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 what we will call its reference value. That is, what matters for the truth value is only what a term names if it names something; and, if it names nothing, that is all that matters. The particular way it refers to what it refers to, or the way in which it fails to refer if reference fails, do not matter. Both truth values and reference values are extensions in the sense discussed in 2.18, so the predicates we will consider are like truth-functional connectives in being extensional operators: 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 identify an expression as an individual term filling a place of a predicate only when that occurrence of the expression is referentially transparent. Occurrences that are referentially opaque must remain part of the predicate because more than just their reference values are needed for determining the truth value of a predication.

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.

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.

Further questions arise when we recognize the fact that some terms do not refer. Such terms still have a reference value but one of special sort that we will describe as a nil value. A term which has such a reference value will be said to be an undefined term. In general, we will treat undefined terms just as we treat other terms, but they require special consideration for a couple of reasons. First, this is one of the places where the issue of semantic presuppositions arises; and, after implicatures, the non-deductive inferences associated with semantic presuppositions are the ones that are most difficult to distinguish from deductive inference. The second reason is related: we will eventually consider the logical properties of definite descriptions and, as was noted in 1.3.7, it is not universally agreed which inferences concerning them are deductive and which derive from semantic presupposition.

Although it is far from universally agreed, we will assume that it is built into the idea of extensionality that all terms that fail to refer should have the same reference value. That is the basis for assigning them not merely a special sort of value—i.e., a nil value, but assigning to all of them the same reference value, which we will refer to as the the Nil. We assume that sentences have truth values in all possible worlds, even when they contain terms, like definite descriptions, that do not refer to anything under certain circumstances. This means that we must assume that predicates yield a truth value as output even when the Nil 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 is usually not settled by the ordinary meaning of an English predicate. Indeed, when non-referring terms are understood to have a nil reference value, this case is like the case of a category mistake (again see 1.3.7 for this idea); the Nil is just not the sort of thing of which most predicates are naturally true or false. As in other cases where truth values are not determined solely by sentences and possible worlds, we will assume that they are somehow stipulated for predications of the Nil but we will avoid considering relations among sentences that depend on the way these values are stipulated.