1.4.4. Relative exhaustiveness

We can use the idea of separation to generalize entailment to a relation between sets. And it is useful to do this because we can capture all the deductive properties and relations of sentences by using a relation that simply says that one set cannot be separated from another. That’s what the following relation does:

For reasons to be discussed shortly, when this relation holds, we will say that Γ renders Σ exhaustive, and we will use the phrase relative exhaustiveness to refer to this relation. We have reused the notation for entailment because entailment is the special case where the set Σ consists of a single sentence (just as implication is the special case of entailment where Γ consists of a single sentence).

While tautologousness is an unconditional guarantee of truth, entailment guarantees the truth of its conclusion only given the truth of a set of assumptions. Entailment is thus a guarantee of truth for a single sentence only given the conditions set out in the assumptions, and we can think about an analogous conditional guarantee that a set is exhaustive. Saying that Σ is exhaustive unconditionally tells us that its cumulative coverage includes all possibilities whatsoever. We can say that a set Σ is exhaustive given a set Γ when the cumulative coverage of Σ includes the shared coverage of Γ. For example, while the two alternatives The glass is full and The glass is empty are not jointly exhaustive, they are exhaustive given the assumption The glass is not partly full since it rules out all possibilities where they are both false. It is this sort of conditional exhaustiveness that is asserted by the relation above: relative exhaustiveness is exhaustiveness relative to a set of assumptions that limit the possibilities that the relatively exhaustive set must cover.

In cases of relative exhaustiveness that are not cases of entailment, what is rendered exhaustive is either a set of several alternatives or the empty set. In these cases, it does not make sense to speak of a conclusion, for when the set on the right has several members, these sentences need not be valid conclusions from the set that renders them exhaustive. Indeed, a jointly exhaustive pair of sentences will be rendered exhaustive by any set, but often neither member of the pair will be entailed by that set. This is particularly clear in the case of sentences like The glass is full and The glass is not full that are both jointly exhaustive and mutually exlcusive—i.e., that are contradictory. Although the set consisting of such pair is rendered exhaustive by any set, only an inconsistent set could entail both of these alternatives. So the term conclusion will be reserved for cases where there is a single sentence on the right of the sign ⊨. When there is more than one, we will speak of these sentences as alternatives. This term is also appropriate when the set on the right is empty, for then the claim being made is that no alternatives are needed to cover the possibilities where the assumptions are all true—that is, they form an inconsistent set.

A case Γ ⊨ Σ of relative exhaustiveness indicates relations of both content and coverage, but the ideas of content and coverage must be applied in different senses to Γ and Σ. The cumulative content of Γ is said to include the shared content of Σ. We wouldn’t expect it to include more than this because the members of Σ may go off in different, even incompatible directions to cover the possibilities left open by the members of Γ. And the cumulative coverage of Σ is said to include all of the shared coverage of Γ, but only its shared coverage because the members of Γ may contribute in different ways to narrowing the range of possibilities that Σ is said to cover.