2.3.1. When enough is enough

So far we have seen only derivations whose gaps all close, derivations which show that arguments are valid. But not all arguments are valid, so there ought to be derivations whose gaps do not all close. If there is no point at which the gaps of a derivation all close, we will eventually have to give up work on it even though it still has open gaps. So we should ask what might lead us to give up work and what, if anything, we can conclude if we do have to stop.

The short answer to the first of these two questions is that we must give up on a derivation when we run out of rules to apply, either to develop a gap or close it. Here’s a simple example of a derivation for which that has happened.

The gap that is marked with the empty circle ○ has C as its goal, and we currently have no rule to plan for such a goal. There are conjunctions among the available resources of the gap; but they were exploited in the course of developing this gap, so they are no longer active. Also, none of the rules for closing gaps apply here: not QED because the goal is not one of available resources, not EFQ because ⊥ is not a resource, and not ENV because the goal is not ⊤. In short, no rule of any of the three sorts can be applied at this point. Notice that the resources added by exploiting A ∧ ⊤ at stage 2 were never used later (hence there are no line numbers to their right). As a result, this exploitation could have been postponed the end. However, the resource A ∧ ⊤ must be exploited before we end work on the derivation because, until it is exploited, there is a way of developing the derivation further.

We will describe an open gap to which no more rules apply as a dead-end gap. (Although the qualification dead-end will be reserved for open gaps—indeed, a gap that has been closed is in one sense no longer a gap—we will often speak somewhat redundantly of dead-end open gaps.) In these terms, we can say that we are forced to abandon a derivation when every open gap has reached a dead end. When we consider the significance of dead-end open gaps, we will see that we may abandon a derivation as soon as one open gap has reached a dead-end. As in the example above, we will use the empty circle to mark open gaps that have reached a dead end and are thus permanently open. And, also as is done in that example, to the right of this sign, we will use the sign ⊭ (negated double right turnstile) to say that, with respect to the analysis of them displayed in the derivation, the active resources do not entail the goal. (The reason for qualifying this by reference to the displayed analysis will be discussed in 2.3.8.)

The way the gaps have developed in this derivation is shown in the following tree:

The gap that remains open at the end had reached a dead end at stage 3, but it is shown to continue at the next stage because it remains open as the derivation develops elsewhere. As we will see, a single dead-end gap in a derivation for a claim of entailment tells us that the claim fails, so work may be stopped as soon as a dead-end is reached. But there is nothing wrong with continuing as long as there are rules to be applied to other gaps, and we will often do so in examples. In general we will not assume that a derivation stops as soon as there is a dead-end gap, so to say that gap has reached a dead-end is not to say that it does not continue at later stages; it is to say rather that we can be sure it will never close.

From one point of view, the function of a derivation is to transform the question whether an argument is valid into an analogous question about one or more simpler arguments. This is the aspect of a derivation that is displayed in the growth of its argument tree, which is shown below for the argument we have been considering.

The proximate argument of a dead-end open gap is the end of the line in this process; it will not be developed further though it may be repeated. We will call the argument whose validity we initially asked about, the one at the root of the tree, the ultimate argument of the derivation. It is the proximate argument of the initial gap of the derivation. The contrast between the proximate argument of a gap and the ultimate argument of a derivation is the source of our use of the term proximate: the proximate argument of a gap is our immediate concern while our final goal is to decide whether the ultimate argument of the derivation is valid.

In discussing the significance of dead end gaps, we will look first at what reaching a dead-end tells us about the proximate argument of the gap that has stopped developing and then consider the connection between the validity of the ultimate argument of a derivation and the existence of dead-end gaps. In terms of the argument trees, this means we will look first at the tips of unclosed branches and then ask about the connection between the tips of branches and the root of the tree.