Gentzen-Style Sequent Calculus for Semi-intuitionistic Logic

The variety \({\mathcal{SH}}\) of semi-Heyting algebras was introduced by H. P. Sankappanavar (in: Proceedings of the 9th “Dr. Antonio A. R. Monteiro” Congress, Universidad Nacional del Sur, Bahía Blanca, 2008) [13] as an abstraction of the variety of Heyting algebras. Semi-Heyting algebras are the algebraic models for a logic HsH, known as semi-intuitionistic logic, which is equivalent to the one defined by a Hilbert style calculus in Cornejo (Studia Logica 98(1–2):9–25, 2011) [6]. In this article we introduce a Gentzen style sequent calculus GsH for the semi-intuitionistic logic whose associated logic GsH is the same as HsH. The advantage of this presentation of the logic is that we can prove a cut-elimination theorem for GsH that allows us to prove the decidability of the logic. As a direct consequence, we also obtain the decidability of the equational theory of semi-Heyting algebras.     

The Lattice of Subvarieties of {\sqrt{\prime}} quasi-MV Algebras

In the present paper we continue the investigation of the lattice of subvarieties of the variety of ${\sqrt{\prime}}$ quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that the variety generated by the standard disk algebra D _r is not finitely based, and we provide an infinite equational basis for the same variety.     

Frontal Operators in Weak Heyting Algebras

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.     

Classical Modal De Morgan Algebras

In this note we introduce the variety ${{\mathcal C}{\mathcal D}{\mathcal M}_\square}$ of classical modal De Morgan algebras as a generalization of the variety ${{{\mathcal T}{\mathcal M}{\mathcal A}}}$ of Tetravalent Modal algebras studied in [11]. We show that the variety ${{\mathcal V}_0}$ defined by H. P. Sankappanavar in [13], and the variety S of Involutive Stone algebras introduced by R. Cignoli and M. S de Gallego in [5], are examples of classical modal De Morgan algebras. We give a representation theory, and we study the regular filters, i.e., lattice filters closed under an implication operation. Finally we prove that the variety ${{{\mathcal T}{\mathcal M}{\mathcal A}}}$ has the Amalgamation Property and the Superamalgamation Property.     

A Preliminary Study of MV-Algebras with Two Quantifiers Which Commute

In this paper we investigate the class of MV-algebras equipped with two quantifiers which commute as a natural generalization of diagonal-free two-dimensional cylindric algebras (see Henkin et al., in Cylindric algebras, 1985). In the 40s, Tarski first introduced cylindric algebras in order to provide an algebraic apparatus for the study of classical predicate calculus. The diagonal–free two-dimensional cylindric algebras are special cylindric algebras. The treatment here of MV-algebras is done in terms of implication and negation. This allows us to simplify some results due to Di Nola and Grigolia (Ann Pure Appl Logic 128(1-3):125–139, 2004) related to the characterization of a quantifier in terms of some special sub-algebra associated to it. On the other hand, we present a topological duality for this class of algebras and we apply it to characterize the congruences of one algebra via certain closed sets. Finally, we study the subvariety of this class generated by a chain of length n + 1 (n <  ω). We prove that the subvariety is semisimple and we characterize their simple algebras. Using a special functional algebra, we determine all the simple finite algebras of this subvariety.     

Products of Modal Logics. Part 3: Products of Modal and Temporal Logics

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.     

Sentential logics and Maehara Interpolation Property

With each sentential logic C, identified with a structural consequence operation in a sentential language, the class Matr ^* (C) of factorial matrices which validate C is associated. The paper, which is a continuation of [2], concerns the connection between the purely syntactic property imposed on C, referred to as Maehara Interpolation Property (MIP), and three diagrammatic properties of the class Matr* (C): the Amalgamation Property (AP), the (deductive) Filter Extension Property (FEP) and Injections Transferable (IT). The main theorem of the paper (Theorem 2.2) is analogous to the Wroński's result for equational classes of algebras [13]. It reads that for a large class of logics the conjunction of (AP) and (FEP) is equivalent to (IT) and that the latter property is equivalent to (MIP).     

On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics.     

A Note on Neat Reducts

SC, CA, QA and QEA denote the class of Pinter’s substitution algebras, Tarski’s cylindric algebras, Halmos’ quasi-polyadic and quasi-polyadic equality algebras, respectively. Let . and . We show that the class of n dimensional neat reducts of algebras in K _m is not elementary. This solves a problem in [2]. Also our result generalizes results proved in [1] and [2]. Presented by Robert Goldblatt     

Nelson algebras through Heyting ones: I

The main aim of the present paper is to explain a nature of relationships exist between Nelson and Heyting algebras. In the realization, a topological duality theory of Heyting and Nelson algebras based on the topological duality theory of Priestley ([15], [16]) for bounded distributive lattices are applied. The general method of construction of spaces dual to Nelson algebras from a given dual space to Heyting algebra is described (Thm 2.3). The algebraic counterpart of this construction being a generalization of the Fidel-Vakarelov construction ([6], [25]) is also given (Thm 3.6). These results are applied to compare the equational category N of Nelson algebras and some its subcategories (and their duals) with the equational category H of Heyting algebras (and its dual). It is proved (Thm 4.1) that the category N is topological over the category H. The main results of this article are a part of theses of the author's doctoral dissertation at the Nicholas Copernicus University in 1984 (cpmp. [24]).Research partially supported by Polish Government Grant CPBP 08-15.     

