1.4.4. A general framework

It was noted in the last section that conditional exhaustiveness does not merely generalize entailment and unconditional 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 Δ are false is a claim 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 conditional exhaustiveness. In the case of the properties and relations we will consider, only equivalence and contradictoriness involve more than one claim of conditional exhaustiveness.

The table below summarizes the deductive properties and relations that involve only one claim of conditional exhaustiveness along with the vocabulary we have used for various special cases. The ideas discussed in the last subsection appear in the three columns at the right. Moving down one of these columns, we move from an unconditional guarantee of truth somewhere in a set of alternatives to a conditional guarantee that is hedged with one or more assumptions. Moving left to right in a one of the rows, we move from a guarantee of truth that focuses on a single alternative, a definite conclusion, to one that applies to a set of two or more alternatives.

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

The column to the left of these three covers the cases where the set of alternatives is empty. There can be no unconditional guarantee of this sort, so there is no entry in the first row. The entry would not be a property or relation but instead the false statement ∅ ⊨ ∅ (which asserts an unconditional guarantee that some member of the empty set is true).

Since there are no alternatives in question, the ideas in the first column are really properties of sets of assumptions (just as those in the first row are properties of sets of alternatives). Absurdity, mutual exlucsiveness, and inconsistency are negative properties, each of which guarantees that a certain group of assumptions cannot all be true. They do this indirectly by making these assumptions conditions of a guarantee of something that is bound to be false—i.e., that the empty set of alternatives exhausts all possibilities. That is, they use the same device as a sentence like If that’s a good book, then I’m the King of France which denies something by stating it as a sufficient condition for an absurd claim.

So, in each of these columns, movement down from one row to the next is a matter of making a guarantee of truth conditional on further assumptions. It is possible to think of movement to the right within each row in a somewhat analogous way: adding alternatives modifies a guarantee by adding exceptions. To claim that It is raining and It isn’t raining are jointly exhaustive is not to guarantee the truth of either sentence, but such a claim does assert the existence of a guarantee that for each sentence that it is true apart from cases where the other is true. Similarly, a claim of entailment is a guarantee that the premises of an argument are not all true unless the conclusion is, so it can be seen to differ from a claim of inconsistency by adding an exception.

Terms like except and unless carry implicatures that can interfere with understanding this idea. It is important to understand them as you would in a guarantee. A guarantee that a product will function for three years unless it has been abused merely makes the guarantee conditional on the absence of abuse. It does not guarantee in addition that the product will not function if it has been abused although the statement The product will function unless it has been abused might suggest this under other circumstances.

The ideas of division and conditional exhaustiveness also provide ways of extending to any set the idea of logical independence introduced in 1.2.6 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 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.

Conditional 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 some way of dividing them is not possible, 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 conditional 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

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.