1.4.4. A general framework
It is not surprising that the relative exhaustiveness should encompass deductive properties and relations if these are understood to 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, and this means that it 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 of the right of the arrow, the set in question is the empty set.)
| Concept | 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.
The definition of this idea can be read off its description in terms of relative exhaustiveness. 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, any possible world makes all its members true because there is no member to provide a counterexample to the claim that the world has made them all true. 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. That is, 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 these properties and relations can be read off the right side of the table by applying the definition of relative exhaustiveness to the case or cases indicated. When the set on one side or the other of the arrow has 0, 1, or 2 members, a direction application of the definition can be simplified as we just saw in the case of exhaustiveness.
The ideas of division and relative exhaustiveness also provides ways of extending the idea of logical independence introduced in 1.2.3 to speak of the absence of any deductive property or relation. Let us 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. When the set has two members, this is the same as the earlier idea. When a set {φ} containing a single sentence φ is logically independent in this sense, we can say that φ is logically contingent because there is at least one possible world in which it is true and at least one where it is false.
Relative exhaustiveness provides another way of looking at the same 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 when sets 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 contingency of members and 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.