2.3.3. Validity through the generations

The connection between the proximate arguments of dead-end gaps and the ultimate argument of a derivation lies in the properties of the rules for developing and closing gaps. We will begin to look at these properties in this section and then look at them more closely in the next.

It will help to have some ways of talking about the relations between gaps at various stages of a derivation. It is common to extend some genealogical vocabulary from family trees to trees in general. In our use of this vocabulary, we will say that any gap that results from applying a rule is a child of the gap to which the rule is applied and that the latter gap is its parent. It will be convenient to apply the same terminology to gaps that continue unchanged while others develop: a gap at one stage that is open but unchanged at the next stage is understood to have a single child. Looking farther up or down a line of descent, we will say that some gaps are ancestors or descendants of others. So in the tree of gaps associated with the derivation discussed in 2.2.5,

the lower gap at stage 3 has the gap at stage 2 as its parent and both that and the two earlier gaps as ancestors. Its children are the lower two gaps at stage 4 and its further descendants are the gaps to their right. The line of gaps at the top are neither ancestors or descendants of the gap in question.

In this terminology, the initial gap of a derivation is an ancestor of all gaps of all gaps at each later stage in its development; and they are all its descendants. Only open gaps will be part of these genealogies, so a gap that is closed at the next stage of its development has no children. Dead-end open gaps continue to have children if the derivation is continued at later stages (remember it need not be), yet they have reached a dead end in the sense that these children are always identical to their parents.

Next, let us develop a way of speaking about the effect of derivation rules on the distribution of valid and invalid arguments in the argument tree of a derivation. In the case of QED, we will initially limit ourselves to its use to close a gap whose goal is also among the active resources; the wider use of QED, to close gaps whose goals are among their available but inactive resources, will be considered in the next section.

The derivation rules Ext and Cnj are based on principles of entailment which give necessary and sufficient conditions for an entailment to hold. That is, each principle gives a list of conditions all of which must hold if a given entailment is to hold and which together are enough to insure that it holds. It may seem odd to say the same about the unconditional claims of entailment that lie behind the rules QED, ENV, and EFQ; but, by asserting an entailment unconditionally, they say that an empty list of conditions is sufficient for its truth (and, since an empty list cannot have a member that fails to hold, satisfying the list is trivially necessary since it is bound to be satisfied).

Phrased in terms of arguments, each principle tells us that a certain sort of argument is valid if and only each member of a (perhaps empty) list of arguments is valid. When the corresponding rule is applied to a gap, the gap is provided with children whose proximate arguments are those on the list (so the gap is given no children—that is, it is closed—if the list is empty).

This means that the proximate argument of a gap to which a rule is applied is valid if and only if all the proximate arguments of any children it has are valid. And, of course, the same is true of a parent which acquires a child when the derviation is developed elsewhere because then there is only one child and its proximate argument is the same as its parent’s.

To say that the proximate argument of a gap is valid is to say that the gap is indivisible, so we can say that a gap before the last stage is indivisible if and only if each one of any children it has is indivisible. It is usually more convenient to speak of divisibility (i.e., of the invalidity of the proximate argument), and we can rephrase what we have been saying in these terms as follows.

This gives us necessary and sufficient conditions for the divisibility of a gap in terms of divisibility at the next stage, but it is stated only for cases where there is a following stage (though it does not require that the gap have children) and it is stated only for the immediately following stage. We will go on to consider what can be said of any gap and said with respect to any following stage. That will be enough to tie the divisibility of the initial gap with the state of the derivation after all work is done.

First note what we can say in cases where there are two stages following a gap. For a gap to be divisible in such circumstances, it must have a divisible child, which must itself have a divisible child. That is, a necessary condition for divisibility when there are two following stages is having a divisible grandchild. And that is clearly also sufficient, for a divisible grandchild will have a divisible parent, which will be a divisible child of the grandparent gap. Of course, the same thing will work for great-grandchildren, great-great-grandchildern, and so on, provided there are enough following stages.

In general, we can say this:

Notice that this not only ties the divisibility of a gap to the divisibility of its descendants, however distant, but also holds for a gap when there are no later stages at all. The latter point is analogous to one made above about gap-closing rules: a generalization about an empty collection is bound to be true, no matter what it says, because there is nothing to serve as a counterexample.

These points are illustrated in the diagram below. It shows a sort of schematic argument tree that does not display actual arguments, only their validity or invalidity—i.e., their indivisibility or divisibility. It is intended to depict a derivation that has come to an end, so the one gap that remains open at the top is a dead end.

We can distinguish three sorts of cases in this tree. First of all, we know from the last section that the dead-end gap is divisible. It has no divisible descendent, but it is not a counterexample to the generalization above because there is no later stage. Next, all ancestors of the dead-end gap, right down to the root of the tree, are divisible because each has a divisible descendant. And finally, in the case of any of the other gaps—i.e., the ones whose proximate arguments are valid—there is a following stage (the last stage of the derivation if not an earlier one) at which the gap has no descendant at all, and so certainly has no divisible descendant. Also, notice that, at stages where such a gap does have descendants, all its descendents are indivisible.

There is a fourth sort of case that does not appear here, a gap that has no descendants but has not been closed and is not at a dead end. But this case will appear only in the last stage of an incomplete derivation, and the generalization says nothing about it because there is no later stage.

The generalization we have been considering tells us that the way we have taken the results of a derivation is correct. If there is a dead-end gap—and thus, by sufficiency, a divisible gap—the initial gap must be divisible, so the ultimate argument is invalid. On the other hand, if all gaps close, there is a stage (the one at which the last gap closes) at which the initial gap has no descendants, so it must be indivisible and the ultimate argument must be valid. Although this generalization does represent an important property of the system of derivations, we will not label it (in the way we have labeled the property of sufficiency) because we will go on in the next section to look further at the basis for this property and state (and label) some related properties that can be applied to a wider range of rules, including the extended use of QED that we excluded from consideration here.