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. The impossibility of a T-F pattern of truth values—in this case for φ and ψ—is an idea that will reappear often, and we will say that a possible world that did make φ true and ψ false would have separated φ from ψ. So φ implies ψ when φ cannot be separated from ψ. The separation in question is separation in regards to truth: when φ implies ψ, if φ is true, then ψ will be true as well.
Separation in this sense is asymmetric. Even if φ cannot be separated from ψ, it may be possible to separate ψ from φ. For example The meeting is tomorrow morning cannot be separated from The meeting is tomorrow. But if the meeting is in fact tomorrow afternoon, the latter sentence will have been separated from the former, for it will be true that the meeting is tomorrow even though it is false that it is tomorrow morning. Clearly what is going on here is that The meeting is tomorrow morning says everything that is said by The meeting is tomorrow, and says something more. The first sentence cannot be separated from the second because, for the first sentence to be true, everything said by the second sentence must be true. But the second sentence can be separated from the first because the extra content of the first may be false even though the second sentence is true. The same point can be made in terms of coverage. If φ implies ψ, then ψ must cover all the possibilities that φ does—for otherwise φ could be separated from it—but it may cover others that φ does not. The second sentence in the example covers the possibility of an afternoon meeting but the first does not.
In short, implication is a relation of both content and coverage, but in opposite directions. If φ implies ψ, then the content of φ includes the content of ψ, and the coverage of ψ includes the coverage of φ. When the relation fails to hold in the other direction—in symbolic notation, when ψ ⊭ φ—we know that the content of φ extends beyond that of ψ. That’s why there can be a possibility separating ψ from φ, a possibility where the extra content of φ is false even though the content of ψ is true. And such a possible world will be part of the coverage of ψ but not that of φ, so the coverage of ψ extends beyond that of φ.
As a more extended example of this terminology, consider the following series of successively more informative statements, each implied by the one below it:
Content increases as we go down the list, and coverage decreases. Each sentence above the last covers some possibilities that are ruled out by the sentence below it. And in general, as we add information, we reduce the range of possibilities that are covered. We will often speak of a sentence that rules out possibilities another does not (and thus does not cover possibilities that the other does) as making a stronger claim, and we will speak of sentence that does not rule out possibilities ruled out by another (and thus covers possibilities the other does not) 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 left open by ψ. These possibilities separate ψ from φ and thus show that ψ ⊭ φ. The diagram on the right depicts that same relation by way of possibilities covered rather than possibilities ruled out. While φ covers only those possibilities on the right of the diagram, ψ covers all that are not within the region at the upper left, so ψ covers any possibility that φ does but not vice versa; that is, ψ is true whenever φ—i.e., φ cannot be separated from it—but ψ is true in some possible worlds in which φ is false—i.e., it can be separated from φ.
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, that it leaves fewer or the same possibilities open.
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.