Categories of First-Order Quantifiers

Abstract

One well known problem regarding quantifiers, in particular the 1 storder quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated by the Ajdukiewicz's classical categorial grammar. The 1 st-order quantifiers are typically ambiguous. Every 1 st-order quantifier of the type k > 0 is treated as a two-argument functorfunction defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1 st-order quantifiers based on the principle of categorial compatibility.

What Is the Sense in Logic and Philosophy of Language

“…From the logical point of view, the three notions of the sense of expressions of language L are understood as follows [62,68] (see Fig. 1):…”