Journal of Symbolic Logic 60 (2):591-623 (1995)
Treating the existential quantification ∃ν i as a diamond $\diamond_i$ and the identity ν i = ν j as a constant δ 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 
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References found in this work BETA
Varieties of Complex Algebras.Robert Goldblatt - 1989 - Annals of Pure and Applied Logic 44 (3):173-242.
Algebraization of Quantifier Logics, an Introductory Overview.István Németi - 1991 - Studia Logica 50 (3-4):485 - 569.
On Varieties of Cylindric Algebras with Applications to Logic.I. Németi - 1987 - Annals of Pure and Applied Logic 36 (3):235-277.
Citations of this work BETA
Algebraic Logic, Where Does It Stand Today?Sayed Ahmed Tarek - 2005 - Bulletin of Symbolic Logic 11 (4):465-516.
The Range of Modal Logic.Johan van Benthem - 2012 - Journal of Applied Non-Classical Logics 9 (2-3):407-442.
Complexity of Equations Valid in Algebras of Relations Part I: Strong Non-Finitizability.Hajnal Andréka - 1997 - Annals of Pure and Applied Logic 89 (2):149-209.
Taming Logic.Maarten Marx, Szabolcs Mikul & István Németi - 1995 - Journal of Logic, Language and Information 4 (3):207-226.
On the Complexity of Axiomatizations of the Class of Representable Quasi‐Polyadic Equality Algebras.Tarek Sayed Ahmed - 2011 - Mathematical Logic Quarterly 57 (4):384-394.
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