1.2.4. Equivalence in content

If the relation of implication holds in both directions—if both φ ⊨ ψ and ψ ⊨ φ—then each of the two sentences says everything the other does, so they provide exactly the same information and cover the same possibilities, 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 ≃ (tilde equal)—gets used for many different kinds of equivalence, but we will use it only for logical equivalence. The idea of logical equivalence can also be described directly in terms of truth values and possible worlds. When φ ≃ ψ, we know that neither can be separated from the other, so φ and ψ must have the same truth value as each other in any possible world. And that means that equivalent sentences have the same truth conditions and express the same proposition.

Since relations of entailment depend only on possibilities of truth and falsity, equivalent sentences entail and are entailed by the same sentences. That means that entailment can be thought of as a relation between the propositions they express. It provides a sort of ordering of propositions by their content that can be compared to the ordering of numbers by ≤ and ≥. Whether entailment seems more like ≤ or ≥ depends on whether we think of it as a comparison of possibilities left open or of possibilities ruled out. When a choice needs to be made, we’ll general adopt the former perspective. In any case, the analogy is with ≤ or ≥ rather than < or > because φ ⊨ ψ tells us that φ says more or the same as ψ—or, in other words, that it covers fewer or the same possibilities.

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 two properties that implication shares with many other relations. First of all, 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: if φ ⊨ ψ and ψ ⊨ χ, then φ ⊨ χ. 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 relations) but simply report the existence of a chain no matter what its length (as is done with ancestor of, which is transitive).

Just about any relation that we would be ready to call an “ordering” is transitive. Implication also shares with certain orderings the more special property of being reflexive in the sense that every sentence implies itself. Reflexivity is what distinguishes orderings like ≤ and as strong as or stronger than from < and stronger than. In the first two, examples reflexivity is achieved by tacking on a second reflexive relation (= in one case and equally strong as in the other) as an alternative. The analogous relation in the case of implication (i.e., one amounting to equal in content to) is equivalence, but that is an alternative already built into implication (i.e., one sort of case in which a sentence φ implies a sentence ψ is when they are equivalent), so it does not need to be added.

Relations like equality (=), the relation equally strong as, and the relation of logical equivalence are reflexive and transitive, but they are not very effective in ordering things because they have no direction: if they hold between a pair of things in one direction, they hold in the other direction, too. In particular, if φ ≃ ψ then ψ ≃ φ. A relation with this property is said to be symmetric. Relations with the three properties of transitivity, reflexivity, and symmetry are said to be equivalence relations. Any equivalence relation points to equivalence or equality in some respect, and different equivalence relations point to different sorts of equality or equivalence. Logical equivalence between sentences points to equivalence in content or in the proposition expressed.