Truth conditions and possible worlds

To answer this question, I am going to use a theory about necessity that has been developed in recent years, which starts from an idea of the eighteenth-century German philosopher, Gottfried Wilhelm Leibniz.

Leibniz was able to use the idea of possible worlds to answer a number of important philosophical questions. In particular, he was able to say what it was for a sentence to be necessarily true. His explanation was that a sentence was necessarily true if it was true in every possible world. There is no way the universe could have been in which a necessary sentence was not true. Thus, “2 and 2 is 4” is true in every possible world.

Leibniz believed that God, at the Creation, had chosen among all the possible worlds and chosen the best one. (It is from Leibniz that we get the expression “the best of all possible worlds.” I'll discuss this thesis again in 8.12, in connection with arguments for the exis­tence of God.) Since he thought that there were no possible worlds in which “2 and 2 is 4” is false, he held that even God could not have created a world in which two and two did not make four.

We shall examine some other problems we can treat in terms of possible worlds in the chapters on science and metaphysics. But we can use Leibniz's idea of a possible world here to build on Frege's theory of meaning. Frege said the meaning of a sentence was its truth conditions. We could formulate his theory as saying that what a sentence meant was determined by what the universe would have to be like if it was true. So we might propose, in Leibniz's terminol­ogy, that the meaning of a sentence is determined by which possible worlds make it true. How would this theory work out?

Leibniz, as we have seen, thought that all the possible worlds, all the ways the universe might have been, really existed; some other philosophers in recent times have also held this view. It is a difficult question whether possible worlds do exist, and certainly most peo­ple find the idea rather counterintuitive. But whether or not you believe in the existence of possible worlds (apart from the actual one), Leibniz's idea provides a very useful way of thinking about ref­erence. For we can translate Frege's theory about reference very easily into Leibniz's imagery.

Take names. Frege said the reference of “Bucephalus” was the horse it referred to. Well, that horse exists in many possible worlds. (Remember what this means: that the universe could have been dif­ferent in many ways while still containing that horse.) In some of those possible worlds Alexander rides it; in others, Alexander doesn't ride it but instead gives it to his teacher, Aristotle.

Take a simple predicate, such as “--------- was ridden.” Frege said

that the reference of this predicate was its extension, the class of things that were ridden.

world the sentence “Bucephalus was ridden” is false. In fact, there are many possible worlds in which Bucephalus was not ridden. In some he stays in Macedonia; in others he gallops off into Russia.

The general idea of explaining reference in terms of possible worlds is simple: a subject-predicate sentence is true in a world if and only if the referent of the subject is in the extension of the pred­icate in that world. For a sentence to be true in the actual world— in other words, for it to be simply true—the referent of the subject must be in the extension of the predicate in this universe.

Using the idea of possible worlds in this way to understand ref­erence and meaning is called “possible-world semantics.” Given this possible-world semantics for reference, we can understand at once why “It is necessary that———” produces an intensional con­text. For this semantics says that

N: It is necessary that 2 and 2 is 4

is true if and only if

S: 2 and 2 is 4

is true in every possible world. If we substitute for S another sen­tence with the same reference, then we are simply substituting a sentence that is true in the actual world. So there is no guarantee that, in this context, substituting co-referring expressions will pre­serve the truth of N.

So far as reference is concerned, then, the possible-world seman­tics is easy. But what about sense? We have already seen that it is natural to say that the meaning of a sentence is determined by which possible worlds make it true.

To know the meaning of “Bucephalus,” on this theory, would be to know what the reference of “Bucephalus” was in every possible

world. So to determine the meaning of “Bucephalus” would be to identify the referent of “Bucephalus” in the actual world and its ref­erent in every other world as well. To know the meaning of “    

was ridden.” So to determine the meaning of “------------ was ridden” is

to identify the extension of “---------- was ridden” in this world along

with the extension of “---------- was ridden” in every other world.

Though this is, indeed, a natural way to apply possible-world thinking to Frege's theory of meaning, it turns out that this way of thinking about sense and reference is not equivalent to Frege's. (Indeed, Frege never dealt with possible worlds, even though Leibniz had already proposed them as a way of thinking about necessity and possibility.) For this reason, I shall say that the possi­ble-world explanation gives words and sentences not senses but “intensions,” using what has now come to be the standard term. The intension of a word is determined once we fix its reference in every possible world. Intensions, unlike senses, are not meanings, which is why I said earlier it is confusing that the sense of a predicate is sometimes called its intension. To understand the distinction between senses and intensions, we must return to Leibniz' answer to this question: What is it for a sentence to be necessarily true?

3.8