1.4.6. Reduction to entailment

Relative exhaustiveness generalizes entailment by allowing cases in which we have, instead of a single conclusion, multiple 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.

When a claim of relative exhaustiveness offers no alternatives, it asserts the inconsistency of the assumptions; and that comes to the same thing as entailing the specific absurdity ⊥. That is, we can state the following:

This law holds because rendering exhaustive the empty set and entailing ⊥ both offer conditional guarantees of a truth that cannot exist, so each has the effect of ruling out the possibility of meeting the conditions of the guarantee. Alternatively (but equivalently) we can note that ⊥, since it cannot be true, covers the same possibilities as the empty set, so, like the empty set, it covers the possibilities in the shared coverage of Γ only when that is empty. And to say that the shared coverage of a set is empty is to say that its members cannot all be true.

To express the idea of rendering exhaustive multiple alternatives using entailment we need help from the concept of contradictoriness. Contradictoriness comes in here because having an exception in a guarantee—which is what an added alternative provides—comes to the same thing as having the contradictory of this exception as a condition. For example, the guarantee The product will funciton for three years unless it is abused is equivalent to The product will function for three years if it hasn’t been abused, and the guarantee The product will function for three years if it is serviced regularly is equivalent to The product will function for three years unless it is not serviced regularly. In the first case we move from an exception to a condition, and in the second we move in the opposite direction. To make this intuitive point more formally, note first that when sentences are contradictory, they always have opposite truth values. So making one true comes to the same thing as making the other false, and that means that contradictory sentences play opposite roles when one set is being separated from another. More specifically, if φ and φ^⋈ are contradictory sentences, then

because each of these separations requires that φ be made false and φ^⋈ be made true. Since a claim of relative exhaustiveness asserts that a separation is not possible, 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, we can move a set of alternatives to the left of the turnstile, and the direct justification for doing that is the same: having a collection of the sentences in the right comes to the same thing, as far as separation is concerned, as having on the left sentences contradictory to the members of the collection. If we assume there is no limit on the number of times this can be done, we get the following law:

In short, we can remove alternatives if we put sentences contradictory to them among the assumptions.

The laws we have seen give us two approaches to restating claims of relative exhaustiveness as entailments. A claim with no alternatives—i.e., a claim of inconsistency—can be turned into an entailment by adding ⊥ as the conclusion. And we may replace any alternatives by assumptions contradictory to them to reduce multiple alternatives to a single conclusion. The two may be combined by replacing all alternatives by contradictory assumptions and then adding ⊥ as conclusion.

The following table uses these two approaches to restate all the deductive properties shown in the table of the last subsection:

		alternatives
		any no.	two	one	none
	any no.	Γ ⊨ Σ Γ, Σ⋈ ⊨ ⊥	Γ ⊨ ψ, ψ′ Γ, ψ⋈ ⊨ ψ′	entails Γ ⊨ ψ (same)	inconsistent Γ ⊨ Γ ⊨ ⊥
	two	φ, φ′ ⊨ Σ φ, φ′, Σ⋈ ⊨ ⊥	φ, φ′ ⊨ ψ, ψ′ φ, φ′, ψ⋈ ⊨ ψ′	φ, φ′ ⊨ ψ (same)	mutually excl. φ, φ′ ⊨ φ, φ′ ⊨ ⊥
	one	φ ⊨ Σ φ, Σ⋈ ⊨ ⊥	φ ⊨ ψ, ψ′ φ, ψ⋈ ⊨ ψ′	implies φ ⊨ ψ (same)	absurd φ ⊨ φ ⊨ ⊥
	none	exhaustive ⊨ Σ Σ⋈ ⊨ ⊥	jointly exh. ⊨ ψ, ψ′ ψ⋈ ⊨ ψ′	tautologous ⊨ ψ (same)	⊨ ⊨ ⊥

The natural statement of the property or relation in terms of relative exhaustiveness is shown first, followed by a statement in terms of entailment if that is different. The alterations are made in the same way for each column. In the last column, ⊥ is added to get a conclusion; in the second, the alternative ψ is removed and its contradictory ψ^⋈ is added as an assumption; and, in the first, the set of alternatives Σ is replaced by ⊥ and the contradictories Σ^⋈ of the members of Σ are added as assumptions.

There are other says of stating most of these ideas in terms of entailment, absurdity, and contradictoriness. Any time ⊥ appears as the conclusion and there is at least one assumption, ⊥ could be replaced as the conclusion by a sentence contradictory to some assumption, which is then dropped from the assumptions. That is, Γ, φ ⊨ ⊥ if and only if Γ ⊨ φ^⋈. And whenever ⊥ is not the conclusion, it could be made the conclusion if the a sentence contradictory to the previous conclusion is added to the assumptions—i.e., Γ ⊨ φ if and only if Γ, φ^⋈ ⊨ ⊥. So, in particular, saying that φ^⋈ ⊨ ψ comes to the same thing as saying that φ^⋈, ψ^⋈ ⊨ ⊥, which comes to the same thing as saying that ψ^⋈ ⊨ φ. In particular, the claim of relative exhaustiveness beginning the following list can be restated as any of the claims of entailment after it:

(This is an instance of the third row, second column of the table.)

It may seem pointless to define the relation of contradictoriness in terms of entailment, as is done in the last row of the table, since we need to use the idea of contradictoriness in order to do this. But the definition does mean that, once we know a single sentence contradictory to a given sentence, we can say what other sentences are contradictory to it using only the ideas of entailment and absurdity.