Showing papers on "Modal operator published in 1977"

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.

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.

Modal Operators and Functional Completeness, II

Abstract: In the Kripke semantics for propositional modal logic, a frame W = (W, <) represents a set of "possible worlds" and a relation of "accessibility" between possible worlds. With respect to a fixed frame IW, a proposition is represented by a subset of W (regarded as the set of worlds in which the proposition is true), and an n-ary connective (i.e. a waybf forming a new proposition from an ordered n-tuple of given propositions) is represented by a function fW: (p(W))y -* P(W). Finally a state of affairs (i.e. a consistent specification whether or not each proposition obtains) is represented by an ultrafilter over W. {To avoid possible confusion, the reader should forget that some people prefer the term "states of affairs" for our "possible worlds".} In a broader sense, an n-ary connective is represented by an n-ary operator f = {fw 'I E Fr}, where Fr is the class of all frames and each fW (P(W))n -* P(W). A connective is modal if it corresponds to a formula of propositional modal logic. A connective C is coherent if whether C(Pi, * *, Pn) is true in a possible world depends only upon which modal combinations of P1, * * , Pn are true in that world. (A modal combination of P1, * * *, Pn is the result of applying a modal connective to P1, * * , Pn.) A connective C is strongly coherent if whether C(P1, * * , Pn) obtains in a state of affairs depends only upon which modal combinations of P1, * * *, Pn obtain in that state of affairs. In ?1 we study the classes of modal, coherent, and strongly coherent connectives, via the corresponding classes of operators. We shall characterize model-theoretically (i.e. without reference to the formulas of modal logic) these classes of operators, and we shall see that "modal" and "strongly coherent" are equivalent, and that they imply "coherent" but not conversely. A (normal modal propositional) logic L is functionally complete if every coherent operator on Fr(L), the class of all frames for L, is modal. The usual functional completeness theorems for the classical propositional calculus (every truth table is realized by a formula) and for S5 (every array of partial truth tables is realized by a formula [3, ?38.0]) may be interpreted as asserting the functional completeness, in the present sense, of CPC (i.e. K + (p <-+ Lp)) and of S5. In ?2 we shall determine which logics are functionally complete.

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.