Journal of Symbolic Logic

Cylindric Modal Logic

Yde Venema

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Treating the existential quantification $\exists\nu_i$ as a diamond $\diamond_i$ and the identity $\nu_i = \nu_j$ as a constant $\delta_{ij}$, 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].

Article information

J. Symbolic Logic, Volume 60, Issue 2 (1995), 591-623.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Secondary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45} 03G15: Cylindric and polyadic algebras; relation algebras


Venema, Yde. Cylindric Modal Logic. J. Symbolic Logic 60 (1995), no. 2, 591--623.

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