A Temporal Semantics for Basic Logic

In the context of truth-functional propositional many-valued logics, Hájek’s Basic Fuzzy Logic BL [14] plays a major rôle. The completeness theorem proved in [7] shows that BL is the logic of all continuous t-norms and their residua. This result, however, does not directly yield any meaningful interpretation of the truth values in BL per se. In an attempt to address this issue, in this paper we introduce a complete temporal semantics for BL. Specifically, we show that BL formulas can be interpreted as modal formulas over a flow of time, where the logic of each instant is ?ukasiewicz, with a finite or infinite number of truth values. As a main result, we obtain validity with respect to all flows of times that are non-branching to the future, and completeness with respect to all finite linear flows of time, or to an appropriate single infinite linear flow of time. It may be argued that this reduces the problem of establishing a meaningful interpretation of the truth values in BL logic to the analogous problem for ?ukasiewicz logic.     

Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality.In part III we discuss applications in logic of the framework developed. Specifically, logics with restricted structural rules give rise to lattices with normal operators (in our sense), such as the Full Lambek algebras (F L-algebras) studied by Ono in [36]. Our Stone-type representation results can be then used to obtain canonical constructions of Kripke frames for such systems, and to prove a duality of algebraic and Kripke semantics for such logics.     

Basic properties of the equivalence

In this paper we investigate some basic semantic and syntactic conditions characterizing the equivalence connective. In particular we define three basic classes of algebras: the class of weak equivalential algebras, the class of equivalential algebras and the class of regular equivalential algebras (see [12]).Weak equivalential algebras can be used to study purely equivalential fragments of relevant logics and strict equivalential fragments of some modal logics (for investigations of strict implicational fragments of modal logics see [20]). Equivalential algebras are suitable to study purely equivalential fragment of BCI and BCK logic (see [21], p. 316). A subclass of the class of regular equivalential algebras is suitable to study equivalential fragments of ukasiewicz logics. Some subvarieties of the class of regular equivalential algebras provide natural semantics for equivalential fragments of the intuitionistic prepositional logic and various intermediate logics (see [13]).The last chapter is a selection of research problems which in the author's opinion are worth to be solved.     

On a Definition of a Variety of Monadic ℓ-Groups

In this paper we expand previous results obtained in [2] about the study of categorical equivalence between the category IRL ₀ of integral residuated lattices with bottom, which generalize MV-algebras and a category whose objects are called c-differential residuated lattices. The equivalence is given by a functor ${{\mathsf{K}^\bullet}}$ , motivated by an old construction due to J. Kalman, which was studied by Cignoli in [3] in the context of Heyting and Nelson algebras. These results are then specialized to the case of MV-algebras and the corresponding category ${MV^{\bullet}}$ of monadic MV-algebras induced by “Kalman’s functor” ${\mathsf{K}^\bullet}$ . Moreover, we extend the construction to ?-groups introducing the new category of monadic ?-groups together with a functor ${\Gamma ^\sharp}$ , that is “parallel” to the well known functor ${\Gamma}$ between ? and MV-algebras.     

Priestley duality for some subalgebra lattices

Priestley duality can be used to study subalgebras of Heyting algebras and related structures. The dual concept is that of congruence on the dual space and the congruence lattice of a Heyting space is dually isomorphic to the subalgebra lattice of the dual algebra. In this paper we continue our investigation of the congruence lattice of a Heyting space that was undertaken in [10], [8] and [12]. Our main result is a characterization of the modularity of this lattice (Theorem 2.12). Partial results about its complementedness are also given, and among other things a characterization of those finite Heyting algebras with a complemented subalgebra lattice (Theorem 3.5).     

Paraconsistent and Paracomplete Logics Based on k-Cyclic Modal Pseudocomplemented De Morgan Algebras

Studia Logica - The study of the theory of operators over modal pseudocomplemented De Morgan algebras was begun in papers [20] and [21]. In this paper, we introduce and study the class of modal...     

Expansions of Semi-Heyting Algebras I: Discriminator Varieties

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [48] and [50] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH. We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH. Finally, we apply these results to give bases for ??small?? subvarieties of BDQDSH, DQSSH, and DblSH.