1.2.2. Ordering by content

When we judge the validity of an argument we are comparing the content of the conclusion to the information contained in the premises, and the ideas of truth values and possible worlds are designed to help us speak about the basis for that comparison. We can see more of what this sort of comparison involves and what similar comparisons are possible by focusing on comparisons of two sentences.

The term implies is a more common English synonym of entails, and we will use it often when considering an argument that has only one premise (i.e., an immediate inference in traditional terminology noted in 1.1.2). Thus φ implies (or entails) ψ when there is no risk that ψ will be in error without any error in φ—i.e., when there is no logically possible world in which ψ is false even though φ is true. When φ implies ψ, the content of ψ can be extracted from the content of φ, so to say that φ ⇒ ψ is to say that φ includes the content of ψ. Thus the relation of implication orders sentences according to their content.

If this relation holds in both directions—if both φ ⇒ ψ and φ ⇐ ψ—then each of the two sentences says everything the other does, so they provide exactly the same information, differing at most in their wording. For example, although one of the sentences Sam lives somewhere in northern Illinois or southern Wisconsin and Sam lives somewhere in southern Wisconsin or northern Illinois might be chosen over the other depending on the circumstances, they allow the same possibilities for Sam’s residence and thus provide the same information about it. We will say that sentences that have the same informational content are (logically) equivalent (usually dropping the qualification logically since we will not be considering other sorts of equivalence). Our notation for logical equivalence—the sign ⇔ (left right double arrow)—reflects its tie to mutual implication.

The idea of logical equivalence can also be described directly in terms of truth values and possible worlds. When two sentences say the same thing there is no way for one to be in error when the other is not. That is to say, sentences are equivalent when there is no possible world in which they have different truth values. To put it another way, no what things are like, either both sentences will be accurate or both will be in error. So, when φ ⇔ ψ, we know that in every possible world φ and ψ will both have the same truth value.

Notice that ⇒ is related to ⇔ in much the way that ≥ is related to =. That is, when φ ⇒ ψ, either φ says everything that ψ does as well as something more or the two sentences are equivalent. When φ does say something more than ψ, it will rule out some possibilities that ψ leaves open. To see an example of this, consider the following series of successively more specific statements, each implied by the one below it:

Each of the first two sentences leaves open some possibilities that are ruled out by the sentence below it. And in general, as we add information, we reduce the range of possibilities left open and increase the range that are ruled out. We will often speak of a sentence that rules out more and leaves open less as making a stronger claim and of one that rules out less and leaves open more as making a weaker claim. So, in the list above, the sentences closer to the bottom make the stronger claims and those closer to the top make the weaker ones.

We have been employing analogies between implication and numerical ordering and the related sorts of comparison that are associated with terms like stronger and weaker. These analogies rest on properties of implication that can be made explicit in two basic laws:

The first says implication is reflexive in the sense that any sentence φ implies itself, and second says it is transitive in the sense that implication by a premise φ carries over from a valid conclusion ψ to any sentence χ implied by that conclusion. That is, we do not count steps in a chain of related items (as is done with parent of, grandparent of, etc., which are not transitive) but simply report the existence of a chain no matter what its length (as is done with ancestor of, which is transitive).

Equivalence inherits the laws governing implication and obeys one further one:

To say that a relation is symmetric is to say that it is reversable, and equivalence is reversable because it amounts to implication in both directions. For the same reason, the reflexivity of equivalence simply states the reflexivity of implication twice over; and, for transitivity, the fact that φ ⇔ ψ and ψ ⇔ χ gives us implications in two directions between each of the pairs, and we need only splice together the ones going in the same direction to get the two implications needed to have φ ⇔ χ.

However, not everything that can be said about implication holds also for equivalence. For example, the principle of transitivity for implication tells us that implication is transferable by implication—though in different directions for premises and conclusions. That is, anything χ implied by something ψ that is a valid conclusion from φ is itself a valid conclusion from φ; and if φ implies a premise ψ that implies χ, then φ is itself a premise implying χ. Or, in other words, implication is preserved even if we weaken a conclusion (replacing it by something it implies that says less), and it is also preserved if we strengthen a premise (replacing it by something that implies it and says more).

But equivalence is not transferable in either direction by implication (though both implication and equivalence are transferable by equivalence). This failure of transferability is probably not too surprising; although implication is transferable by implication, we can’t assume that another relation would be transferable by implication. However, the reason that transferability by implication fails for equivalence is an important feature of the connection between the two relations. When we say that φ ⇔ ψ, we are looking at each of φ and ψ as both a premise and a conclusion. We are saying that each is both at least as strong as and at least as weak the other. This means that we say the two have the same logical strength, so we can replace one of them by another sentence only if the latter sentence is both at least as strong and at least as weak as the one it replaces. That is, we can replace each by a sentence it implies or is implied by only if the relation of implication holds in both directions, only if the replaced sentence is equivalent to the sentence replacing it.