On Intermediate Predicate Logics of some Finite Kripke Frames,I. Levelwise Uniform Trees

An intermediate predicate logic L is called finite iff it is characterized by a finite partially ordered set M, i.e., iff L is the logic of the class of all predicate Kripke frames based on M. In this paper we study axiomatizability of logics of this kind. Namely, we consider logics characterized by finite trees M of a certain type (levelwise uniform trees) and establish the finite axiomatizability criterion for this case.     

Monadic Fuzzy Predicate Logics

Two variants of monadic fuzzy predicate logic are analyzed and compared with the full fuzzy predicate logic with respect to finite model property (properties) and arithmetical complexity of sets of tautologies, satisfiable formulas and of analogous notion restricted to finite models.     

Loosely Guarded Fragment of First-Order Logic has the Finite Model Property

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.     

About finite predicate logic

We say that an n-argument predicate P ⁿ is finite, if P is a finite set. Note that the set of individuals is infinite! Finite predicates are useful in data bases and in finite mathematics. The logic DBL proposed here operates on finite predicates only. We construct an imbedding for DBL in a special modal logic MPL. We prove that if a finite predicate is expressible in the classical logic, it is also expressible in DBL. Quantifiers are not necessary in DBL. Some simple algebraic properties of DBL are indicated.     

On the Existence of Continua of Logics Between Some Intermediate Predicate Logics

A method for constructing continua of logics squeezed between some intermediate predicate logics, developed by Suzuki [8], is modified and applied to intervals of the form [L, L+ ¬¬S], where Lis a predicate logic, Sis a closed predicate formula. This solves one of the problems from Suzuki's paper.     

Incompleteness and the Barcan formula

A (normal) system of prepositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systemswith the Barcan Formula which are incomplete, while those without are complete. In the second group it is thosewithout the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence.     

On the logic of adverbs

In the paper I investigate aspects of adverbial modification as an operation applying an adverb or adverbial phrase to a predicate and thereby creating a new predicate. The logic of adverbial modification, on this view, belongs to the logic of predicate modifiers. The theory I present is intended to cover not only adverbial modification but also attributive modification, but problems concerning the latter will not be given any special attention.     

The Logics of Strict-Tolerant Logic

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egré, Ripley and van Rooij have put forward an approach based on a non-transitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this logic. In this paper, we show that this logic, once it is adequately understood, is weaker than classical logic. Moreover, the logic is in a way similar to the paraconsistent logic LP.     

Glivenko Type Theorems for Intuitionistic Modal Logics

In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior's MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Intto MIPC. As a result we obtain two different versions of Glivenko's theorem for logics over MIPC. Since MIPCcan be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko's theorem for logics over MIPCis closely related to that for intermediate predicate logics obtained by Umezawa [27] and Gabbay [15]. Another one is rather surprising.     

Eine Auch Sich Selbst Missverstehende Kritik: Über das Reflexionsdefizit Formaler Explikationen

The criticism formulated by L. B. Puntel concerning the theory of dialectic proposed by the author is rejected. Puntel's attempt at explicating predication by means of (second order) predicate logic fails: It misjudges predication being already presupposed for the possibility of predicate logic, thus belonging to the transcendental conditions of formal predicate logic, so that predication itself cannot be further explicated by means of such logic. What is in fact criticized by Puntel is something like an artefact of formalization. The unreflected application of formal logic here generates problems instead of solving them.     

Existence and Predication from Aristotle to Frege

One of the characteristic features of contemporary logic is that it incorporates the Frege-Russell thesis according to which verbs for being are multiply ambiguous. This thesis was not accepted before the nineteenth century. In Aristotle existence could not serve alone as a predicate term. However, it could be a part of the force of the predicate term, depending on the context. For Kant existence could not even be a part of the force of the predicate term. Hence, after Kant, existence was left homeless. It found a home in the algebra of logic in which the operators corresponding to universal and particular judgments were treated as duals, and universal judgments were taken to be relative to some universe of discourse. Because of the duality, existential quantifier expressions came to express existence. The orphaned notion of existence thus found a new home in the existential quantifier.     

Embedding Logics into Product Logic

We construct a faithful interpretation of ukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.We prove a completeness theorem for product logic extended by a unary connective of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic.We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Gödel's logic).     

