1.4.3. Relative exhaustiveness

We can use the idea of division to define a relation between sets rather than between a set and a sentence. And it is useful to do this because the relation we define in this way constitutes a single fundamental idea that encompasses all the deductive properties and relations of sentences. We have focused on entailment and will continue to do so, but it doesn’t suffice by itself to capture all the ideas of deductive logic. In particular, we needed the idea of absurdity in 1.2.4 to capture the idea of inconsistency, and we have not yet seen how to say, in terms of entailment, when sentences are jointly exhaustive.

The more general relation we will define using division is relative exhaustiveness. When it holds between a pair of sets, we will say that one set renders the other set exhaustive. Our notation for this idea will extend the use of the entailment turnstile to allow a set to appear on the right. The negative and positive forms of its definition are as follows:

Or, in terms of division, Γ ⊨ Δ if and only if there is no possible world that divides Γ from Δ. Entailment is the special case of this idea where the set Δ consists of a single sentence: to say that φ is entailed by Γ comes to the same thing as saying that φ is rendered exhaustive by Γ.

In cases of relative exhaustiveness that are not cases of entailment, what is rendered exhaustive is either a set with several members 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 sentences.

This means that we need new terminology for sentences on the right of the turnstile when they appear in groups. We will say that such sentences are alternatives. The conditional guarantee provided by a claim Γ ⊨ Δ of relative exhausitiveness is a guarantee that the alternatives Δ are not all false—i.e., that at least one is true—provided the premises Γ are all true. In particular, when Γ ⊨ φ, ψ, we have a guarantee that, if the members of Γ are all true, then either φ or ψ is true.