1.1.6. Entailment, exclusion, and inconsistency

When the conclusion of an argument merely states information extracted from the premises and is therefore risk free, we will say that the conclusion is entailed by the premises. Using this vocabulary, cases of extraction of information may characterized by a relation of entailment between the initial data and the information extracted from it. If we speak in terms of arguments, entailment is a relation that may or may not hold between given premises and a conclusion, and we will say that an argument is valid if its premises do entail its conclusion. We will say also that the conclusion of an argument with this property is a valid conclusion from its premises. Figure 1.1.6-1 summarizes these ways of stating the relation of entailment between a set of premises or assumptions Γ and a conclusion φ.

We will use the sign ⊨ (double right turnstile) as shorthand for the verb entails, so we add to the English expressions in Figure 1.1.6-1 the claim Γ ⊨ φ as a symbolic way of saying that the assumptions Γ entail the conclusion φ. Using the sign ⊨, we can express the validity of argument in Figure 1.1.2-2 by writing

The relation of entailment represents the positive side of deductive reasoning. The negative side is represented by the idea of a statement φ that cannot be accurate when a set Γ of statements are all accurate. In this sort of case, we will say that φ is excluded by Γ, and we will say that cases of this sort are characterized by the relation of exclusion. We will see later that it is possible to adapt the notation for entailment to express exclusion, so we will not introduce special notation for this relation.

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.

One aspect of the notation we will use for arguments and entailment deserves a final comment. The signs / and ⊨ differ not only in their content but also in their grammatical role. A symbolic expression of the form Γ / φ is a noun phrase since it abbreviates the English expression the argument formed of premises Γ and conclusion φ, so it is comparable in this respect to an expression like x + y (which abbreviates the English the sum of x and y). On the other hand, an expression of the form Γ ⊨ φ is a sentence, and it is thus analogous to an expression like x < y. In short, ⊨ functions as a verb, but the sign / functions as a noun. In Γ / φ, the symbols Γ and φ appear not as subject and object of a verb but as nouns used to specify the reference of a term, much as the names Linden and Crawfordsville do in the term the distance between Linden and Crawfordsville. And the relation between the claims

is analogous to the relation between the claims

(Of course, there are also many respects in which these pairs of claims are not analogous; for example, the relation expressed by ⊨ has a direction while that expressed by is close to is reversible.)