Universal Algebra and Mathematical Logic

In this paper, first-order logic is interpreted in the framework of universal algebra, using the clone theory developed in three previous papers [1] [2] and [3]. We first define the free clone T pL, Cq of terms of a first order language L over a set C of parameters in a standard way. The free right algebra FpL, Cq of formulas over the clone T pL, Cq of terms is then generated by atomic formulas vis a binary operation ñ and a unary operation . The classical unary operations x1, x2, ... are derived from via substitutions. Structures for L over C are represented as perfect valuations of FpL, Cq, and theories of L are represented as filters of FpL,Hq. Finally Godel’s completeness theorem and first incompleteness theorem are stated as expected.