Syntactical and semantical properties of generalized quantifiers
In the first-order language, quantifiers (∀x) and (∃#) are understood “to say” that “for all elements” and “there is at least one element such that …”, respectively. We are interested in changing the interpretation to “for all elements with fewer than κ exceptions” and “there are at least κ elements such that”, respectively, where κ is a cardinal. We call this the κ-interpretation of the quantifiers.1 The first question which presents itself is “What is the relationship between the κ-interpretation and the λ-interpretation?” For instance, is a formula valid under one interpretation also valid in all other interpretations? In the second section, it will be shown that as far as infinite interpretations, i.e. κ-interpretations for infinite cardinals κ, are concerned, the validity of a formula is preserved. Actually, a more general result is obtained there by model theoretic methods.