6.2.4. Expanded and reduced forms

We will use the ideas of expansion and reduction and of expanded and reduced form in connection with symbolic analyses as well as English sentences. (The context will usually indicate which use of the terms is intended but, if necessary, we can speak of symbolic expansion on the one hand or expansion using such that on the other.) As it applies to symbolic analyses, expansion is the process of restating an analysis using an abstract as we did when we moved from the analysis Tbab of Bill told Ann his name to [Txa(nx)]_xb.

From one point of view, there is no need to use expansion to study the ambiguity of Bill told Al his name. A pair of simple reduced forms like Tba(nb) and Tba(na) is quite sufficient. And even the point that Barb told Ann her name shares the same ambiguity can be captured by referring to a pair of logical forms Tτυ(nτ) and Tτυ(nυ) that are exhibited by each of the two pairs.

Of course, this simpler approach would ignore the fact that the ambiguity lies in the pattern of coreference marked by anaphoric pronouns. Abstracts capture this, but in a rather crude way since they introduce extra pronouns to do so. While the English sentence Bill told Al his name has a single pronoun, our analyses each had three bound variables. The notation could be modified to be more subtle if our main interest was in anaphoric pronouns with individual terms as antecedents. However, the prime application of abstracts will be in later chapters where we will use abstracts in connection with our analysis of quantifier phrases.

In order to analyze a sentence as a truth-functional compound, we must be able to identify components that function independently. In particular, a pronoun in one component cannot have its antecedent in another. The approach we took before employing abstracts was to simply replace a pronoun by its antecedent when this was possible and avoid analysis when it was not. The prime example of a pronoun we could not replace is one whose antecedent is a quantifier phrase. The sort of analysis we will eventually use in this case employs abstracts behind the scenes, and the use of abstracts for cases where pronouns have individual terms as antecedents brings those cases closer to our handling of cases where the antecedents are quantifier phrases.

Still, one of the key points to be made about abstracts with regard to individual terms is the very fact that they are dispensible, so let us look more closely at how to dispense with them once we have used them in an analysis. For example, consider the sentence Ann visited the class and she spoke to Davie. If we use an abstract to capture the coreference of she and Ann, we can analyze this as follows:

The formula x visited the class and x spoke to Davie can be analyzed as a truth-functional compound because the two occurences of the variable x are independent of each other (though each is bound to the abstractor).

The approach we used earlier would have led us to analyze the sentence as the compound Ann visited the class ∧ Ann spoke to Davie in which she is replaced by Ann, and this sentence would receive a symbolic analysis of the form Vac ∧ Sad. Now, if we compare the symbolic analyses

[Vxc ∧ Sxd]xa

Vac ∧ Sad

we can see that the second is the result of putting the term a in place of the variable x in the body of the abstract in the first. That is, the second is the reduced form of the first.

When we reduce the predication of an abstract, we take the body of the abstract and put the term of which it is predicated in the blanks marked by the variable. An analogous description applies to the reduction of compound terms formed by applying functor abstracts, and the description can be extended to apply to abstracts on any number of variables. Schematically, the general pattern is as follows:

[---x1---…---xn---]x1…xnτ1…τn

---τ1---…---τn---

When interpreting the schema, remember that the variables of the abstractor can appear in the body in any order and may each appear any number of times (including not at all). The expression on the right is the result of using each term τ_i to replace all occurrences of the corresponding variable.

Special care is needed when performing a reduction if the body contains abstracts and a term to be substituted contains free variables. The short account of this sort of case is that no free variable should become bound as a result of reduction and that abstracts should be replaced by alphabetic variants as necessary to avoid this happening. The easiest way to insure this is to choose bound variables so that they are all different from each other and from any free variables. However, our use of abstracts will be limited to much simpler situations, so a detailed rule is not important. Moreover, we will regard reduced and expanded expressions as two ways of writing the same formula or term, so no rule at all is needed as part of our rules for derivations, where sentences will be written only in fully reduced form.

Let us now return to the issue of pronouns and truth-functional connectives. From our present point of view, the fact that pronouns can always be replaced by individual term antecedents can be seen as the result of the fact the reduction is always possible. The analyses of sentences involving quantifier phrases that we will go on to develop in the next couple of chapters will employ predicate abstracts but not by way of predication, so nothing analogous to reduction will be in question. That can be cited as the reason a pronoun often cannot be replaced by a quantifier phrase antecedent—as in A mother visited the class and she spoke to Davie, which is not equivalent to A mother visited the class and a mother spoke to Davie. In cases where replacement by a quantifier phrase antencedent is possible without changing the meaning—as in A mother visited the class or she spoke to Davie on the phone—this will be due to special interactions between the quantifier phrase and other logical constants in the sentence.

Finally, although our focus has been on pronouns, much of what we have seen applies also to sentences containing compound predicates and other compound phrases. The sentence Ann visited the class and spoke to Davie can also be analyzed as [Vxc ∧ Sxd]_xa. While this analysis introduces the symbolic analogues pronouns that do not appear in the English, it does capture the form of the English in one respect: it treats it as a predication whose predicate contains the connective. And the possibility of restating the sentence as Ann visited the class and Ann spoke to Davie can be seen as due to the reduction of this form to Vac ∧ Sad.