Frame Based Formulas for Intermediate Logics

In this paper we define the notion of frame based formulas. We show that the well-known examples of formulas arising from a finite frame, such as the Jankov-de Jongh formulas, subframe formulas and cofinal subframe formulas, are all particular cases of the frame based formulas. We give a criterion for an intermediate logic to be axiomatizable by frame based formulas and use this criterion to obtain a simple proof that every locally tabular intermediate logic is axiomatizable by Jankov-de Jongh formulas. We also show that not every intermediate logic is axiomatizable by frame based formulas. Presented by Johan van Benthem     

因果型连动结构的逻辑刻画

逻辑学以及形式语义学对汉语连动结构的研究一般都停留在一阶逻辑的层面上,即通过一阶逻辑中的形式化方法给出连动结构的形式化构造。但是在对不同类型连动结构进行更为深入的研究时,一阶逻辑这一工具就显的不够了。本文中,作者将以因果型连动结构为研究对象,利用一阶模态逻辑为其构建形式系统并尝试给出因果型连动结构和途径–目的型连动结构之间的区别和联系。     

The contraction rule and decision problems for logics without structural rules

This paper shows a role of the contraction rule in decision problems for the logics weaker than the intuitionistic logic that are obtained by deleting some or all of structural rules. It is well-known that for such a predicate logic L, if L does not have the contraction rule then it is decidable. In this paper, it will be shown first that the predicate logic FL_ec with the contraction and exchange rules, but without the weakening rule, is undecidable while the propositional fragment of FL_ec is decidable. On the other hand, it will be remarked that logics without the contraction rule are still decidable, if our language contains function symbols.     

Craig's interpolation theorem for the intuitionistic logic and its extensions—A semantical approach

A semantical proof of Craig's interpolation theorem for the intuitionistic predicate logic and some intermediate prepositional logics will be given. Our proof is an extension of Henkin's method developed in [4]. It will clarify the relation between the interpolation theorem and Robinson's consistency theorem for these logics and will enable us to give a uniform way of proving the interpolation theorem for them.The results in this paper were reported at Symposium on Mathematical Logic held at Shizuoka in December 1982.     

Faithfulness for naive validity

Nontransitive responses to the validity Curry paradox face a dilemma that was recently formulated by Barrio, Rosenblatt and Tajer. It seems that, in the nontransitive logic ST enriched with a validity predicate, either you cannot prove that all derivable metarules preserve validity, or you can prove that instances of Cut that are not admissible in the logic preserve validity. I respond on behalf of the nontransitive approach. The paper argues, first, that we should reject the detachment principle for naive validity. Secondly, I show how to add a validity predicate to ST while avoiding the dilemma.

     

Constructing a continuum of predicate extensions of each intermediate propositional logic

Wajsberg and Jankov provided us with methods of constructing a continuum of logics. However, their methods are not suitable for super-intuitionistic and modal predicate logics. The aim of this paper is to present simple ways of modification of their methods appropriate for such logics. We give some concrete applications as generic examples. Among others, we show that there is a continuum of logics (1) between the intuitionistic predicate logic and the logic of constant domains, (2) between a predicate extension ofS4 andS4 with the Barcan formula. Furthermore, we prove that (3) there is a continuum of predicate logics with equality whose equality-free fragment is just the intuitionistic predicate logic.Dedicated to the memory of the late Professor S. MaeharaThis research was supported in part by Grant-in Aid for Encouragement of Young Scientists No. 06740140, Ministry of Education, Science and Culture, Japan.Presented byHiroakira Ono     

Dialektik Und Formalisierung

Dialectics and Formalization. In an appendix to his article Can the concept of dialectic be made clear? (JGPS 27: 131–165) the author critically examined an attempt by D. Wandschneider of reconstructing Hegelian dialectical logic by displaying antinomic structures. The main steps of this reconstruction were formalized within the framework of second order predicate logic. In his reply (JGPS 27: 347–352) Wandschneider argues that formalization is the wrong way of interpreting and judging his reconstruction project on the ground that the understanding of predication is already presupposed by predicate logic and, therefore, cannot be formalized. In the present contribution it is shown that Wandschneider's criticism is based on a fundamentally mistaken conception of formalization and formal logic and that the problems posed by his project remain completely unresolved.     

On maximal intermediate logics with the disjunction property

For intermediate logics, there is obtained in the paper an algebraic equivalent of the disjunction propertyDP. It is proved that the logic of finite binary trees is not maximal among intermediate logics withDP. Introduced is a logicND, which has the only maximal extension withDP, namely, the logicML of finite problems.