1.4.3. Content and coverage

In the case of implication, the idea of separation was tied to relations of content and coverage between sentences. When φ can be separated from ψ, we know that it is possible to have φ true and ψ false, so the content of φ does not include that of ψ and ψ does not cover every possibility that ψ does. On the other hand, when φ cannot be separated from ψ—i.e., when φ ⊨ ψ—we know that the content of ψ is part of the content of φ and that ψ covers any possibility that φ does.

These relations of content and coverage are closely tied to the ideas of extracting content or of having a conditional guaranteed of truth, so it is natural to extend them to entailment generally. But, to do this, we need to see how to apply the ideas of content and coverage to sets. We will actually apply them each in two different ways. Initially, we will use one approach for content and the other for coverage, but we will bring in the remaining two options shortly. The cumulative content of a set Γ—or, alternatively, its content as a set of assumptions—is set of all possible worlds in which every member of Γ is false. That means that the cumulative content of Γ includes the content of each of its members—it is the result of accumulating their individual contents—and this is the content we are interested in when asking what follows from a set of assumptions. The coverage of a set Γ as a set of assumptions consists of the remaining possible worlds, the ones that are not ruled out by any member of Γ and thus appear in the coverage of each of its members. We will describe this set also as the shared coverage of Γ since it consists of overlap in the coverages of the individual members of Γ. And when we use a deductively valid argument as support for our confidence in the coverage of a conclusion, what we know is that its coverage is at least as great as the shared coverage of the premises.

It’s no surprise that the cumulative content of an empty set is empty, for there is nothing to accumulate. But it may seem surprising that its shared coverage is the full range of possibilities. This another instance of a vacuous generalization: the shared coverage of a set includes a possible world when every member of the set covers that world, and that is bound to be so when the set has no members. But it makes sense in its own right. We would expect that the shared coverage might decrease as we add members to a set since the new coverage in this sense will be limited to any overlap between what the set did cover and what is covered by a new member. But a set with one member φ has as its shared coverage just the coverage of φ, and it is the result of adding φ to the empty set. So the shared coverage of the empty set must have been at least that of φ, and that means it must have included every possible world because φ might be a tautology and have all possibilities in its coverage.

Of course it is possible to add up possibilities covered by the members of a set just as we have added up their contents. When we do this we will say that we are looking at the set as a set of alternatives. The idea is that, from this perspective, adding new members to the set can allow it to cover possibilities it did not before, so it adds new alternative ways of covering possibilities. So the coverage of a set Γ as a set of alternatives is the cumulative coverage of the set; it is the full range of possibilities covered by any of the set’s members. When we look at a set as a set of alternatives, its content is the set of possibilities that are ruled out no matter what alternative we consider; that is, it is the set’s shared content.

	content	coverage

Fig. 1.4.3-1. A set of three sentences with the content of each sentence shaded on the left and its coverage shaded on the right. The top row shows, by hatching, the content and coverage of the set as a set of assumptions, and the bottom row shows its content and coverage as a set of alternatives. Cumulative content or coverage is hatched horizontally, and shared content or coverage is hatched vertically.

Figure 1.4.3-1 illustrates the application of ideas we been surveying to the case of a set of sentences with three members. Notice that the hatched areas side by side in each row are exactly opposite. The full range of possible worlds is divided between the content and coverage of any given sentence, and the same is true of the content and coverage of any set, provided we consider the set in both cases either as a set of assumptions or as a set of alternatives.

The application of these ideas to an empty set should now be no surprise. An empty set provides no alternatives, so its cumulative coverage is empty; and it has no members whose content must be respected in determining an overlap, so its shared coverage is the full range of possibilities.

Notice also that if we say that Γ is separated from Σ, we speak of a possibility that is in the shared coverage of Γ (because all of Γ’s members are true) that is not in the cumulative coverage of Σ (because all of Σ’s members are false). So if we reject the possibility of separation in this case, we say that the shared coverage of Γ is included in the cumulative coverage of Σ or, equivalently, that the cumulative content of Γ includes the shared content of Σ. In short, these two different perspectives on sets are built into the idea of separability that is at the root of all deductive properties and relations.