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Copeland algebras1

  • Daniel B. Demaree (a1)

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It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.

Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.

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1

Results described in this paper are contained in the second chapter of the author's Ph.D. thesis submitted to the University of California, Berkeley, in May, 1970. We express our gratitude to Professor J. D. Monk for the suggestion to study the Copeland formulation, and for his encouragement and guidance along the way.

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[1]Copeland, A. H. Sr., Note on cylindric algebras and polyadic algebras, The Michigan Mathematical Journal, vol. 3 (19551956), pp. 155157.
[2]Halmos, P. R., Algebraic logic, Chelsea, New York, 1962, 127 pp.
[3]Henkin, L., Monk, J. D. and Tarski, A., Cylindric algebras. Part I, North-Holland, Amsterdam, 1971.
[4]Johnson, J. S., Amalgamation of polyadic algebras and finitizability problems in algebraic logic, Doctoral Dissertation, University of Colorado, Boulder, Colorado, 1968, vi + 129 pp.

Copeland algebras1

  • Daniel B. Demaree (a1)

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