2.3.8. Formal validity

As was noted earlier, the use of the term valid in connection with derivations requires some qualification. In the context of derivations, as in the context of analyses, Roman capital letters are used to stand for particular sentences that are not analyzed further, and such sentences need not be logically independent. That means that a given extensional interpretation of unanalyzed sentences need not be realized in any possible world. So, in the example of 2.3.1, even though the appearance of a dead-end gap leads us to write B, A, ⊤ ⊭ C, it might be that the particular sentences A and B do together entail the particular sentence C, and it could even be that C is tautology or that A and B are mutually exclusive. In short, knowing that there is an extensional interpretation of analyzed sentences that assigns them certain truth values does not show that it is logically possible for the sentences to have those truth values.

On the other hand, our interest in derivations is as a way of applying general principles of formal logic. And, even though these principles are applied to particular sentences, their application depends only on the features of these sentences that are displayed in symbolic analyses. In particular, the use of derivation rules does not depend on the specific identity of unanalyzed components. This means that when the gaps of a derivation do all close we know not only that its premises entail its conclusion but also that the same is true for any argument having the same form. One way of putting this is to say that we know the argument to be formally valid or, more precisely, to be valid in virtue of the form exhibited in the particular analysis we have used. Since formal validity is a stronger property than simple validity, knowing that an argument is formally valid is enough to tell us it is valid; and we will usually drop the qualification formal for this reason. But it is important to remember that when an argument is labeled invalid on the basis of a derivation, this judgment is relative to a particular analysis of it. Indeed, if this were not so, we could stop after studying conjunction: the point of considering further logical forms is to recognize the validity of arguments that count as formally invalid when considered solely in terms of conjunction.

The idea of validity in virtue of form can itself be spelled out by saying that an argument is formally valid with respect to a given analysis when any way of associating sentences with its unanalyzed components produces a valid argument. So when the derivation of 2.2.5 showed us that (A ∧ B) ∧ C, D ⊨ C ∧ (A ∧ D), this told us something not only about the specific sentences (A ∧ B) ∧ C, D, and C ∧ (A ∧ D) but about any sentences that are related in the way indicated by these analyses—that is, about the sentences could be formed in these ways from any choice of sentences, A, B, C, and D. Such choice of actual sentences, one for each of a group of unanalyzed components, is an intensional interpretation in the sense discussed in 2.1.8, so we can say that an analyzed argument is formally valid when every intensional interpretation of it is valid.

When a derivation leads to a dead-end gap, what we know, speaking most strictly, is that its ultimate argument is not formally valid. That is because one test of formal validity is whether there is an extensional interpretation of the argument that separates its premises from it conclusion. And we will look more closely at why that is so.

First, if there is an extensional intepretation that provides a counterexample to an argument, we can construct an intensional interpretation by assigning to each component an actual sentence with the truth assigned by the extensional interpretation, and this interpretation will yield an actual argument having the same form as the original one but with actually true premises and an actually true conclusion. In example from 2.3.1, the counterexample given by the dead-end gap assigns T to A and B and F to C. So we might associate English sentences with these unanalyzed components as follows:

If so, the proximate argument of the dead-end gap will be

and the ultimate argument of the derivation will be

To get something completely in English, we can replace ⊤ by any tautology. If we use Atlanta is Atlanta, we get

Each of these particular arguments has a false conclusion along with true premises not merely in some possible world but in the actual world, so they are certainly invalid. Because the latter two have the same form as the ultimate argument of the derivation, that ultimate argument is not valid with respect to the form displayed in its analysis. If in that argument, the unanalyzed A, B, and C happen to be sentences such that A, B ⊨ C, the argument will in fact be valid. For example, it might be

But it will remain true that it is not valid with respect to the form displayed in the symbolic analysis, and we have shown it is not by giving another interpretation of this form that is not valid.

We have seen that an argument whose premises are separated from its conclusion by an extensional interpretation is not formally valid. The converse is also true. That is, if an argument is not formally valid, its premises are separated from its conclusion by some extensional interpretation. The claim that an argument is formally valid is a generalization about both intensional interpretations and possible worlds, and a counterexample to this generalization is provided an intensional interpretation and a possible world with the property that the actual argument that results from the intensional interpretation has its premises separated from its conclusion by the possible world. But any intensional interpretation and possible world will determine an assignment of truth values to the unanalyzed components of the argument. In the example above the value T is assigned to the unanalyzed component A by associating the sentence Atlanta is in Georgia with A and considering the truth value of this sentence in the actual world. Since any intensional interpretation and possible world will determine an extensional interpretation in this way, any counterexample to the formal validity of a symbolic argument will provide an extensional interpretation that separates its premises from its conclusion.

This means that even if we do not define formally validity directly in terms of inseparability of premises from conclusion by extensional interpretations but instead in terms of validity under any intensional interpretation, it will still be true that an argument is formally valid if and only if no extensional interpretation separates its premises from its conclusion.