Hiroakira Ono

Logics Without the Contraction Rule

if we formulate our logics in a Gentzen-type formal system. Some syntactical properties of these logics have been studied firstly by the second author in [11], in connection with the study of BCK-algebras (for information on BCK-algebras, see [9]). There, it turned out that such a syntactical method is a powerful and promising tool in studying BCK-algebras. Using this method, considerable progress has been made since then (see, e.g., [8], [18], [27]). In this paper, we will study these logics more comprehensively. We notice here that the distributive law

Substructural Logics and Residuated Lattices — an Introduction

This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. Then, residuated lattices are introduced as algebraic structures for substructural logics, and some recent developments of their algebraic study are presented. Based on these facts, we conclude at the end that substructural logics are logics of residuated structures, and in this way explain why sequent systems are suitable for formalizing substructural logics.

Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

On the size of refutation Kripke models for some linear modal and tense logics

LetL be any modal or tense logic with the finite model property. For eachm, definerL(m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionrL determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofrL(m) for some linear modal and tense logicsL.

Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo's Logic MTL∀

The present paper deals with the predicate version MTL∀ of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Onos Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL∀ and classical predicate logic is undecidable. Finally, we prove that MTL∀ is complete with respect to the standard semantics, i.e., with respect to Kripke frames on the real interval [0,1], or equivalently, with respect to MTL-algebras whose lattice reduct is [0,1] with the usual order.

Structural Rules and a Logical Hierarchy

Gentzen-type sequent calculi usually contain three structural rules, i.e., exchange, contraction and weakening rules. In recent years, however, there have been various studies on logics that have not included some or any of these structural rules. The motives or purposes of these studies have been so diverse that sometimes close connections between them have been overlooked. Here we will make a brief survey of recent results on these logics in an attempt to make these interrelationships clearer.

Algebraic aspects of cut elimination

We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17].

Interpolation and the Robinson property for logics not closed under the Boolean operations

A variant of the Robinson property (ROB*) is introduced. The property ROB* is equivalent to the usual Robinson property and hence is equivalent also to the interpolation property, in any compact logic closed under the Boolean operations. On the other hand, it will be shown that ROB* is not always equal to the interpolation property, if a logic is not closed under them. Next, we will study ROB* and various interpolation properties for equational logics, which are typical examples of compact logics not closed under the Boolean operations. It will be shown that an equational logic has ROB* if and only if it has the amalgamation property for isomorphic embeddings.

Interpolation Properties, Beth Definability Properties and Amalgamation Properties for Substructural Logics

This article develops a comprehensive study of various types of interpolation properties and Beth definability properties (BDPs) for substructural logics, and their algebraic characterizations through amalgamation properties (APs) and epimorphisms surjectivity. In general, substructural logics are algebraizable but lack many of the basic logical properties that modal and superintuitionistic logics enjoy [Gabbay and Maksimova (2005, Oxford Logic Guides, Vol. 46)]. In this case, careful examination is necessary to see how these logical and algebraic properties are related. To describe these relations exactly, many variants of interpolation properties and BDPs, and also corresponding algebraic properties, are introduced. Because of their generality, the results reported here hold not only for substructural logics, but can also be extended to a more general setting such as abstract algebraic logic [Andreka, Nemeti and Sain (Handbook of Philosophical Logic, Vol. 2, 2nd edn, pp. 133–247) and Czelakowski and Pigozzi (1999, Vol. 203 of Lecture Notes in Pure and Applied Mathematics, pp. 187–265)].

Closure Operators and Complete Embeddings of Residuated Lattices

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.