Axiomatization of Some Natural Quantifiers

This paper extends an existing logic, £n, to some of the generalized quantifiers of natural language. In contrast to the usual approach, this extension does not require the identity relation. Sommers has suggested that the identity is unnecessary in a logic that properly treats singular terms. This paper lends support to Sommers position. £n is a logic designed for natural language reasoning (see [3]). This paper defines an extension, LNQ, of that logic to include the cardinal quantifiers, at least n, and the second-order quantifier, most. Because of the limited expressiveness of first-order languages, a complete axiomatization for most is not possible. However incompleteness does not negate the usefulness of the axiomatization for natural language reasoning. Theorems, generalizing those of [3], are given. These theorems establish the properties of monotonicity, conservativity, and conversion for LNQ. These results are of interest in connection with Sommers position that by endowing singular terms with wild quantity, identity as a logical operator is not needed. This in turn results in a logic that is simpler and more closely conforms to natural language