2.3.2. Dead ends and counterexamples

Now, let’s look more closely at what we can say in general about the significance of dead-end open gaps. First of all, recall what led us to conclude that the gap in the example of the last section could not be developed further. A dead-end gap must not have a conjunction either as its goal or among its active resources, for otherwise we could apply the rules Cnj or Ext. Moreover, it must not have ⊤ as a goal or ⊥ as a resource, or else we could apply the rules ENV or EFQ. Finally, its goal must not be among its resources because then we could apply the rule QED. So the active resources of dead-end gaps are limited to unanalyzed components and ⊤ and their goals are limited to unanalyzed components and ⊥; and no dead-end gap can contain an unanalyzed component both as an active resource and as its goal.

This means that we can assign truth-values to the unanalyzed components appearing in a dead-end gap in a way that makes its active resources true and its goal false. Since no unanalyzed component appears both as a resource and as the goal, we can make any that appears as a resource T and any that appears as the goal F. While we are not free to assign values to ⊤ and ⊥, the first can appear only as a resource and the second only as the goal so they will not keep us from having true resources and a false goal. In short, we can assign truth values in a way that separates the active resources of a gap from its goal. That is, there is a counterexample to the gap’s proximate argument.

In noting this, we described an assignment of truth values to unanalyzed sentences. This is an extensional interpretation in the sense discussed in 2.1.8, and it can be presented in a table. The following table displays the interpretation defined by the dead-end gap of the example we have been considering.

The extensional interpretation of unanalyzed components appears on the left of the table. On the right are the resulting truth values of resources and goals of the gap, which mainly just repeat the assignments. (No value is assigned to ⊤ on the left because its truth value is stipulated by the meaning of the sign. That is, the sentence ⊤ has its value by general stipulation rather than by assignment in specific interpretations.)

Although we have more to show before we know that the system of derivations does what it is supposed to, we can say already that it has enough rules in a certain sense, for we know that, whenever the proximate argument of a gap is valid, some rule can be applied to either develop or close the gap. For if there is no rule allowing us to develop the gap, it has reached a dead end, and we have just seen that the proximate argument of a dead-end gap is not valid. We will indicate this sort of completeness in our rules by saying that a system of derivations is sufficient when every dead-end open gap has its active resources separated from its goal by some extensional interpretation. Of course, in saying that our system is sufficient, we do not say that every gap whose proximate argument is invalid has already reached a dead end. We would not expect this to be true since it would mean that we would never need to apply any rules at all in the case of an invalid argument. Indeed, one of the things we have yet to show is that any gap whose proximate argument is not valid will eventually reach a dead end.