7.7.2. Soundness

A strict rule, in the sense introduced in 2.3.4, does not throw away lurking counterexamples as it develops a gap. That is, any counterexample lurking in a gap to which the rule is applied will still lurk in at least one child gap produced by the rule. In applying this idea to the rules for universals, we are faced with a problem caused by rules that introduce new vocabulary. New vocabulary is introduced always by the planning rule UG, which introduces independent terms, and new vocabulary must be introduced by the exploitation rule UI if generalization would otherwise go unexploited.

Now, a counterexample that lurks in a gap before a rule is applied may fail to lurk in any child gap produced by the rule simply because it gives no interpretation at all to new vocabulary that the rule introduces. And, even if it does happen to interpret this new vocabulary, the interpretations it gives have played no role in making it a counterexample before the vocabulary was introduced, so we may need to revise these interpretations as we go on. In short, if a counterexample lurking in a gap is to lurk in any of its children, we may need to provide new interpretations of new vocabulary appearing in that child.

To begin to handle this problem, let us first be more explicit about the conditions under which a structure counts as an interpretation of a gap. In previous chapters we took it for granted that the interpretations we considered interpreted all vocabulary appearing anywhere in the derivation since all such vocabulary appeared in the initial premises and conclusion and we wanted all our interpretations to give truth values to these sentences. Now we need to be more flexible, so let us say that a structure interprets a gap if it assigns interpretations to all the non-logical vocabulary that appears in resources or goals of the gap or any of its ancestors. Such a structure must interpret all vocabulary in the initial premises and conclusion of the ultimate argument of the derivation and also interpret all independent terms introduced along the way to the gap in question, but it need not interpret independent terms whose occurrences are boxed off from the gap we are considering. Notice that we allow an interpretation of a gap to provide an interpretations of vocabulary not appearing in a gap. This means that any interpretation of a gap not only interprets the vocabulary of all its ancestor gaps but in fact counts as an interpretation of those gaps. Among the structures that interpret a gap, we distinguish those that are counterexamples lurking in it in the same way we have in the past—that is, as the structures that make its active resources true and its goal false.

In order to adapt the definition of soundness to the possibility of changing vocabulary, we can no longer require that, when counterexample lurks in a gap, an identical counterexample can be found lurking in at least one child since we may need to extend or modify the counterexample to accommodate new vocabulary. Let us that say that two interpretations agree for a gap when they have the same referential range and give the same interpretation to all vocabulary appearing in the gap and all its ancestors. This idea is motivated by a principle concerning structures that should seem plausible but that we will not argue for: if two structures have the same range and agree on the interpretation of all vocabulary in a sentence, then they each assign the same truth value to that sentence. It follows that if two interpretations agree for a gap, then one will be a counterexample lurking in the gap if and only if the other is (and this will be true also for each ancestor of the gap). Notice also that, if an interpretation interprets only vocabulary appearing in a gap or its ancestors, any interpretation agreeing with it will simply extend it by adding interpretations to further vocabulary.

Given these ideas, we will redefine strictness and say that a rule is strict when, for any counterexample lurking in a gap before the rule is applied, we can find a counterexample that agrees with the given one for that gap and that lurks in at least one child gap resulting from the rule. According to this definition, a strict rule need not preserve lurking counterexamples unchanged; but it must preserve what was essential to the function of such a structure in constituting a counterexample to the proximate argument of the parent gap, though it may force it to be elaborated or altered in order to interpret a gap resulting from the rule. We will say that a rule is sound when it preserves (in this way) counterexamples that lurk in both the gap to which the rule is applied and in all of its ancestors. The idea of a path provides an alternative way of expressing the idea of soundness. In terms of it, a rule is sound when, for any counterexample that lurks throughout a path before the rule is applied to its final gap, we can find an interpretation that agrees with given interpretation on each gap in this path and that constitutes a counterexample lurking throughout at least one path that results from applying the rule.

The rules UG and UI are strict in the new sense. The actual arguments showing this are not very surprising, and we will look at only the case of UG.

Suppose S is a counterexample lurking in the gap on the left. Since S makes the goal ∀x θx false, it must assign θ an extension that does not include the whole referential range. Let S′ be like S except in assigning to the independent term a some value outside the extension of θ. Then S′ will agree with S for the gap at the left (since a does not appear before UG is applied), and it will make θa false. So S′ (like S) will make all active resources of the two gaps true, and it will make the new goal false (whether or not S does). So, given a counterexample S lurking in the old gap, the essentials of the way it constitutes a lurking counterexample are preserved in a structure S′ that is a counterexample lurking in the new gap; and that means that UG is strict.