6.1.3. Extensionality
The only restriction on an analysis of a sentence into a predicate and individual terms is that the contribution of an individual term to the truth value of a sentence must lie only in its reference value. That is, all that matters is what a term names if it names something; and, if it names nothing and thus has the nil reference value mentioned in 1.3.7, that is all that matters. Both truth values and reference values are extensions in the sense introduced in 2.18, so the predicates we will consider are like truth-functional connectives in being extensional operations: the extension of their output depends only on the extensions of their inputs.
In the specific case of predicates, this requirement is sometimes spoken of as a requirement of referential transparency. When it is satisfied, we can look through individual terms and pay attention only to their reference values when judging whether a sentence is true or false; in other cases, we might need to pay attention to the terms themselves or to the ways in which they refer to their values in order to judge the truth value. For example, in deciding the truth of The U. S. president is over 40, all that matters about the individual term the U. S. president is who it refers to. On the other hand, the sentence For the past two centuries, the U. S. president has been over 35 is true while the sentence For the past two centuries, Barack Obama has been over 35 is false—even when the terms the U. S. president and Barack Obama refer to the same person. So, in this second case, we must pay attention to differences between terms that have the same reference value. When this is so the occurrences of these terms are said to be referentially opaque; that is, we cannot look through them to their reference values. The restriction on the analysis of sentences into predicates and individual terms is then that we can separate an individual term from a predicate and count it as filling a place of the predicate only when that occurrence is referentially transparent. Occurrences that are referential opaque cannot be separated from the predicate and must remain part of it.
Hints of idea of a predicate as an incomplete expression can be found in the Middle Ages, but it was first developed explicitly by Gottlob Frege in the late 19th century. Frege applied the idea of an incomplete expression not only to predicates but also to mathematical expressions for functions. Indeed, Frege spoke of predicates as signs for a kind of function, a function whose value is not a number but rather a truth value. That is, just as a function like + takes numbers as input and issues a number as output, a predicate is a sign for a function that takes the possible references of individual terms as input and issues a truth value as output by saying something true or false about the input.
We will speak of the truth-valued function associated with a 1-place predicate as a property and speak of the function associated with a predicate of two or more places as a relation. Thus a predicate is a sign for a property or relation in the way a truth-functional connective is a sign for a truth function.
Just as a truth-functional connective can be given a truth table, the extensionality of predicates means that a table can capture the way the truth values of the their output sentences depend on the reference values of their input. For example, consider the predicate __ divides __ (evenly). Just as there can be addition or multiplication tables displaying the output of arithmetic functions for a limited range of input, we can give a table indicating some of the output of the relation expressed by this predicate. For the first half dozen positive integers, we would have the table shown below. Here the input for the first place of the predicate is shown by the row labels at the left and the input for the second place by the column labels at the top. The first row of the table then shows that 1 divides all six integers evenly, the second row shows that 2 divides only 2, 4, and 6 evenly, and the final column shows that each of 1, 2, 3, and 6 divides 6 evenly.
| _ divides _ | 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|---|
| 1 | T | T | T | T | T | T | |
| 2 | F | T | F | T | F | T | |
| 3 | F | F | T | F | F | T | |
| 4 | F | F | F | T | F | F | |
| 5 | F | F | F | F | T | F | |
| 6 | F | F | F | F | F | T |
Of course, this table does not give a complete account of the meaning of the predicate; and, for many predicates, no finite table could. But such tables like this will still be of interest to us because we will consider cases where there are a limited number of reference values and, in such cases, tables can give full accounts of predicates.
As was noted in 1.3.7, we assume that sentences have truth values even when they contain terms that do not refer to anything. This means that we must assume that predicates yield a truth value as output even the nil value is part of their input; that is, we assume that predicates are total. The truth value that is issued as output when the input includes the nil value is usually not settled by the ordinary meaning of an English predicate. It is analogous to the supplements to contexts of use suggested in 1.3.6 as a way of handling cases of vagueness. As in that case, we try to avoid making anything depend on the particular output in cases of undefined input but instead look at relations among sentences that hold no matter how such output is stipulated.