1.4.5. Reduction to entailment
Relative exhaustiveness relaxes the restriction to a single conclusion found in entailment to allow several alternatives or none at all. To express the ideas captured by relative exhaustiveness in terms of entailment, we need to add ways of capturing each of these added cases.
We have already seen a way to use entailment to say what it is to render exhaustive the empty set of alternatives. In 1.2.4, we characterized inconsistency in terms of entailment and absurdity by what was called the Basic Law for Inconsistency. If we restate that law by 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 ⊥. Rendering exhaustive an empty set and entailing ⊥ are both conditional guarantees of something that cannot happen, so each has the effect of ruling out the possibility of meeting the conditions of the guarantee (i.e., of having all members of Γ true).
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. But 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 by the claim of relative exhaustiveness. 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 turnstile.
Basic law for relative exhaustiveness. Suppose Δ′ is the result of replacing each member of Δ by a sentence contradictory to it. Then Γ ⊨ Δ, Σ if and only if Γ, Δ′ ⊨ Σ.
That is, we can remove alternatives 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 assumptions contradictory to them or (ii) we may replace all alternatives by asssumptions contradictory to them 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 | |
Either of the two further ways of stating exhaustiveness shown in the next-to-last row could be used instead of the second entailment in the last row. 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, the idea that ⊨ Γ, φ could be restated not only as Γ′, φ′ ⊨ ⊥ but also as Γ′ ⊨ φ. That is, a set containing a sentence φ is exhaustive if we have a guarantee that φ is true when the others members of the set are false. (It may seem pointless to define the relation of contradictoriness in terms of entailment when we need to use the relation in order to do this, but the definition does mean that, once we know a single sentence contradictory to a given sentence, we say what other sentences are contradictory to it using only the ideas of entailment and absurdity.)