2.3.6. Reaching decisions

We know that if a system of derivations has individual rules that are both sound and safe and is, as a whole, sufficient, it will never give us an incorrect answer regarding the validity of an argument. But it is entirely possible that such a system will give us no answer at all. Of course, if we ever run out of rules to apply, we will have an answer. For then either all gaps will have closed or we will have an open gap that has reached a dead end, and both results provide an answer. However, without some guarantee that we will eventually run out of rules, we have no guarantee that we will eventually have an answer. And such a guarantee is not trivial; in fact, once we get to the last two chapters, we will be working in a system some of whose derivations do go on forever.

We will say that a system is decisive when we always reach a point where either all gaps are closed or there is a dead-end open gap. It should be clear that our system so far is decisive. The rules Ext and Cnj replace conjunctions among the resources and goals of a gap by simpler sentences and must therefore eventually eliminate all conjunctions. And when the proximate argument of a gap contains no conjunctions, the only rules that might apply are QED, ENV, and EFQ. Each of these closes a gap and there will be only a limited number of gaps to close, so we must eventually run out of things to do.

But we will go on to consider further rules, and some of these will be sufficiently differently from those we have considered so far that, even when a system is decisive, it may not be as easy to see that it is. So let’s look at some questions that arise in making this judgment. As we do this, it is worth remembering that, in assessing decisiveness, we are not really interested in whether a system reaches some valuable goal, only in whether we are bound to run out of things to do when we apply its rules.

One way to judge whether that is so is to provide some count of how much there is that might be done, and see whether each rule of the system reduces that count. However, it is not always easy to describe a single quantity that is always reduced, and the reason can be seen even with our current system. The rules QED, ENV, and EFQ reduce the number of open gaps, and that is certainly a relevant quantity. The rules Ext and Cnj, on the other hand, reduce the complexity of proximate arguments, something else that cannot go on for ever. While complexity may seem too abstract to be reduced to a single number, the simple expedient of counting the number of connectives in a proximate argument actually provides a useful quantity in the present setting. So far, so good, but the real problem arises in putting these two numbers together.

This problem is easiest to see by considering Cnj. While the proximate arguments of both its children are simpler than that of their parent, it adds to the total number of open gaps. It is tempting to say that this is acceptable because the increase in the number of open gaps is no greater than the decrease in the complexity, so the sum of the two is not increased. But this would be wrong on two counts. First, it is not enough that we avoid increasing the quantity we are watching: rules that merely kept it the same might go on for ever doing that. Second, our system would still be decisive if Cnj added 10, 100, or even a million new gaps when it eliminated a single connective. For, in the absence of a rule that added connectives, it would eventually run out of connectives to eliminate, and we would be forced to use other rules which did reduce one quantity without increasing the other.

This is not to say that there is no way of putting the number of open gaps and the complexity of proximate arguments together to produce a useful quantity, but any way of doing that must recognize their asymmetry: we can add gaps as we reduce the number of connectives but only provided we add no new connectives when we close gaps. However, we will not look at ways of actually combining these quantities. We will simply employ the abstract idea of a rule moving things along. We will call a rule that does this progressive, understanding that whether a rule is progressive depends not only on what quantities it might reduce but also on what other rules are present. The common idea associated with our various uses of this term progressive will be that, if all our rules are progressive, each moves us far enough along that we can never apply them more than a limited (though perhaps very large) number of times before we run out of things to do.

So a system all of whose rules are progressive will be decisive; that is, we will always reach a point at which no more rules can be applied. At that point, any gap that is left open will have reached a dead end, and the derivation will have provided an answer about the validity of the original system. And we saw earlier that if a system is sufficient and conservative, the existence or non-existence of an open gap when no more rules apply provides a correct answer regarding validity of the ultimate argument. A system that always eventually provides an answer and a correct one, can be said to provide a decision procedure for validity.