1.4.5. Reduction to entailment
Relative exhaustiveness generalizes relaxing the restriction to a single conclusion to allow several alternatives or none at all. To express the ideas captured by relative exhaustiveness in terms of entailment, we need add ways of expressing both of these ideas.
We have already seen a way of expressing the idea of rendering exhaustive the empty set of alternatives. In 1.2.5, we characterized inconsistency in terms of entailment and absurdity by what was called the Basic Law for Inconsistency. If we restate that law expressing inconsistency in terms of relative exhaustiveness, it says
so a set renders exhaustive an empty set of alternatives if and only if it entails the absurdity ⊥. Both of these are conditional guarantees of something that cannot happen, so they have the effect of ruling out the possibility that the conditions of the guarantee (i.e., the truth of all members of Γ) can ever be met.
To express the idea of rendering exhaustive multiple alternatives using entailment we need help from the concept of contradictoriness. When sentences φ and ψ are contradictory (i.e., when φ ⊗ ψ), they always have opposite truth values. so making one true comes to the same thing as making the other false. Since the difference between having a sentence as an assumption and having it as an alternative lies in the truth value assigned to it in the pattern that is being ruled out. This means that having a sentence as an alternative comes to the same thing as having a sentence contradictory to it as an assumption; that is,
If we apply this idea repeatedly (perhaps infinitely many times), we can move any set of alternatives to the left of the arrow. To make it easier to state the result of doing this, we will use Γ for the result of replacing each member of Γ by a sentence contradictory to it.
Basic law for relative exhaustiveness. Γ ⇒ Δ, Σ if and only if Γ, Δ ⇒ Σ.
That is, extra alternatives can be removed if we put sentences contradictory to them among the assumptions. This gives us two ways of restating claims of relative exhaustiveness as entailments: (i) we may replace all but one alternative by contradictory sentences among the assumptions or (ii) we may replace all alternatives by contradictory sentences and replace the resulting empty set of alternatives by ⊥.
The following table summarizes the application of these ideas to state all the deductive properties we have considered using entailment, absurdity, and contradictoriness:
| Concept | in terms of entailment and other ideas |
| Γ entails φ | Γ ⇒ φ |
| φ is a tautology | ⇒ φ |
| φ and ψ are equivalent | both φ ⇒ ψ and ψ ⇒ φ |
| Γ excludes φ (i.e., Γ, φ ⇒ ) | Γ, φ ⇒ ⊥ |
| Γ is inconsistent (i.e., Γ ⇒ ) | Γ ⇒ ⊥ |
| φ and ψ are mutually exclusive (i.e., φ, ψ ⇒ ) | φ, ψ ⇒ ⊥ |
| φ is absurd (i.e., φ ⇒ ) | φ ⇒ ⊥ |
| Γ is exhaustive (i.e., ⇒ Γ) | Γ ⇒ ⊥ |
| φ and ψ are jointly exhaustive (i.e., ⇒ φ, ψ) | φ, ψ ⇒ ⊥ (or φ ⇒ ψ or ψ ⇒ φ) |
| φ and ψ are contradictory (i.e., both φ, ψ ⇒ and ⇒ φ, ψ) | both φ, ψ ⇒ ⊥ and φ, ψ ⇒ ⊥ |
| Here φ is any sentence contradictory to φ, and Γ is the result of replacing each member of Γ by a sentence contradictory to it | |
Of course, either of the two further ways of stating exhaustiveness could be used instead of the second entailment required for two sentences to be contradictory. And, when a non-empty set Γ is said to be exhaustive, we could leave one member behind as a conclusion rather than adding ⊥; that is, ⇒ Γ, φ when Γ ⇒ φ.