1.2.1. Truth values and possible worlds

When an inference is deductive, its conclusion cannot be in error unless there is an error somewhere in its premises. The sort of error in question lies in a statement being false, so to know that an argument is valid is to know that its conclusion must be true unless at least one premise is false. Similarly, to know that a set of sentences is inconsistent—to know that it’s members are deductively incompatible—is to know that these sentences cannot all be true. This means that the ideas of truth and falsity have a central place in deductive logic, and it will be useful to have some special vocabulary for them.

It is standard to speak of truth and falsity together as truth values and to abbreviate their names as T and F, respectively. So, to say that an argument is valid is to claim that there is no risk of the pattern of truth values for its premises and conclusion shown in Figure 1.2.1-1 occurring. That is (using some of the other terminology we have available), a conclusion is entailed by a set of assumptions if the truth value of the conclusion cannot be F when each of the assumptions has the truth value T.

And a set is inconsistent if the truth values of its members cannot all be T.

Since to speak of no risk of error is to speak of no possibility of error, it is also useful to have some vocabulary for speaking of possibility and impossibility. The sort of possibility in question in deductive logic is very weak and the corresponding sort of impossibility is very strong. We will refer to this as logical possibility and impossibility. A description of a situation that runs counter to the laws of physics (for example, a locomotive floating 10 feet above the earth’s surface without any abnormal forces acting on it) might be said to be physically impossible; but it need not be logically impossible, and we must consider many physical impossibilities when deciding whether a conclusion is deductively valid. For, otherwise, anything following from the laws of nature, including the laws themselves, would be a valid conclusion from any premises whatsoever, and these laws would not say anything more than mere descriptions of the facts they were designed to explain. In short, if there is any set of premises such that a sentence φ says something that they do not, then it is logically possible for φ to be false.

We can say that something is impossible by saying that there is no possibility of it being true. In saying this, we use a form of words analogous to one we might use to say that there is no photograph of Abraham Lincoln chopping wood. That is, in saying there is no possibility, we speak of possibilities as if they were things like photographs. This way of speaking about possibilities is convenient, so it is worth spending a moment thinking about what sort of things possibilities might be. The sort of possibility of chief interest to us is a complete state of affairs or state of the world, where this is understood to include facts concerning the full course of history, both past and future. Since Leibniz, philosophers have used the phrase possible world as a particularly graphic way of referring to possibilities in this sense. For instance, Leibniz held that the goodness of God implied that the actual world must be the best of all possible worlds, and by this he meant that God made the entire course of history as good as it was logically possible for it to be.