Algebraic semantics for deductive systems
The notion of an algebraic semantics (a.s.) was introduced, and the study of it was initiated, by \textit{W. J. Block} and \textit{D. Pigozzi} [Algebraizable logics. Mem. Am. Math. Soc. 396 (1989; Zbl 0664.03042)]. It was shown there that a deductive system may have many different algebraic semantics, while a so-called equivalent a.s., if a deductive system has it at all, is essentially unique. A deductive system possessing an equivalent a.s. was said to be algebraizable. While the main subject of the monograph cited above was algebraizable systems and their equivalent semantics, the paper under review deals with the property of possessing an arbitrary a.s., which is much harder to capture. The paper begins with a detailed overview of some basic facts concerning the general concepts of a deductive system, an equational consequence relation, and an algebraic semantics. It is shown that any extension of a deductive system possessing an a.s. also has this property. Some necessary conditions for a deductive system to have an a.s. are established; using these, an example of a non-trivial protoalgebraic deductive system that has no a.s. is provided. The main theorem of the paper gives some sufficient condition for the existence of an a.s. It is used to demonstrate that various particular classes of deductive systems, even non-algebraizable, admit an a.s.
scientific article; zbMATH DE number 2042712