6.2.3. Variables and pronouns

English has devices which function like bound variables. The force of the abstract

or the equivalent

can be captured in English by the expression

which uses expressions like the first, the second, and so on, instead of subscripted variables. (Parentheses were used in the English displayed above simply to make the comparison easier.) No particular group of people is in question here, and the expressions the first, etc., do not refer to anything outside the sentence. Instead, these expressions function here much like pronouns that have three people as their antecedant. We have used a different word order from the that used in 6.2.1 to read abstracts in English in order to put this antecedent before the pronouns that refer to it.

In the case of a one-place predicate abstract the corresponding English can be stated with a genuine pronoun:

The blank that is marked by x in the body of the symbolic abstract is filled in the English with the pronoun it, which has a thing as its antecedent. Since a thing makes no definite reference, neither does the pronoun; the pronoun refers back to its antecedent only in the sense that their references are linked in their indefiniteness and cannot be indefinite in independent ways. The general moral is that the variables used in the bodies of abstracts are like pronouns, and the ones in abstractors are like their antecedents. One consequence of this reiterates a point made in the last subsection: you should not expect variables bound to different abstractors to be linked in their reference any more than you would expect this of pronouns that have different antecedents.

We can also move in the other direction and use abstracts to represent the contribution of pronouns to the logical form of a sentence. We can get a hint of how they might do this by looking at a particular English rendering of the sample predicate abstract discussed in 6.2.1

where we use another common pronoun-like device. Now consider the following restatement:

The reflexive pronoun in this expression corresponds to the repeated variable in the symbolic abstract.

Of course this English expression was a rather artificial one constructed to correspond to an abstract, but there are ways to apply abstracts more broadly. To see how, let us look at three further English expressions corresponding to the abstracts we have been considering. This time the English expressions are predicates (rather than noun phrases that refer to the contents of predicates):

Of course, these predicates themselves are also artificial, but they represent a device that may be applied to virtually any English sentence to restate it (in English) in a way that corresponds to a symbolic analysis that uses an abstract.

Because the first element of a sentence often indicates the topic under discussion, languages have many devices for restating sentences with various elements at the front. One common device in English is the use of passive voice. If we wish to say who wrote a book but focus attention on the book rather than its author, we might say something like Moby Dick was written by Melville. He we take the direct object of Melville wrote Moby Dick and move it to the front of the sentence by changing the verb from active to passive voice. Passive voice can be used similarly to move more than direct objects to the front, but it has limitations, as do many of the other devices English has for making noun phrases into subjects. The device we will most often use—which we will call expansion—allows a great variety and arbitrary number of nouns phrases to be made into a subject by following the subject by the phrase is such that which is itself followed by the result of replacing the noun phrases in the original sentence by pronouns or pronoun-like devices. For example, Melville wrote Moby Dick can be converted into any of

The result of expansion is an expanded form, and we will often write it, as has been done here, with the residue of the original sentence in parentheses. When we need to distinguish among alternative ways of expanding a sentence, we will speak of expanding on a particular noun phrase. The opposite of expansion is reduction, and we will describe the original sentence as being in reduced form relative to that expansion. The idea of reduced form is relative because, in principle, expansion can be applied more than once, and a reduced form may be reduced still further. For example, the first expanded form above is also the result of reducing Moby Dick is such that (Melville is such that (he wrote it)). Expansion will serve us in a number of different ways. For now, the fact that it uses pronouns and is analogous to the use of abstracts will help in using abstracts to analyze the role of pronouns in a sentence.

To see how, let us analyze the sentence Bill told Ann his name in a way that employs a predicate abstract to reflect the use of a pronoun.

Once the sentence as a whole has been analyzed as the predication of an abstract, the formula x told Ann x’s name that is the body of the abstract is analyzed in the same way as Bill told Ann Bill’s name would be. This second analysis departs from the original sentence in having the equivalents of two pronouns instead of one (as does Bill is such that (he hold Ann his name) but it is like the original in having only a single occurrence of Bill, and it is in this way that it reflects the use of pronouns.

We might also have expanded on both Bill and Ann to get Bill and Ann are such that he told her his name, with the analysis

That would have added no enlightenment in the case of that sentence, but consider the following ambiguous sentence with abbreviated analyses of two interpretations of it. (You can imagine the second concerns a case of amnesia.)

In each of these analyses, the names Bill and Al are separated completely from the abstracts, which show any patterns of coreference using variables. The advantage of this sort of analysis is that it enables us to identify the same ambiguity in sentences that involve different names, such as Barb told Ann her name.