Predicates and open sentences

Once Frege has an explanation of the sense and reference of sen­tences he can explain the sense and reference of other words and phrases, relying always on the compositionality thesis, applied now both to sense and to reference.

In traditional grammar, sentences were said to consist of a sub­ject and a predicate. Thus, the sentence “Susan is in Canada” was said to consist of the subject, “Susan,” and the predicate, “is in Canada.” The subject—in this case a name—fixed what the sen­tence was about, and the predicate fixed what was being said about it. Suppose we are trying to determine what is the reference of “is in Canada.” Since Canada is the largest country on the North American continent, any sentence that says that something or some­body is in the largest country on the North American continent will have the same truth value as a sentence that says that somebody is in Canada. Using the compositionality thesis for reference, we can say that the property that the predicate “is in Canada” shares with the predicate “is in the largest country on the North American con­tinent” is their common reference.

That shared reference, on Frege's theory, was the class of things in Canada. So, just as a name refers to an object, a predicate refers to a class of objects. That class is called the “extension” of the pred­icate. If you want to find out if something is in the extension of a predicate, you simply make a sentence with the name of that thing followed by the predicate and see if that sentence is true.

Now we know what the reference of a predicate is. We can apply the general rule that the sense of a word or phrase is a mode of pres­entation of the reference. The predicates “is in Canada” and “is in the largest country on the North American continent” are different modes of presentation of the same class of objects: the class, namely, of things in the country whose capital is Ottawa. The sense of a predicate is sometimes referred to as its “intension.” As we shall see in 3.8, however, this terminology could lead to confusion, so I'll stick to talking of “senses.”

Now, Frege knew that not all sentences fitted the simple subject­predicate pattern. After all, is the sentence

S: John and Mary, who are friends of Peter's, sat in the garden and ate strawberries

about John or Mary or something called “John and Mary” or even something called “John and Mary, who are friends of Peter's”? So Frege suggested that we should replace the traditional notion of a predicate with the notion of what is called an “open sentence.” To get an open sentence from S, you simply remove one or several of the names. Thus

S₁:------ and Mary, who are friends of Peter's, sat in the gar­

den and ate strawberries

and

S₂:------ and Mary, who are friends of----------- 's, sat in the gar­

den and ate strawberries

are both open sentences.

We can easily see how to apply Frege's suggestion to S₁.

Frege suggested that the reference of S₂ was the class of ordered pairs of things such that if you put the name of the first member of the pair in the first blank and the name of the second member in the second blank, you got a true sentence. An ordered pair is just a pair of things taken in a particular order. (So is a different ordered pair from , even though the pairs have the same members.) Obviously, it can be true that

John and Mary, who are friends of Peter's, sat in the garden and ate strawberries

when it is false that

Peter and Mary, who are friends of John's, sat in the garden and ate strawberries.

So which name you put in which blank is important, and that is why the pair has to be ordered. It is clear that this idea can be general­ized: if you took out three names, then the open sentence would be satisfied not by ordered pairs but by ordered triples, and so on. However complex a sentence is, and however many names it con­tains, Frege's theory can say what the reference and the sense are of the open sentence produced by removing all the names.

In Chapter 1, you will remember, I introduced the idea of a vari­able to explain the Ramsey-sentences that functionalists use to set up their theory of the mind. There is a simple connection between variables and these open sentences. When you create an open sen­tence, you introduce one variable for each name you remove. So instead of writing the open sentence

------- sat in the garden and ate strawberries

style='font-size:10.0pt;line-height: 112%'>you write

X sat in the garden and ate strawberries.

(If you remove the same name more than once from a sentence, you can replace it each time with the same variable.) Frege showed that using this device, you could then explain how the words “some” and “all”—and related words like “somebody” and “everybody,” all of which are called quantifiers—worked in English.

For all X, X sat in the garden and ate strawberries

and for “Somebody sat in the garden and ate strawberries,”

There exists an X such that X sat in the garden and ate straw­berries

which is what we did with the Ramsey-sentences. “For all X, X...” is the universal quantifier; we use it to make the claim that every­thing—in the universe!—satisfies an open sentence. “There exists an X, such that X F” is the existential quantifier; we use it to make claims about the existence of something that satisfies an open sen­tence. (Here “F” is standing in for some particular open sentence. So if “F” is “laughs,” then “There exists an X such that X laughs” is true just in case some object satisfies the open sentence “——— laughs,” that is, just in case somebody laughs.) Given the way we dealt with open sentences with two blanks just now, you can see how Frege could have gone on to handle sentences with more than one quantifier, in the sort of way we did in the Ramsey-sentences of Chapter 1.

3.6