate this in the logic K. We call such logics binary expansions of the unary modal logics. There are many more such binary expansions than the ones given by the axiom |$U$|⁠. We initiate an investigation of the properties of these expansions and in particular of the maximal binary expansions of a logic. Our results directly imply that all sub- and superintuitionistic logics with a standard modal companion also have binary modal companions. The latter also applies to the weak subintuitionistic logic WF of our previous papers. This logic doesn’t seem to have a unary modal companion. (shrink)

The present paper introduces and studies the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋn of n-linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋ2 that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic algebras introduced in [2].

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 (...) in [ 10 ]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces $${\langle X, \leq, T, R \rangle}$$ where $${\langle X, \leq, T \rangle}$$ is a WH -space [ 6 ], and R is an additional binary relation used to interpret the modal operator. We will also study the WH -algebras with successor and the WH -algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH -spaces. (shrink)

An l-hemi-implicative semilattice is an algebra \\) such that \\) is a semilattice with a greatest element 1 and satisfies: for every \, \ implies \ and \. An l-hemi-implicative semilattice is commutative if if it satisfies that \ for every \. It is shown that the class of l-hemi-implicative semilattices is a variety. These algebras provide a general framework for the study of different algebras of interest in algebraic logic. In any l-hemi-implicative semilattice it is possible to define an (...) derived operation by \ \wedge \). Endowing \\) with the binary operation \ the algebra \\) results an l-hemi-implicative semilattice, which also satisfies the identity \. In this article, we characterize the commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element. Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices. (shrink)

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A built up of atoms p1,..., pn, BPC⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A if and only if for all arithmetical sentences B1,..., Bn, BA⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A. The technique used in our proof can easily be applied to some known extensions of BA.

A latarre is a lattice with an arrow. Its axiomatization looks natural. Latarres have a nontrivial theory which permits many constructions of latarres. Latarres appear as an end result of a series of generalizations of better known structures. These include Boolean algebras and Heyting algebras. Latarres need not have a distributive lattice.

Kolmogorov established the principle of the double negation translation by which to embed Classical Predicate Logic |${\operatorname {CQC}}$| into Intuitionistic Predicate Logic |${\operatorname {IQC}}$|⁠. We show that the obvious generalizations to the Basic Predicate Logic of [3] and to |${\operatorname {BQC}}$| of [12], a proper subsystem of |${\operatorname {IQC}}$|⁠, go through as well. The obvious generalizations of Kuroda’s embedding are shown to be equivalent to the Kolmogorov variant. In our proofs novel nontrivial techniques are needed to overcome the absence of (...) full modus ponens in Basic Predicate Logic. In [3] we argued that |${\operatorname {IQC}}$| is not the logic of constructive mathematics. Our doubts were far from new. New was that we put forward an alternative, |${\operatorname {BQC}}$|⁠. One concern is that |${\operatorname {BQC}}$| is too weak for serious mathematics, or even trivial. This paper is one step to alleviate such concerns. (shrink)

Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.

We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra.

In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.

We generalize the double negation construction of Boolean algebras in Heyting algebras to a double negation construction of the same in Visser algebras. This result allows us to generalize Glivenko’s theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras.

We study Basic algebra, the algebraic structure associated with basic propositional calculus, and some of its natural extensions. Among other things, we prove the amalgamation property for the class of Basic algebras, faithful Basic algebras and linear faithful Basic algebras. We also show that a faithful theory has the interpolation property if and only if its correspondence class of algebras has the amalgamation property.

We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.

This article is about Avicenna’s account of syllogisms comprising opposite premises. We examine the applications and the truth conditions of these syllogisms. Finally, we discuss the relation between these syllogisms and the principle of non-contradiction.

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 (...) to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues. (shrink)

We establish a completeness theorem for first-order basic predicate logic BQC, a proper subsystem of intuitionistic predicate logic IQC, using Kripke models with transitive underlying frames. We develop the notion of functional well-formed theory as the right notion of theory over BQC for which strong completeness theorems are possible. We also derive the undecidability of basic arithmetic, the basic logic equivalent of intuitionistic Heyting Arithmetic and classical Peano Arithmetic.

