1.4.3. Conditional exhaustiveness

We can use the idea of division to generalize entailment to a relation between sets. And it is useful to do this because the more general relation encompasses all the deductive properties and relations of sentences. Although we have focused on entailment and will continue to do so, it doesn’t suffice by itself to capture all the ideas of deductive logic. In particular, we need the idea of the absurdity ⊥ to describe inconsistency in terms of entailment, and we have not yet seen how to say, in terms of entailment, when sentences are jointly exhaustive. But the more general relation can serve to define both of these ideas.

This new relation associated with joint exhaustiveness in much the way entailment is associated with tautologousness. Actually, it is associated in this way with a more general idea of exhautiveness that concerns any number of sentences, not merely two. Just as a pair of sentences are jointly exhaustive when we can be sure that, no matter what, at least one of the two is true, we will say that a set Δ of any size is exhaustive when we can be sure that at least one of its members is true. We will speak of these members as alternatives, so a set of alternatives is exhaustive when we can be sure that always at least one of these alternatives is true.

For example, the alternatives The glass is full, The glass is empty, and The glass is partly full form a set that is exhuastive in this sense. You might notice that it happens that any two of these alternatives are mutually exclusive, but that is an accident of this example. Replacing the first two alternatives with The glass is at least 90% full and The glass is no more than 10% full would not damage exhaustiveness since the new alternatives are true in even more possibilities, and neither of them excludes the claim that the glass is partly full. For another, more artificial, example, consider The book is not red, The book is not green, and The book is not blue. It is possible for all three of these alternatives to be true, so certainly no two of them are mutually exclusive; and if one is false the other two are true, so we are bound to have at least two of them true and the three are certainly an exhaustive set of alternatives.

We will use the notation ⊨ Δ for this general idea of exhaustiveness and define it more formally (in a negative and positive form, respectively) as follows:

The notation for exhaustiveness provides notation for tautologousness; for, if φ is the sole member of Δ, a guarantee that at least one alternative from Δ is true is a guarantee that φ is true. So we can write ⊨ φ to say that φ is a tautology—i.e., that φ ≃ ⊤. The extended use of the entailment turnstile also provides us with a new notation for the idea of joint exhaustiveness: φ ▿ ψ if and only if ⊨ φ, ψ.

Now let us return to the project of generalizing entailment. 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 Δ is exhaustive unconditionally tells us that ranges of possibilities left open by its alternatives taken together cover all possibilities whatsoever. We can say that a set Δ is exhaustive given a set Γ when the ranges of possibilities left open by the alternatives in Δ taken together cover all possibilities in which every assumption in Γ is true. When this is so we have a guarantee that in any possible world in which all assumptions in Γ are true at least one alternative in Δ is true. 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.

Our notation for conditional exhaustiveness will again use the entailment turnstile, writing Γ ⊨ Δ with the set of assumptions on the left and the set of alternatives on the right. It will help in reading this notation to have vocabulary that makes Γ the subject, so we will say that Γ renders Δ exhaustive when Δ is exhaustive given Γ. The negative and positive forms of the definition of this idea are as follows:

And, as promised, this idea can be stated very directly 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 where the set Δ consists of a single sentence, for to say that φ is entailed by Γ comes to the same thing as saying that φ is rendered exhaustive by Γ. Either way we are claiming that there is no possible world that divides Γ from φ.

In cases of conditional 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 alternatives 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 alternative.