1.4.1. A closer look at entailment

In section 1.2 we looked at implication, the special case of entailment that applies to single-premised arguments or immediate inferences, and we looked at it in the context of other deductive relations between individual sentences. Now we will return to entailment in its full generality, as applying also to multiple-premised arguments, and consider also the full range of deductive properties and relations. These relations were each defined (in a way summarized in 1.2.8) as the rejection of the possibility of one or more patterns of truth values, and the properties of tautologousness and absurdity were each defined as the rejection of the possibility of a certain truth value for the sentence in question. Our task is now to extend this idea to collections of more than two sentences, and we will begin with entailment.

Entailment holds in the case of implication, φ ⊨ ψ, when there is no possible world that separates φ from ψ. That definition can be stated more explicitly in either of the two following forms:

φ ⊨ ψ	if and only if	there is no logically possible world in which ψ is false while φ is true
	if and only if	ψ is true in every logically possible world in which φ is true

These are not two different ideas, for the two statements to the right of if and only if say the same thing. Still, they provide different perspectives on implication. The second—which we will speak of as the positive form of the definition—is closely tied to the idea of a conditional guarantee of truth and thus to the reason why deductive inference is valuable. The first form—the negative form—makes the content of the guarantee especially clear—telling us that it is a guarantee against new error in moving from φ to ψ—and this form of definition will generally be the more useful when we try to prove things concerning the concept. The other deductive properties and relations we have discussed or will go on to discuss can be given analogous pairs of definitions, a negative form ruling out certain patterns of truth values and another form stating a more positive generalization.

The equivalence of the two forms of the definition reflects a feature of all generalizations. When a generalization is false, it is because of a counterexample. This is an example of the sort about which we generalize but that does not have the property we have said that all such things have, so a counterexample to the claim that all birds fly is a bird that does not fly. In the positive definition of implication, the generalization is about all possible worlds in which φ is true and such worlds are said to all have the property that ψ is true in them. A counterexample to such a generalization is then a world in which φ is true but ψ is not. The negative form of the definition then affirms the same generalization but by saying that no such counterexample exists. The added clarity of the negative definition reflects a rule of thumb applying to all generalizations: a good way to clarify a generalization is always to ask what sort of counterexample is being ruled out.

The analogous pair of definitions for entailment more generally characterize that relation as a guarantee against new error when adding the conclusion to a set of assumptions, or as a guarantee of the truth of the conclusion conditional on the truth of the assumptions:

Γ ⊨ ψ	if and only if	there is no logically possible world in which ψ is false while all members of Γ are true
	if and only if	ψ is true in every logically possible world in which all members of Γ are true

These differ from the corresponding definitions of implication by replacing a reference to a single assumption φ by a reference to a set Γ. And since a set of sentences does not have a truth value, we need to speak of the truth of the assumptions by speaking of all members of Γ (a phrase whose significance we will return to in the next section).

Since we call an argument valid when its premises entail its conclusion, and validity is a good property for an argument to have, it is important to remember that validity is not all that we might ask of an argument. Compare a variant of the example in 1.1.3 with another argument having the same logical form:

All human beings are mortals Socrates is a human being Socrates is a mortal

All dogs are reptiles Socrates is a dog Socrates is a reptile

Since these arguments have the same form, they are equally valid and, indeed, valid for the same reason. But the second is clearly not a very good argument on other grounds. This is an instance of the general point that deductive logic is not concerned with the specific truth values of individual sentences (except in the special cases of tautologies or absurd sentences) but instead with ways in which the truth values of sentences are tied to one another. More specifically, the example emphasizes the fact that the relation of entailment rules out only one pattern of truth values, a false conclusion along with premises that are all true, and all other patterns can be found among deductive arguments. To take one further example, substituting god for human being in the first argument above shows that a valid argument may have a true conclusion even when its premises are all false.