This paper is a comparative study of the propositional intuitionistic (non-modal) and classical modal languages interpreted in the standard way on transitive frames. It shows that, when talking about these frames rather than conventional quasi-orders, the intuitionistic language displays some unusual features: its expressive power becomes weaker than that of the modal language, the induced consequence relation does not have a deduction theorem and is not protoalgebraic. Nevertheless, the paper develops a manageable model theory for this consequence and its extensions (...) which also reveals some unexpected phenomena. The balance between the intuitionistic and modal languages is restored by adding to the former one more implication. (shrink)

It is shown that all the provably total functions of Basic Arithmetic BA, a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which BA is sound, is introduced. This realizability is similar to Kleene's recursive realizability, except that recursive functions are restricted to primitive recursives.

In the present paper we study systematically several consequence relations on the usual language of propositional intuitionistic logic that can be defined semantically by using Kripke frames and the same defining truth conditions for the connectives as in intuitionistic logic but without imposing some of the conditions on the Kripke frames that are required in the intuitionistic case. The logics so obtained are called subintuitionistic logics in the literature. We depart from the perspective of considering a logic just as a (...) set of theorems and also depart from the perspective taken by Restall in that we consider standard Kripke models instead of models with a base point. We study the relations between subintuitionistic logics and modal logics given by the translation considered by Došen. Moreover, we classify the logics obtained according to the hierarchy considered inAlgebraic Logic. (shrink)

This paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation (...) called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic . Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: it is injective, it is isotonic, it commutes with the inverse image operator of any algebraic homomorphism, and it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties –. (shrink)

We use the left self‐distributive axiom to introduce and study a special class of weak Heyting algebras, called self‐distributive weak Heyting algebras (SDWH‐algebras). We present some useful properties of SDWH‐algebras and obtain some equivalent conditions of them. A characteristic of SDWH‐algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH‐algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety of basic algebras, the (...) variety of subresiduated lattices, the variety of reflexive WH‐algebras (RWH‐algebras), and the variety of transitive WH‐algebras (TWH‐algebras). (shrink)

Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras.algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety ������ the lattice of congruences of A is isomorphic to the lattice of deductive filters on (...) A of the τ-assertional logic of ������. Moreover, if ������ has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ ������ of the 1-assertional logic of ������ coincide with the ������-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics. (shrink)

We introduce a Gentzen style formulation of Basic Propositional Calculus(BPC), the logic that is interpreted in Kripke models similarly tointuitionistic logic except that the accessibility relation of eachmodel is not necessarily reflexive. The formulation is presented as adual-context style system, in which the left hand side of a sequent isdivided into two parts. Giving an interpretation of the sequents inKripke models, we show the soundness and completeness of the system withrespect to the class of Kripke models. The cut-elimination theorem isproved (...) in a syntactic way by modifying Gentzen's method. Thisdual-context style system exemplifies the effectiveness of dual-contextformulation in formalizing various non-classical logics. (shrink)

This paper introduces sequent systems for Visser's two propositional logics: Basic Propositional Logic (BPL) and Formal Propositional Logic (FPL). It is shown through semantical completeness that the cut rule is admissible in each system. The relationships with Hilbert-style axiomatizations and with other sequent formulations are discussed. The cut-elimination theorems are also demonstrated by syntactical methods.

Most Gentzen systems arising in logic contain few axiom schemata and many rule schemata. Hilbert systems, on the other hand, usually contain few proper inference rules and possibly many axioms. Because of this, the two notions tend to serve different purposes. It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation ‘for’ the logic may be sought—or vice versa. Where this has occurred, the word ‘for’ has taken (...) on several different meanings, partly because the Gentzen separator ⇒ can be interpreted intuitively in a number of ways. Here ⇒ will be denoted less evocatively by ⊲.In this paper we aim to discuss some of the useful ways in which Gentzen and Hilbert systems may correspond to each other. Actually, we shall be concerned with thededucibility relationsof the formal systems, as it is these that are susceptible to transformation in useful ways. To avoid potential confusion, we shall speak of Hilbert and Gentzenrelations. By aHilbert relationwe mean any substitution-invariant consequence relation onformulas—this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By aGentzen relationwe mean the fully fledged generalization of this notion in whichsequentstake the place of single formulas. In the literature, Hilbert relations are often referred to assentential logics. Gentzen relations as defined here are their exactsequentialcounterparts. (shrink)

Two cut-free sequent calculi which are conservative extensions of Visser's Formal Propositional Logic are introduced. These satisfy a kind of subformula property and by this property the interpolation theorem for FPL are proved. These are analogies to Aghaei-Ardeshir's calculi for Visser's Basic Propositional Logic.

This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.