To build a formal model of predication and to express lexical meaning, I will use the lambda calculus. The lambda calculus is the oldest, most expressive, and best understood framework for meaning representation; and its links to various syntactic formalisms have been thoroughly examined from the earliest days of Montague Grammar to recent work like that of de Groote (2001), Frank and van Genabith (2001). Its expressive power will more than suffice for our needs.

The pure lambda calculus, or λ calculus, has a particularly simple syntax. Its language consists of variables together with an abstraction operator λ. The set of terms is closed under the following rules: (1) if v is a variable, then v is a term; (2) if t is a term and v a variable, then λvt is also a term; (3) if t and t′ are terms, then the application of t to t′, t[t′], is also a term. We can use this language to analyze the predication involved when we apply a predicate like an intransitive verb to its arguments. The meaning of an intransitive verb like sleeps is represented by a lambda term, λx sleep′(x); it is a function of one argument, another term like the constant j for John that will replace the λ bound variable x and yield a logical form for a larger unit of meaning under the operation of β reduction.