6.2.s. Summary
We adapt the notation of lambda abstraction to provide a flexible way of linking the places of a predicate to blanks in an English sentence. An expression formed using our notation—which will have the general form [… x1 … xn …]x1 … xn—is an abstract (in this use, a predicate abstract); it consists of a abstractor applied to a parenthesized body. In English notation, a predicate abstract takes the form what … x1… xn … says of x1 … xn, and a functor abstract takes the form … x1… xn … for x1 … xn. (Variables in an abstractor that do not appear in the body are cases of vacuous abstraction.)
A variable in the body of an abstract that appears in an abstractor is bound to it, provided it is not already bound to one with narrower scope. Bound variables may be thought of as pronouns whose antecedents are in the abstractor. Expressions that establish the same patterns of binding using different variables are alphabetic variants. A expression that has variables not bound to any abstractor (such as the body of an abstract considered by itself) is open; otherwise, it is closed. A sentence-like expression that is open is not a sentence in the strict sense, but it does count as a formula. Formulas have many of the syntactic properties of sentences; in particular, they can be built from other formulas using connectives. And we can distinguish as atomic formulas not only unanalyzed sentences but all formulas that are predictions. (Indeed, unanalyzed sentences can be thought of as predications of zero-place predicates.)
Many pronouns in English function like the bound variables of the symbolic notation for abstracts, and the phrase is such that can be used to expand an English sentence by introducing them. The resulting expanded form is analogous to the predication of an abstract and can be reduced to a sentence in which the pronouns introduced by expansion are replaced by their antecedents. Because of the analogy between variables and anaphoric pronouns, abstracts can be used to represent the contribution of such pronouns to logical form.
Processes analogous to the expansion and reduction of English sentences apply to symbolic forms. In the simplest case, the application of an abstract can be reduced by replacing variables bound to it by the terms filling the corresponding places of the predicates. And a symbolic form may be expanded to introduce the predication of an abstract. Both operations help in comparing sentences in reduced form to logical forms studied in later chapters in which abstracts appear in contexts other than predication.