1.4.4. A general framework
It is not surprising that relative exhaustiveness should encompass deductive properties and relations if these properties and relations are understood to all consist in guarantees that certain parterns of truth values appear in no possible world. For to say that there is no world where certain sentences Γ are true and other sentences Δ are false is to say that Γ ⊨ Δ. 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, no more than two claims of relative exhaustiveness are ever required, as can be seen in the following table. (When nothing appears to the left or the right of the turnstile, the set on that side is the empty set.)
| Concept | described in terms of relative exhaustiveness |
| Γ entails φ | Γ ⊨ φ |
| φ is a tautology | ⊨ φ |
| φ and ψ are equivalent | both φ ⊨ ψ and ψ ⊨ φ |
| Γ excludes φ | Γ, φ ⊨ |
| Γ is inconsistent | Γ ⊨ |
| φ and ψ are mutually exclusive | φ, ψ ⊨ |
| φ is absurd | φ ⊨ |
| Γ is exhaustive | ⊨ Γ |
| φ and ψ are jointly exhaustive | ⊨ φ, ψ |
| φ and ψ are contradictory | both φ, ψ ⊨ and ⊨ φ, ψ |
This list adds only one concept to those already discussed, a generalization of the idea of a pair of jointly exhaustivess sentences to the exhaustiveness of a set, and this is a good example of how these descriptions work. A direct definition of this new idea can be read off its description in terms of relative exhaustiveness in the following way. To say that ⊨ Γ is to say that there is no possible world that divides the empty set and Γ. That is, there is no possible world that makes every member of the empty set true and every member of Γ false. But, since the empty set has no members, there is no way for any possible world to fail to make all its members true because there is nothing to serve as an exception. This means that the property of making all members of the empty set true adds nothing to the description of the sort of world ruled out by the claim that ⊨ Γ, and this claim can be stated more simply by saying that there is no possible world that makes all members of Γ false. To state the definition in positive form, a set Γ is exhaustive when, in every possible world, at least one member of Γ is true. That is, if we take the sets of possible worlds left open by the various members of Γ and put them all together, they will all exhaust all possibilities. In the same way, the definition of each of the properties and relations in the table above can be read off the right side of the table by applying the definition of relative exhaustiveness to the case or cases indicated.
The ideas of division and relative exhaustiveness also provide ways of extending to a set the idea of logical independence introduced in 1.2.6 to speak of the absence of any deductive property or relation in 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 divide that part from the rest of the set. When the set has two members, this is the same as the earlier idea. When the set {φ} containing a single sentence φ is logically independent in this sense, the sentence φ is said to be logically contingent because there is at least one possible world in which it is true and at least one where it is false, so its truth or falsity is not settled by logic.
Relative exhaustiveness provides an alternative way of describing this idea. 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 that is so, the set contains at least one pair of non-overlapping subsets Γ and Δ such that Γ ⊨ Δ. 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 there is no way of dividing such sets.)
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
|
X is fast and strong |
X has skill and stamina |
X is fast and has stamina |
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.