Predicates

Consider the following statements:

x is a star

x has two moons

x has n moons

x is between Earth and y

x is between y and z

These statements aren't propositions, because they aren't unequivocally either true or false. And the reason they aren't is that they involve parameters (also known as placeholders or free variables). For example, the statement "x is a star" involves the parameter x, and we obviously can't say whether it's true or false unless we're told what that x stands for (at which point we're no longer dealing with the given statement anyway but a different one instead see the next paragraph).

Now, we can substitute arguments for the parameters and thereby obtain propositions from the statements. For example, if we substitute the argument The Sun for the parameter x in "x is a star," we obtain "The Sun is a star." And this statement is indeed a proposition because it's unequivocally either true or false (in fact, of course, it's true). But the original statement as such ("x is a star") is, to say it again, not itself a proposition. Rather, it's a predicate, which as you'll recall from Chapter 4 is a truth-valued function; that is, it's a function that, when invoked, returns a truth value. Like all functions, a predicate has a set of parameters; when it's invoked, arguments are substituted for the parameters; substituting arguments for the parameters effectively converts the predicate into a proposition; and we say the arguments satisfy the predicate if and only if that proposition is true. For example, the argument The Sun satisfies the predicate "x is a star," whereas the argument The Moon does not.

As an aside, I remind you from Chapter 4 that logicians speak not of invoking a predicate but rather of instantiating it. In fact, for reasons that need not concern us here, their concept of instantiation is slightly more general than that of the familiar notion of function invocation. However, I'll stay with the terminology of invocation in this appendix.

I also remind you from Chapter 4 that a proposition can be regarded as a degenerate predicate. To be precise, it's a predicate for which the corresponding set of parameters is empty (and the function thus always returns the same result, either TRUE or FALSE, every time it's invoked). In other words, all propositions are predicates, but most predicates aren't propositions.

Now consider the example "x has n moons." Obviously, this predicate involves two parameters, x and n. Substituting the arguments Mars for x and 2 for n yields a true proposition; substituting the arguments Earth for x and 2 for n yields a false one.

Predicates can conveniently be classified according to the cardinality of their set of parameters. That is, we often speak of an n-place predicate, meaning a predicate with exactly n parameters; for example, "x is between y and z" is a 3-place predicate, while "x has n moons" is a 2-place predicate. A proposition is a 0-place predicate.

NOTE

An n-place predicate is also called an n-adic predicate. If n = 1, the predicate is said to be monadic; if n = 2, it's said to be dyadic.

Given a set of predicates, we can combine predicates from that set in a variety of ways to form further predicates, using the logical connectives already discussed NOT, AND, OR, and so forth (in other words, those connectives are logical operators that operate on predicates in general, not just on the special predicates that happen to be propositions). A predicate that involves no connectives is called simple; a predicate that isn't simple is called compound. Here's a trivial example of a compound predicate:

(x is a star) OR (x is between Earth and y)

This predicate is also dyadic not because it involves two simple predicates, but because it involves two parameters, x and y.