1.1.7. Inconsistency and exhaustiveness

While the bounds on inference marked by entailment and exclusion are the most important topics for deductive logic, it is concerned also with bounds that are not directly bounds on inference. One example comes from looking in a different way at the negative bounds we have just considered. When a possible conclusion is absolutely incompatible with the data, adding it to the data yields a collection of claims that cannot all be true. When a group of sentences are incompatible in this way we are forced to choose among them. So there can be upper bounds on the collections of claims that can be maintained to together. An opposed lower bound comes by trading cannot maintain all for can maintain at least one. And the latter idea lies behind adding None of the above to a list of choices, for we can be sure that, if none of the others can be maintained, it can be, so any list containing it provides at least one acceptable choice. (Notice that, although the relation of this alternative to the others is not one of inference, it can be inferred from the denials of the other alternatives, and that will provide the basis for studying this sort of bound by way of bounds on inference.)

Entailment and exclusion are natural opposites, but the nature of the opposition means that the clear distinction between premises and conclusion is no longer found when we consider exclusion. When we say that Γ ⊨ φ, we are saying that there is no chance that φ will fail to be accurate when the members of Γ are all accurate. When we say that Γ excludes φ, we are saying that there is no chance that φ will succeed in being accurate along with the members of Γ. In the latter case, we are really saying that a set consisting of sentence consisting of the members of Γ together with φ cannot be wholely accurate, so it is natural to trace the relation of exclusion to a property of inconsistency that characterizes such sets: we will say that a set of sentences is inconsistent when its members cannot be jointly accurate. Then to say that φ is excluded by Γ is to say that φ is inconsistent with (or given) Γ in the sense that adding φ to Γ would produce an inconsistent set. The symmetry in the roles of terms in a relation of exclusion is reflected in ordinary ways of expressing this side of deductive reasoning: the difference between saying That hypothesis is inconsistent with our data and Our data is inconsistent with that hypothesis is merely stylistic.

If we turn the idea of inconsistency around—or perhaps inside out—we get the lower bound mentioned above. A set is inconsistent when its members cannot all be true, when at least one must be false. A set is said to be exhaustive when its members cannot all be false, when at least one must be true. Such a set exhausts all possibilities in the sense that there is some truth in it no matter what. And the point of adding None of the above is then to provide an exhaustive list of choices, a list that provides at least one option fitting any possibility.