How Non-Context Free is Variable Binding?

Abstract

Within and across theoretical frameworks, linguists have debated whether various phenomena related to variable binding should be handled within syntax, as part of a syntax-to-semantics mapping, or possibly by one or more separate levels of rules or principles specifically dedicated to such phenomena. Relevant phenomena include the interpretation of pronouns and reflexives as bound variables and the binding of gaps or traces by WH-phrases or other operators. Given current interest in the generative capacity of alternative syntactic theories, it is important to note that whatever level of grammatical description deals with variable binding must involve some non-context-freeness if certain natural restrictions on well-formedness (spelled out below) are taken to belong to grammar at all. In this paper we investigate the formal properties of variable binding via an investigation of two restricted versions of predicate logic. We first review the familiar fact that the set L of formulas of first-order predicate logic (FOPL) (allowing free variables and vacuous quantifiers) is a context free (CF) language. We then consider two restricted sublanguages of L, the set S of Sentences of FOPL (formulas with no free variables), and the set U of formulas of FOPL with no vacuous quantifiers. Both S and U can be shown by established methods to be non-CF.