Nelson’s Logic of Constructible Falsity demands that falsity be treated symmetrically with truth, that is, to say that a sentence is false requires more than mere reductiones, it requires a construction. E.g., to disprove a conditional requires a (constructive) proof of the antecedent and a (constructive) disproof of the consequent.

But this isn’t the only way to go. While the falsity conditions for, say, disjunction can be read easily from the connective, implication (as well as coimplication) do not so readily suggest their falsity conditions. Wansing has introduced a sequence of sixteen “completions” of Nelson’s logic; there is little reason—other than a purely philosophical motivation—to accept one over the other.

I suggest that the data concerning many-valued extensions of these logics do weigh in on this question.

The heredity property of intuitionistic logic—that truth persists through all accessible points—yields a tidy property on finite, linearly ordered frames. In an \(n\)-pointed frames, there are only \(n+1\) distributions for any formula. The familiar Goedel-Dummett logics \(\mathsf{G}_{n}\) abuse this to yield finitely-valued extensions of intuitionistic logic. A truth value in the interval \((0,1)\) can be read as the proportion of points at which the sentence is true with respect to the number of points in the frame; this can be read as a “degree of truth.”

Nelson’s logic—as well as Wansing’s logics—likewise have this property, save for the fact that not only truth persists but falsity as well. Thus, rather than a single ratio, we must track a pair of points in \((0,1)\). Call such a pair \((x_{0},x_{1})\).

A very attractive, semantical constraint is that the degree of truth and the degree of falsity ought to add to \(1\). On, e.g., an epistemic or doxastic reading, belief in the truth of a proposition with probability \(1\) ought to imply that the belief in its falsity is \(0\). If one is 75% sure that a sentence is true, one is 25% sure that it is false, etc.

A property of the Goedel-Dummett logics is that their operations preserve this property, as the degree of falsity is implicit in the degree of truth. When one segues to the finitely-valued extensions of Wansing’s logics, however, only one of the sixteen can be maintained under this constraint. This is to say that for the set of pairs of positive rationals \((x_{0},x_{1})\) such that \(x_{0}+x_{1}=1\) is not closed under the operations of fifteen of the sixteen logics, including that suggested by Nelson.

The only logic under whose operations this set of truth values is closed is Wansing’s \(\mathsf{I}_{4}\mathsf{C}_{4}\) and this is very unusual. Whereas the other fifteen have rather straightforward conditions for the falsity of a conditional, those discovered in the logic \(\mathsf{I}_{4}\mathsf{C}_{4}\) seem slightly artificial. For example, the logics \(\mathsf{I}_{1}\mathsf{C}_{i}\) treat a negated conditional as a conjunction between the antecedent and the negation of the consequent (cf. Nelson’s reading). \(\mathsf{I}_{2}\mathsf{C}_{i}\) treat it connexively, so that \(\neg(A\rightarrow B)\) is defined simply as \(A\rightarrow\neg B\). And \(\mathsf{I}_{3}\mathsf{C}_{i}\) treats the negation of \(A\rightarrow B\) simply as the corresponding coimplication \(A\leftarrow B\). \(\mathsf{I}_{4}\mathsf{C}_{i}\), on the other hand, defined the negation as contraposed coimplication, which increases the complexity of the formula. (\(\mathsf{I}_{j}\mathsf{C}_{4}\) on the other hand treats negated coimplication as contraposed implication.)

But if we calculate the values of the functions associated with these logics on finite, linearly ordered frames, all of the others go berserk under the above constrain; only \(\mathsf{I}_{4}\mathsf{C}_{4}\) is left standing. From a Goedel-Dummet perspective, the negation of this logic can be seen to be the extension of Goedel-Dummet logics with Lukasiewicz negation—a strange convergence.

Why is this the case? Why is it the least obvious that works? Why should Lukasiewicz and Nelson negation converge at this point? It’s a very odd situation, I think.