Abstract

The first part of this paper presents the fundations and some properties of the inclusive theory of quantification and analyses the relationship between inclusive and standard quantification theory. Some arguments concerning application of the inclusive theory instead of the standard theory have been discussed too. The second part of the article presents some versions of the inclusive theory of quantification formulated by: Mostowski (1951), Hailperin (1953), Quine (1954), Meyer and Lambert (1968). In the third part, a new conception of the inclusive theory of predicates has been presented. This conception is based, analogically to Meyer and Lambert theory, on the idea of undersignating names the first-order language. Four consequence relations have been semantically defined, and the way, how the fundations concerning realizing in the domain of individual names condition the form of the quantification theory has been presented. The discussed consequence relations have been axiomatically characterized. The only difference between the axiomatics is the way of dealing with the formulas of the following type: ∀xα –> α [x/t] where t is the name (term) of the first-order language.