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Cylindric modal logic

Published online by Cambridge University Press:  12 March 2014

Yde Venema
Yde Venema*
Affiliation:
Department of Mathematics and Computer Science, Free University, de Boelelaan 1081, 1081 HV Amsterdam, E-mail: yde@cs.vu.nl
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Abstract

Treating the existential quantification ∃νi as a diamond ♢i and the identity νi = νj as a constant δij, we study restricted versions of first order logic as if they were modal formalisms. This approach is closely related to algebraic logic, as the Kripke frames of our system have the type of the atom structures of cylindric algebras; the full cylindric set algebras are the complex algebras of the intended multidimensional frames called cubes.

The main contribution of the paper is a characterization of these cube frames for the finite-dimensional case and, as a consequence of the special form of this characterization, a completeness theorem for this class. These results lead to finite, though unorthodox, derivation systems for several related formalisms, e.g. for the valid n-variable first order formulas, for type-free valid formulas and for the equational theory of representable cylindric algebras. The result for type-free valid formulas indicates a positive solution to Problem 4.16 of Henkin, Monk and Tarski [16].

Keywords

Primary 03B2003B4503G15.Algebraic logicmodal logiclogic with finitely many variablescompletenessderivation rules
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 2 , June 1995 , pp. 591 - 623
Copyright
Copyright © Association for Symbolic Logic 1995

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