1.4.5. Reduction to entailment
Conditional exhaustiveness relaxes the restriction to a single conclusion found in entailment to include cases where there are several alternatives or none at all. To express the ideas captured by conditional exhaustiveness in terms of entailment, we need to add ways of capturing each of these added cases.
When a claim of conditional 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:
Inconsistency via Absurdity. Γ ⊨ (i.e., Γ ⊨ ∅) if and only if Γ ⊨ ⊥.
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.
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 comes to the same thing as having its contradictory 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. 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 contradictory sentences play opposite roles when sets are being divided. More specifically, if φ and φ are contradictory sentences, then
| Γ | is divided from | (Δ together with φ) |
| if and only if | ||
| (Γ together with φ) | is divided from | Δ |
because each of these divisions requires that φ be made false and φ be made true. Since a claim of conditional exhaustiveness asserts that a division 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 (perhaps infinitely many times), we can move any set of alternatives to the left of the turnstile, and that is the basis of the following law:
Alternatives via assumptions. Let Δ be 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.
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 conditional 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 state all the deductive properties we have considered in terms of the general ideas of entailment and contradictoriness and of the specific absurdity ⊥:
| Concept | in terms of entailment and other ideas |
| φ is a tautology | ⊨ φ |
| Γ entails φ | Γ ⊨ φ |
| φ is absurd—i.e., φ ⊨ | φ ⊨ ⊥ |
| φ and ψ are mutually exclusive—i.e., φ ▵ ψ (or φ, ψ ⊨) | φ, ψ ⊨ ⊥ |
| Γ excludes φ—i.e., Γ, φ ⊨ | Γ, φ ⊨ ⊥ |
| Γ is inconsistent—i.e., Γ ⊨ | Γ ⊨ ⊥ |
| φ and ψ are jointly exhaustive—i.e., φ ▿ ψ (or ⊨ φ, ψ) | φ ⊨ ψ (or ψ ⊨ φ, or φ, ψ ⊨ ⊥) |
| Γ is exhaustive—i.e., ⊨ Γ | Γ ⊨ ⊥ |
| φ and ψ are equivalent—i.e., φ ≃ ψ | both φ ⊨ ψ and ψ ⊨ φ |
| φ and ψ are contradictory—i.e., φ ⧖ ψ (or both φ, ψ ⊨ and ⊨ φ, ψ) | both φ, ψ ⊨ ⊥ and φ ⊨ ψ (or ψ ⊨ φ, or 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 | |
There are alternative ways of stating each of these ideas in terms of entailment. Any time ⊥ appears as the conclusion and there is at least one assumption, ⊥ could be replaced as the conclusion by a sentence contradictory to an assumption, which is then dropped. 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 ⊨ ⊥. Also, we may replace an assumption and the conclusion both by putting a sentence contradictory to each on the other side of the turnstile—i.e., Γ, φ ⊨ ψ if and only if Γ, ψ ⊨ φ.
It may seem pointless to define the relation of contradictoriness in terms of entailment, as is done in the last row of this 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.