1.4.5. A general framework

It was noted in the last section that relative exhaustiveness does not merely generalize entailment and absolute exhaustiveness but encompasses all deductive properties and relations. It is not surprising that does so if these properties and relations are understood to all consist in guarantees that certain parterns of truth values appear in no possible world. For any claim there is no world where certain sentences Γ are true and other sentences Σ are false is a claim that Γ ⊨ Σ. Of course, a given deductive property or relation may rule out a number of different patterns—i.e., rule out a number of different ways of distributing truth values among the sentences it applies to—but this just means that a deductive property or relation may consist of a number of different claims of relative exhaustiveness. In the case of the properties and relations we will consider, only equivalence and contradictoriness involve more than one claim of relative exhaustiveness.

The table below summarizes the deductive properties and relations that involve only one claim of relative exhaustiveness, and also shows the vocabulary we have used for various special cases. The first row covers cases where the number of assumptions is unspecified, with the next three concerning the cases of specific numbers of assumptions. Similarly, the first column places no constraints on the number of alternatives while the following three columns do. As a result, the first cell in the first row, the one for relative exhaustiveness encompasses all the rest. Notice, for example, that because tautologousness concerns a single alternative and no assumptions, it is a special case of both entailment and exhaustiveness.

		alternatives
		any no.	two	one	none
	any no.	Γ ⊨ Σ	Γ ⊨ ψ, ψ′	Γ ⊨ ψ entails	Γ ⊨  inconsistent
	two	φ, φ′ ⊨ Σ	φ, φ⁠′ ⊨ ψ, ψ⁠′	φ, φ′ ⊨ ψ	φ, φ′ ⊨ (or φ ▵ φ′) mutually exclusive
	one	φ ⊨ Σ	φ ⊨ ψ, ψ′	φ ⊨ ψ implies	φ ⊨  absurd
	none	⊨ Σ exhaustive	⊨ ψ, ψ′ (or ψ ▿ ψ′) jointly exhaustive	⊨ ψ tautologous	⊨

Since there are no alternatives in question, the ideas in the last column are really properties of sets of assumptions (just as those in the last row are properties of sets of alternatives). It does not make a claim about some sentence or set of sentences but about entailment itself, and the claim it makes is false. Since it concerns the case of no assumptions and no alternatives, it might be written more explicitly as the claim that ∅ ⊨ ∅. As was noted at the end of 1.4.2, the empty set ∅ is bound to be separated from itself, so ∅ ⊨ ∅ is bound to be false. And that’s a result we should expect because this case of relative exhaustiveness offers an unconditional guarantee that some member of the empty set is true.

The ideas of separation and relative exhaustiveness also provide ways of extending to any set the idea of logical independence introduced in 1.2.8 in the case of a pair of sentences. First, let us look at this general idea of logical independence directly. We will say that a set Γ of sentences is logically independent when every way of assigning a truth value to each member of Γ is exhibited in at least one possible world. This is the same as saying that, for every part of the set (counting both the empty set and the whole set Γ as parts of Γ), it is possible to separate that part from the rest of the set. When the set has two members, this is the same as the earlier idea of logical independence; and when the set {φ} containing a single sentence φ is logically independent in this sense, it is neither a tautology or absurd—i.e., it is logically contingent in the sense defined in 1.2.5.

Relative exhaustiveness provides an alternative way of describing logical independence of this general sort. For, when the sentences in a set are not independent, not every way of dividing them into a set of true sentences and a set of false sentences is logically possible. And when some way of dividing them is not possible, the set contains at least one pair of non-overlapping subsets Γ and Σ such that Γ ⊨ Σ. And, of course, if the set contains such a pair, its members are not logically independent. So the members of a set are logically independent when the relation of relative exhaustiveness never holds between non-overlapping subsets. (It always holds between sets that overlap because then there clearly is no way of separating one set from the other.)

When a set is logically independent, each member is contingent and any two of its members are logically independent, but the contingency of members and the independence of pairs does not by itself imply that the set as a whole is logically independent. For example, assume that the sentences X is fast, X is strong, X has skill, and X has stamina form an independent set. Then the sentences

are each contingent, and any two of them can be seen to be independent. However, the first two taken together entail the third, so these three more complex sentences do not form an independent set. This also shows that, while it is natural to speak of the members of the set as independent, independence in this sense is really a property of the set as a whole. For we can say that the two sentences X is fast and strong and X is fast and has stamina are independent as a pair, but adding the third gives us a set whose members do not count as independent with respect to the other two.