First-order Logic without bound variables: Compositional Semantics

An attractive format for semantics is that in which composite expressions are built up from atomic ones by means of the operation of concatenation and the concatenation XY expresses the application of a function denoted by X to an argument denoted by Y . The use of relative pronouns presents an obstacle to this form of compositional semantics, since the reference of a relative pronoun in one component may occur in another. In the standard notation of first-order predicate logic this occurs in the form of variablebinding operations of quantification: in the sentence ∀xφ(x), the reference of x in ∀x is in φ(x) and neither component has an independent meaning. Frege, in the interests of compositional semantics, was led by this to declare that the open formula φ(x) is semantically significant: it simply denotes an ‘incomplete object’. We won’t discuss here the many reasons for rejecting this very ugly idea, but reject it we will. So the demands of compositional semantics require that we formalize first-order predicate logic without using bound variables. Of course the use of bound variables is very natural and the means that we use to eliminate them can result in quite complex expressions. Our purpose, therefore, is not the practical one of finding the most readable notation: it is the theoretical one of obtaining a compositional semantics. On the other hand, we shouldn’t be too humble: although the notation φ(v) with free variable v and instances φ(t) where t is a term is quite intuitive, the substitutions involved in actual cases, in substituting t for the possibly multiple occurrences of v in φ(v), can create long expressions. The elimination procedure will consist in showing that φ(v) can be expressed by φ′v, expressing