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 2.2.6,

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 later.)

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 thus 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 if the list is not empty, and the gap is closed if the list is empty. This is shown for individual rules in the following table:

In general, we can 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 given by the rule are valid. When a parent gap acquires a child due to development of another gap, it then acquires only one child and the proximate argument of this child is the same as the parent’s, so in this case, too, the proximate argument of the parent if and only if the proximate argument of each child is valid. Putting these two cases together, we can say this:

Here the claim every child … should be understood to be true when the gap has no children to provide counterexamples to this generalization. And this is the reason for the limitation to cases of a pair of successive stages, the fact that a stage has no children tells us nothing about its validity if it has no children merely because the derivation hasn’t yet been developed beyond that point. On the other hand, there is no need to limit this claim stages that are immediately successive. For what we have seen about children applies equally to grandchildren, great-grandchildren, and so on.

It may be easier to see that if we turn things around and look the conditions under which proximate arguments fail to be valid. In order to have a more compact way to talk about that, let us say that an interpretation that is a counterexample to a gap’s proximate argument lurks in the gap and that the gap has a lurking counterexample. So the proximate argument of a gap fails to be valid just in case there is a counterexample lurking in the gap. And the principle above then comes to the same thing as the following:

This is equivalent to the earlier principle because saying that some child has a lurking counterexample is the same as denying that every child has a valid proximate argument.

Now suppose a gap is followed by two successive further stages. What we have said regarding any children at the first of these applies to grandchildren at the second. For, if a counterexample lurks in the gap, we have seen that a counterexample must lurk in some child and, for the same reason, in some child of that child. And if a counterexample lurks in a grandchild, one must lurk in a child, and therefore one must lurk in the gap itself. The same argument applies to further succeeding stages, so we can say this:

We still speak only of gaps for which there is a succeeding stage, but that is enough to tie the validity of the initial gap’s proximate argument with the state of the derivation after all work is done. And, when all work is done, we know from the last subsection that any remaining open gaps, which will have reached dead ends, must have counterexamples lurking in them.

The diagram below shows how we can put these ideas together. It displays a sort of schematic argument tree that does not indicate actual proximate arguments, only their validity or invalidity—that is, whether or not there is a counterexample lurking in the gap. It is intended to depict a derivation that has come to an end, so the one gap that remains open at the right is understood to be a dead end.

We can distinguish three sorts of cases in this tree. First of all, we know from the sufficiency of the rules that the dead-end gap has a counterexample lurking in it. It has no descendent with a lurking counterexample, but that doesn’t conflict with the principle above because there is no later stage. Next, all ancestors of the dead-end gap, right down to the root of the tree, must have lurking counterexamples because each has a descendant that does. 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 descendant with a lurking counterexample. Also, notice that, at stages where such a gap does have descendants, all its descendents have valid proximate arguments. (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.)

We now know 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 gap with a lurking counterexample—the initial gap must have a counterexample lurking in it, 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 have no lurking counterexample and the ultimate argument must be valid. Although the principle we have been using does represent an important property of the system of derivations, we will not label this property (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 it 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.