1.4.3. Relative exhaustiveness

Clearly, we can use the idea of division in the way we can use it to define entailment to define a relation between sets rather than between a set and a sentence, and there is reason for doing this because the result constitutes a single fundamental idea that encompasses all the concepts of deductive reasoning. 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. We needed to add the idea of absurdity in 1.2.5 to capture the idea of inconsistency.

This more general relation 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 arrow 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 and 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 the other cases of relative exhaustiveness, it is either a set with several members or the empty set that is rendered exhaustive. In these cases, it does not make sense to speak of a conclusion. When the set on the right have several members, they 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 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.

Consequently, we need new terminology for sentences on the right of the arrow 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, either φ or ψ is true.