8. Stage Two: Expanded Verificationism and the Logic N\(_{3}\)

Symbolized as \(-{{A}}\).

Lopez-Escobar (1972); Wansing (1993).

The logic was independently introduced by von Kutschera (1969). For further information on Nelson logic, see Wansing (1993, 1998); Odintsov (2008).

Some care is needed in transferring the condition from intuitionistic logic to Nelson logic. The clause for intuitionistic logic is sometimes given homophonically as

(1) \({ {w}}\Vdash _{1}{{A}}\supset {{B}}\) iff for all \(x\ge { {w}},\) if \(x\Vdash _{1}A\), then \(x\Vdash _{1}B\)

and sometimes in terms of a disjunction, the way we have done above. Here it is again:

(2) \({ {w}}\Vdash _{1}{{A}}\supset {{B}}\) iff for all \(x\ge { {w}},\) \(x\Vdash _{0}A\) or \(x\Vdash _{1}B\)

Given the classical metalanguage and the fact that in intuitionistic logic \(x\nVdash _{1}A\) is equivalent to \(x\Vdash _{0}A\), the two clauses are equivalent to

(3) \({ {w}}\Vdash _{1}{{A}}\supset {{B}}\) iff for all \(x\ge { {w}},\) \(x\nVdash _{1}A\) or \(x\Vdash _{1}B\)

In Nelson logic, this equivalence breaks down, because \(x\nVdash _{1}A\) is not equivalent to \(x\Vdash _{0}A\) . The homophonic clause (1) is equivalent only to the last condition, (3), the one I gave above for Nelson logic.

Why not consider (2), though? First, because it simply does not capture the thought that \({{B}}\) is verified whenever \({{A}}\) is verified, but rather the thought that \({{B}}\) is verified whenever \({{A}}\) is unfalsified, which does not seem promising as an account of a conditional. Second, the hereditary conditions ensure that this is actually equivalent to

(4) \({ {w}}\Vdash _{1}{{A}}\supset {{B}}\) iff \({ {w}}\Vdash _{0}A\) or \({ {w}}\Vdash _{1}B\)

Choosing this clause then makes the only intentional notion in the definitions superfluous, and all our worlds and heredities come to nought: As will become apparent presently, we would just have defined K3 or FDE with a material conditional in a complicated way.

Although Cogburn does not point to any specific logical systems that might be candidate companions to his view, I think that N\(_{4}\) would be well up to the job.

This is the question that Cogburn addresses. Maybe he is going a bit too easy on himself here, for the question could be put more polemically as: “If all evidence is supposed to be defeasible, what else could defeat good evidence for \({{A}}\), if not good evidence for \(\lnot A\)?”.

One finds a non-monotonic system based on Nelson models on p. 144 of Wansing (1998). Some further, very promising ideas in this direction are in Jaspars (1994, 1995). Yet another suggestion, made by Johannes Marti in discussion, is to simply do away with the heredity constraints altogether. While this would indeed ensure a great deal of revisability (maybe even too much), unnegated conditionals would still stay true throughout. It does not seem that this is a promising route to take. Note that alongside a non-monotonic system for verifications and falsifications, we could keep using N\(_{4}\) to keep track of information that supports a statement, with the understanding that such information need not amount to a verification. However, even a piece of information that will be defeated as evidence is a piece of information that we have and that we keep. Therefore, the irreversibility of “support for \({{A}}\) and support against \({{A}}\)” that results from the heredity constraint in N\(_{4}\) would make perfectly good sense here. For an interpretation of N\(_{4}\) that lays more emphasis on such an informational interpretation, see Ref. Wansing (1993).

At least this is the genealogy Vakarelov (2006) offers for the name. A reasonable choice of terminology on my part might have been to call every negation I call “toggle negation,” i.e., every negation that switches between verification and falsification, “strong negation.” I have chosen not to because (a) the definability of intuitionistic negation as well as the validity of \(\vDash -A\supset \sim A\) depends on the conditional as well as the negation, and toggle negation might appear in a system which is missing these features, (b) because there are other uses of strong negation in other parts of the logical literature, and (c) because the name “toggle negation” seems to me more suggestive of the essential role it plays in a verification–falsification semantics.

Or, with the definition of intuitionistic negation above in place, \(\sim A\wedge \sim -A\).

Cf. Wansing (2008), p. 342.

Counter model for both \(A\supset B\vDash -B\supset -A\) and \(A,{{A}}\supset {{B}}\models B\): A model with one world at which \({{A}}\) is verified and \({{B}}\) neither verified nor falsified.

\(\supset _{\text {AND}}\) is mentioned in Chap. 12 of Rasiowa (1974), albeit in a different framework, where it is used alongside the original Nelson conditional.

A related phenomenon is that logical equivalents may not be substituted freely in formulas; only when also their negations are logically equivalent this will be possible. For example, double negations may be introduced and eliminated inside formulas, as both \({{A}}\) and \(--A\) on the one hand, and on the other hand \(-{{A}}\) and \(---A\) are logically equivalent.

The counter model for modus ponens is again the one-world model in which B is falsified and A a gap, the counter model for modus tollens the model in which A is verified and B a gap.

Appendix 1 collects some more information on N\(_{\text {TOL}}\), N\(_{\text {AND}}\) and N\(_{\text {OR}}\).

This view of the matter is defended in McLaughlin (1990), though the analysis of the “even if” conditionals is not pursued very far.

Stalnaker (1984), p. 124.

It is possible to modify these systems to get rid of modus tollens, though.

Unless one also goes on to embrace the modification of logical consequence that is \(\vDash _\mathrm{Contrap }\).

And also in N\(_{\text {AND}}\).

The best place to start an exploration of such matters is still, I think, the classic (Horn 1989).