Generalized Quantifiers in Algebra

Abstract

Several classical twentieth century works in mathematical logic (e. g. by A. I. Malcev, A. Tarski and A. Robinson) have proved the significance of logic formalism, especially of first order predicate calculus, for studying different aspects of algebra. This formalism has become a working instrument of algebra. The attemps to formalize further aspects of algebraic systems on the one hand and a natural development of the theory of logical calculi on the other, gave three main extensions of first order classical calculus: second order logic and its different fragments, infinite languages, and languages with generalized quantifiers. Second order logic is better for reflecting typical algebraic notions and constructions. At the same time, as a more powerful language, it looses attractive properties of first order predicate calculus, such as: compactness, interpolation properties, axiomatizability, and others. Deeper understanding of this, following the development of the first order calculus resulted in the so-called abstract model theory. The abstract model theory, in turn, gave an impuls for developping the theory of generalized quantifiers. At the beginning, in the paper [10] of A. Mostowski, the languages with generalized quantifiers, have been constructed as tools for studying formalization of different aspects of particular algebraic theories. Later they have become a training field for checking ideas and hypotheses of abstract model theory. This is reflected in the general theory of generalized quantifiers of P. Lindström [7] and has resulted in such monsters as Q _cf .