1.2.2. Truth conditions and propositions

When judging the validity of an argument, what we need to know about its premises and conclusion are the truth values of these sentences in various possible worlds. This information about a sentence is an aspect of its meaning that we will call its truth conditions. That is, when we are able to tell, no matter what possible world we might be given, whether or not a sentence is true, we know the conditions under which the sentence is true; and, when we know those conditions, we can tell whether or not it is true in a given possible world.

It will also be convenient to be able to speak of this kind of meaning or aspect of meaning as an entity in its own right. We will do this by speaking of the truth conditions of a sentence as encapsulated in the proposition expressed by the sentence. This proposition can be thought of as a way of dividing the full range of possible worlds into those in which the sentence is true and those in which it is false. And we can picture a proposition as a division of an area representing the full range of possibilities into two regions.

Fig. 1.2.2-1. The proposition expressed by a sentence φ, seen as dividing the full range of possible worlds into possibilities in which it is true and possibilities in which it is false.

Since knowing what possibilities are in one of these regions tells us that the rest are in the other region, we know what proposition is expressed by a sentence when we know what possibilities it rules out—or know what possibilities it leaves open. It might seem that the proposition is really indicated by the line between the two. And that’s right provided we add an indication of which side of the line corresponds to truth and which side to falsity.

We will use several ways of speaking about these two regions. On the one hand, a proposition can be said to divide the possible worlds into the possibilities it rules out and the ones it leaves open. Leaving open a possibility is a failure to rule it out, and it will sometimes be useful to have a more positive way of speaking about the possibilities in which a sentence is true: we can say in such a case that the sentence covers that possibility. So a proposition can also be seen as a division of all possible worlds into possibilities covered and possibilites not covered.

Fig. 1.2.2-2. Three ways of describing the two regions into the proposition expressed by a sentence φ divides the full range of possible worlds.

For reasons that will be discussed in the next subsection, we will speak of the collection of possibilities ruled out by a proposition as its content, and it is natural to refer to the full range of possibilities covered by a proposition as its coverage. So a proposition can be said to divide the possible worlds into its content and its coverage.