Semantic Modeling of Predicate Calculus Based on N-Tuple Algebra

Abstract

For semantic modeling of a first-order language, it is proposed to use the thesis expressed by the recognized specialist in mathematical logic E. Mendelson that in mathematical logic “semantic concepts are of a set-theoretic nature”. This idea was not implemented for the reason that paradoxes were discovered in set theory. Therefore, it is proposed to use set algebra instead of set theory. The report formulates its differences from theory of sets, substantiates impossibility of paradoxes in it. The interpretation of the first-order language proposed by E. Mendelson is considered, its shortcomings are analyzed, and an interpretation variant is proposed closer to modeling of natural reasoning. It is established that the mathematical model of this interpretation is n-tuple algebra developed by the authors, which is isomorphic to algebra of sets. Because of this, modeling of reasoning based on this algebra can be called semantic modeling. The paper shows correspondences between predicate calculus formulas and expressions of n-tuple algebra, explains connections of these expressions with reasoning in natural languages. Positive results of semantic modeling based on n-tuple algebra are shown. A list of unresolved issues is provided.