6.1.2. Logical predicates

We derived the concept of an individual term from a traditional description of the grammatical subject of a sentence by focusing on the semantic idea of naming. As we will see in 6.1.6, the idea of an individual term is much narrower than the idea of a grammatical subject: not every phrase that could serve as the subject of a sentence counts as an individual term. We have seen that the opposite is true of our concept of a predicate: it includes grammatical predicates but many other expressions, too.

Like the definition of an individual term, the definition of a logical predicate is semantic: a predicate says something about the about whatever objects are named by the individual terms to which it is applied. The simplest example of this is a grammatical predicate that says something about an object named by an individual term. But consider a sentence that has not only a subject but also a direct object—Ann met Bill for example. This says something about Ann, but it also says something about Bill. From a logical point of view, we could equally well divide the sentence into the subject Ann on the one hand and the predicate met Bill on other or into the subject-plus-verb Ann met and the direct object Bill. And we will be most in the spirit of the idea that predicates are used to say something about individuals if we divide the sentence into the two individual terms Ann and Bill on the one hand and the verb met on the other. The subject and object both are names, and the verb says something about the people they name. That is why we define a predicate as an operator that forms a sentence when applied to one or more terms. We will speak of the application of this operator as predication and speak of a sentence that results as a predication.

We can present predicates in this sense graphically by considering sentences containing any number of blanks. For example, the predication Jane called Spot might be depicted as follows:

The number of different terms to which a predicate may be applied is its number of places, so the predicate [ _ called _ ] has 2 places while predicates, like [ _ ran] and [ _ barked], that are also predicates in the grammatical sense will have one place. We will discuss our notation for predicates more in 6.2.1, but we will often (as has been done here) indicate a predicate by surrounding with brackets the English sentence-with-blanks that expresses it.

In the example above, the two-place predicate is a transitive verb and the second individual term functions as its direct object in the resulting sentence. The individual terms that serve as input to predicates may also appear as indirect objects or as the objects of prepositional phrases that modify a verb—as in the following examples:

Other examples of many-place predicates are provided by sentences containing comparative constructions or relative terms. Even conjoined subjects can indicate a many-place predicate when and is used to indicate the terms of a relation rather than to state a conjunction:

Although you will rarely run into predicates with more than three or four places, it is not hard make up examples of predicates with arbitrarily large numbers of places. For example, imagine the predicate you would get by analyzing a sentence that begins Sam travelled from New York to Los Angeles via Newark, Easton, Bethlehem, …. and goes on to state the full itinerary of a trans-continental bus trip.

The places of a many-place predicate come in a particular order. For example, the sentences Jane is older than Sally and Sally is older than Jane are certainly not equivalent, so it matters which of Jane and Sally is in the first place and which in the second when we identify them as the inputs of the predicate [ _ is older than _ ]. Even when the result of reordering individual terms is equivalent to the original sentence, we will count the places as having a definite order and treat any reordering of the terms filling them as a different sentence. So Dick is the same age as Jane and Jane is the same age as Dick will count as different sentences even though [ _ is the same age as _ ] is symmetric in the sense that

for any terms σ and τ.