2.3.5. Confirming counterexamples

A dead-end open gap is always has a counterexample lurking in it, and any counterexample lurking in it lurks already in the ultimate argument of the derivation. We will finish off derivations by exhibiting a counterexample lurking in a dead-end open gap and calculating the truth values of the original premises and conclusions on that interpretation. In the case of the example discussed in 2.3.1, this calculation is shown in the following table:

A	B	C	(A	∧	⊤)	∧	B	/	B	∧	(⊤	∧	C)
T	T	F		T	T	Ⓣ				Ⓕ	T	F

Here the values of unanalyzed components have not been repeated on the right, but they are used to calculate the values of compounds containing them, with the order of calculation being guided by parentheses. In performing this calculation we are confirming that the counterexample lurking in the dead-end gap really does constitute a counterexample to the ultimate argument; and we will say that, in constructing the table, we are confirming a counterexample. It will be our standard way of concluding the treatment of an argument whose derivation fails.

It is not always the case that all unanalyzed components of the ultimate argument all appear among the resources and goal of a dead-end gap. When unanalyzed components do not appear there, values must still be assigned to them in order for a truth value to be defined for each sentence in the ultimate argument; but it will not matter what value we assign to these further unanalyzed components. If an interpretation separates the active resources of a gap from its goal, any way we choose to extend it to unanalyzed components not appearing in the gap’s proximate argument will still be a counterexample to that proximate argument and therefore also to the ultimate argument.

The example below is designed to illustrate this. Of the three interpretations shown, the first is a counterexample lurking in only the first dead-end gap (since it assigns the value T to the goal of the second dead-end gap), and the last lurks only in the second open gap (for a similar reason); but the middle one lurks in both open gaps. With 4 unanalyzed components, there are 2×2×2×2 = 2⁴ = 16 possible interpretations, so there are 13 interpretations that do not lurk in either gap. The soundness and safety of our rules insures that the 3 interpretations shown below constitute counterexamples to the ultimate argument and that the other 13 do not.

	│A ∧ B	1
	├─
1 Ext	│A
1 Ext	│B	(4)
	│
	││○	A, B ⊭ C
	│├─
	││C	2
	│
	│││●
	││├─
4 QED	│││B	3
	││
	│││○	A, B ⊭ D
	││├─
	│││D	3
	│├─
3 Cnj	││B ∧ D	2
	├─
2 Cnj	│C ∧ (B ∧ D)

A	B	C	D	A	∧	B	/	C	∧	(B	∧	D)
T	T	F	T		Ⓣ				Ⓕ		T		lurks in first dead-end gap
T	T	F	F		Ⓣ				Ⓕ		F		lurks in both dead-end gaps
T	T	T	F		Ⓣ				Ⓕ		F		lurks in second dead-end gap

While a dead-end gap specifies just one assignment of truth values to the vocabulary actually appearing in its proximate argument, this assignment may be provided by more than one interpretation of the derivation as a whole if the derivations contains further vocabulary. That happens in both gaps here, and it also happens that a single interpretation of the whole derivation is a counterexample lurking in both of the gaps. That’s why we end up with 3 interpretations all told.

Since each of these interpretations lurks in all ancestors of the dead-end gap or gaps in which it lurks, any one of the three is enough to provide a counterexample to the ultimate argument. Beginning with chapter 6, it will prove to be most convenient to assign F to an unanalyzed component whenever we have a choice, and here that would lead us to the middle interpretation in the case of both gaps. But, for now, when an unanalyzed component does not appear in the proximate argument of a dead-end gap, the choice of the value to assign to it is entirely arbitrary.