Algebraic logic. III. Predicates, terms, and operations in polyadic algebras

Abstract

Introduction. The theory of polyadic algebras is an algebraic photograph of the logical theory of first-order functional calculi. It is known, for instance, that the Godel completeness theorem can be formulated in algebraic language as a representation theorem for a large class of simple polyadic algebras, together with the statement that every polyadic algebra is semisimple. The next desideratum is an algebraic study of the celebrated G6del incompleteness theorem. Before that can be achieved, it is necessary to investigate the algebraic counterparts of some fundamental logical concepts (such as the ones mentioned in the title above). The purpose of this paper is to report the results of such an investigation('). Although this is the third of a sequence of papers on algebraic logic, its development does not lean very heavily on the first two papers. The main purpose of the first paper was to study the topological properties of quantification, via the Stone duality theory for Boolean algebras [Algebraic logic I, Compositio Math. vol. 12 (1955) pp. 217-249]. The purpose of the second paper was to study the algebraic properties of polyadic algebras, with main emphasis on their representation theory [Algebraic logic II, to appear in Fund. Math.1(2). Since the methods and results of this paper might be described as combinatorial, neither the duality theory nor the representation theory plays any role; a sympathetic understanding of the basic definitions and of their elementary consequences is sufficient for present purposes. For the convenience of the reader, the basic definitions and theorems are summarized in ?1 below. The most difficult concept introduced in this paper is the concept of a term (?4), and the most difficult theorems are the ones that describe properties of terms (? ?5-8) and the ones that give methods of constructing them (??9, 11, and 13). The climax is reached in ?9; the existence theorem of that section (despite its somewhat complicated statement) turns out to be a most efficient tool for constructing terms satisfying various conditions. The remaining sections are devoted to auxiliary matters and to the more easily