1.1.6. Entailment and exclusion

Time has come to provide names for some of the ideas we have been considering. 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. 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, or that its has a valid conclusion, if the conclusion is entailed by the 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

Notice that 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

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.