1.4.1. A closer look at entailment

Entailment was introduced in 1.1.6 somewhat informally as a relation between premises and a conclusion that merely extracts information from them and thus brings no risk of new error. Another way of putting the latter point is that a relation of entailment provides a conditional guarantee of the truth of the conclusion: it must be true if the premises are all true.

The discussion of entailment in 1.2.1 developed the resources necessary to give a more formal general definition. In fact it is useful to have in mind two equivalent ways of stating one.

These are not two different concepts of entailment, for the two statements to the right of if and only if say the same thing. Still, they provide different perspectives on the concept. 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 to the reason why entailment is valuable. The first form—the negative form—makes the content of the concept especially clear, and this form of definition will generally be the more useful when we try to prove things concerning entailment. 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, something that is the sort of thing about which we generalize but that does not have the property we have said that all such things have. A counterexample to the claim that all birds fly is a bird that does not fly. In the positive definition of entailment, the generalization is about all possible worlds in which the premises are all true and such worlds are said to all have the property that the conclusion is true in them. A counterexample to such a generalization is then a world in which the premises are all true but the conclusion is not. The negative form of the definition then affirms the same generalization but by saying that no counterexample exists. As in the case of the generalization used to define entailment, one good way to clarify a generalization is always to ask what sort of counterexample is being ruled out.

It is important to notice how little a claim of entailment says about the actual truth values of the premises and conclusion of an argument. We can distinguish four patterns of truth values that the premises and conclusion could exhibit. Of these, a claim that an argument is valid rules out only the one appearing at the far right of Figure 1.4.1-1.

So, knowing that an argument is valid tells us about actual truth values only that we do not find the conclusion actually false when the premises are all actually true. The other three patterns all appear in the actual truth values of some valid arguments (though not all are possible for certain valid arguments because they may be ruled out by other deductive properties and relations among the premises and conclusion).

To see examples of this, consider an argument of the simple sort we will focus on in the next chapter:

This argument is clearly valid since its conclusion merely combines two items of information each of which is extracted from one of the premises. Depending on the state of the weather, the premises may be both true, both false, or one true and the other false; and, in any case where they are not both true the conclusion can be either true or false. In particular, if it’s hot and humid but neither sunny nor windy, the conclusion will be true even though both premises are false. This should not be surprising: a false sentence can still contain some true information, so information extracted from a pair of sentences that are not both true might be either true or false.

Of course, seeing one of these permitted patterns does not tell us that the argument is valid; no information that is limited to actual truth values can do that because validity concerns all possible worlds, not just the actual one. In particular, having true premises and a true conclusion does not make an argument valid. For example, the following argument is not valid:

For, although the single premise and the conclusion are both true, there is a logical possibility of the capital of Illinois being different while that of Indiana is as it actually is, so there is a possible world that provides a counterexample to the claim that the argument is valid.