Showing papers on "Normal modal logic published in 1977"

The Computational Complexity of Provability in Systems of Modal Propositional Logic

Abstract: The computational complexity of the provability problem in systems of modal propositional logic is investigated. Every problem computable in polynomial space is $\log $ space reducible to the provability problem in any modal system between K and $S4$. In particular, the provability problem in K, T, and $S4$ are $\log $ space complete in polynomial space. The nonprovability problem in $S5$ is $\log $ space complete in nondeterministic polynomial time.

Propositional modal logic of programs

Abstract: We introduce a fundamental propositional logical system for describing correctness, termination and equivalence of programs. We define a formal syntax and semantics for the propositional modal logic of programs and give several consequences of the definition. Principal conclusions are that deciding satisfiability requires time dn/log nfor some d > 1 and that satisfiability, even in an extended system, can be decided in nondeterministic time cnfor some c. We provide applications of the decision procedure to regular expressions, Ianov schemes, and classical systems of modal logic.

On Some Intuitionistic Modal Logics

Abstract: Some modal logics based on logics weaker than the classical logic have been studied by Fitch [4], Prior [7], Bull [1], [2], [3], Prawitz [6] etc. Here we treat modal logics based on the intuitionistic propositional logic, which call intuitionistic modal logics (abbreviated as IML’s). Let H be the intuitionistic propositional logic formulated in the Hilbertstyle. The rules of inference of H are modus ponens and the rule of substitution. The IML L0 is obtained from H by adding the following three axioms,

On modal logic with an intuitionistic base

Abstract: A definition of the concept of “Intuitionist Modal Analogue” is presented and motivated through the existence of a theorem preserving translation fromMIPC (see [2]) to a bimodalS4–S5 calculus.

A Study of Kripke-type Models for Some Modal Logics by Gentzen's Sequential Method

Abstract: Page INTRODUCTION 382 Chapter 1 THE FORMAL SYSTEMS 385 1.1. Basic Language 385 1.2. Languages 385 1.3. Well Formed Formulas 386 1.4. Hilbert-type Systems 387 1.5. Gentzen-type Systems 390 1.6. Some Metatheorems 393 Chapter 2 TOPOLOGY ON 2 396 2.1. Definition of Topology 397 2.2. Topological Characterization of Syntactical Properties 398 Chapter 3 KRIPKE-TYPE SEMANTICS 400 3.1. Definition of Kripke-type Models 400 3.2. Soundness of KT/-models 401 3.3. Completeness of KT/-mode1s 403 3.4. Cut-free System for S5 , 407 3.5. Cut-elimination Theorem for GT3 and GT4 412 Chapter 4 CATEGORIES OF KRIPKE MODELS 415 4.1. Definition of Jft(£) .... 415 4.2. Properties of .#%(£) 416 4.3. Structure of jrt(Q) 420 Chapter 5 S5 MODEL THEORY 421 5.1. Lindenbaum Algebra of KTi 422 5.2. S5 Model Theory 424 Chapter 6 APPLICATIONS 434 6.1. The Wise Men Puzzle 434 6.2. The Puzzle of Unfaithful Wives 439 6.2.1. Knowledge Set and Knowledge Base 439

Imbedding of the Quantum Logic in the Modal System of Brower

Abstract: We lean upon the usual variant of the logic of quantum mechanics [1] Here the propositions correspond to the results of the quantum experiments A beautiful essay [2] may be connected with a conservation of semantics But we try to broaden the semantics, admitting propositions about the possibility of results of experiments Doing so, we fulfil the old wish of W A Fock, who attracts our attention to the importance of the modal categories for the interpretation of the quantum theory [3] ?1 Modal system We begin with the formal description of the modal system Br' The alphabet contains the signs - , v and D1 (negation, alternative and the sign of necessity), the set V of the propositional variables and parentheses The rules of formation are: if A E V, then A is a proposition; if X and Y are propositions, then - X, X v Y and DX are propositions also; there are no other propositions X3 will design the set of all propositions

On some Modal Logical Systems Defined in Connexion with Jaśakowski's Problem

Abstract: Publisher Summary This chapter focuses on some modal logical systems defined in connection with Jaśkowski's problem. Jaśkowski formulated the problem of finding logical systems that could be employed as underlying logics of deductive systems not devoid of inconsistency and presented one solution of the problem at the level of the propositional calculus. A normal modal system is a set of modal propositional formulas closed under substitution, detachement for material implication, and the rule of Godel. The chapter discusses various systems, properties of defined normal modal system, Kripke's system, axiomatics studies, Henkin semantics, semantical characterization, problems of normal modal system and its counterparts, and various theorems.

David Lewis's Semantics for Deontic Logic

Abstract: Numerous attempts to provide'semantic interpretations for deontic logic are based on the idea that a given state of affairs ought to be the case in this world if and only if it is the case in every morally perfect world. In classical versions of this theory, a 'morally perfect world' is simply defined as one in which all the particular obligations obtaining in the actual world are fulfilled.' Such theories have two disadvantages. First, as Richard Purtill has pointed out, it is extremely difficult to spell out the requirement that all obtaining obligations be satisfied in a morally perfect world.2 If a world contains no agents at all, are all our obligations fulfilled in it? If it contains agents who are not counterparts to the agents in this world, are our obligations fulfilled in it? Second, the value of such theories lies solely in their elucidation of the logical form of obligation statements, e.g., their explanation of what makes a set of obligation statements consistent. The application of such theories depends on prior possession of the list of true statements of particular obligations ('Jones ought to do A', 'Smith ought to do B', etc.). Since they do not offer an independent method for determining whether or not a given obligation obtains, they fail to provide illuminating reductive or eliminative definitions for statements of particular obligation. David Lewis has proposed a semantic theory which falls within this general tradition but employs an alternative definition of the morally best worlds.3 Because of this difference, his theory may escape Purtill's direct objections.4 In addition, this aspect of his theory makes possible the derivation of particular obligations, not from an initial list of such obligations, but rather from a statement of abstract moral principles together with the general notion of a possible world. Thus it may be interpreted as providing not only an account of the logical form of statements of particular obligations, but also a reductive definition of them -in terms of these other two concepts. Promising as this theory appears, I shall argue that it does not succeed, because it neglects the fact that

On detachment-substitutional formalization in normal modal logics

Abstract: The aim of this paper is to propose a criterion of finite detachment-substitutional formalization for normal modal systems. The criterion will comprise only those normal modal systems which are finitely axiomatizable by means of the substitution, detachment for material implication and Godel rules.