David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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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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Johan van Benthem (2012). The Range of Modal Logic. Journal of Applied Non-Classical Logics 9 (2-3):407-442.
Hajnal Andréka, István Németi & Johan van Benthem (1998). Modal Languages and Bounded Fragments of Predicate Logic. Journal of Philosophical Logic 27 (3):217-274.
Hajnal Andréka (1997). Complexity of Equations Valid in Algebras of Relations Part I: Strong Non-Finitizability. Annals of Pure and Applied Logic 89 (2):149-209.
Tarek Sayed Ahmed (2011). On the Complexity of Axiomatizations of the Class of Representable Quasi‐Polyadic Equality Algebras. Mathematical Logic Quarterly 57 (4):384-394.
Maarten Marx, Szabolcs Mikul & István Németi (1995). Taming Logic. Journal of Logic, Language and Information 4 (3):207-226.
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