1.2.3. Ordering by content
When we judge the validity of an argument we are comparing the content of the conclusion to the contents of 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 ψ ⊨ φ—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 ≃ (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 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 yet another way, no matter what things are like, a pair of equivalent sentences will both be accurate or both be in error. This means that, when φ ≃ ψ, we know that in any possible world we might consider, φ and ψ will both have the same truth value. 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, that it leaves fewer or the same possibilities open.
When φ does say something more than ψ—that is, when φ ⊨ ψ but ψ ⊭ φ—the possibilities left open by ψ will include all those left open by φ (because φ ⊨ ψ) but it will leave open some on top of these (because there is some possible world in which ψ is true but φ is false). To see an example of this, consider the following series of successively more specific statements, each implied by the one below it:
Each sentence until the last 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.
The relation between a sentence expressing a stronger proposition and a sentence expressing a weaker can be displayed graphically by using the depiction of a proposition as a line between the possibilities it rules out and those it leaves open.
Fig. 1.2.3-1. The relation between non-equivalent propositions φ and ψ where φ ⊨ ψ, depicted (on the left) by indicating the relation between the possibilities ruled out and (on the right) by indicating the possibilities left open by φ and ψ.
Here φ ⊨ ψ but ψ ⊭ φ, so there are no possible worlds where φ is true and ψ is false, but there are possibilities where φ is false while ψ is true. The possibilities ruled out by ψ, which are also ruled out by φ, are in the small region hatched in both directions on the upper left. The area outside this region but still on the left of the line running through the middle of rectangle are possibilities ruled out by φ that are not ruled out by ψ. These show that the content of φ extends beyond that of ψ and thus that ψ ⊭ φ. The diagram on the right depicts that same relation by way of possibilities left open rather than possibilities ruled out. While φ leaves open only those possibilities on the right of the diagram, ψ leaves open all that are not within the region at the upper left, so ψ leaves open any possibility that φ does but not vice versa; that is, ψ is true whenever φ is but is true in some possible worlds in which φ is false.
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.