Undecidability and multivalent logic

[Auto-Didact]

Are undecidable statements, such as the provability of the continuum hypothesis, natural examples of statements that require a multivalent logic in order for them to be adequately described and/or even properly understood? (NB: by properly I am taking this to mean that undecidable matters such as the provability of the CH are currently not fully understood)

If so, is this many-valuedness only useful trivially as in a possible trivalent description of the situation in terms of yes/no/undecidable, or can a multivalent approach to this actually contribute something much more deep and subtle, such as the direction towards another foundation of mathematics instead of ZFC set theory due to some other desirable criteria?

For example, could the $\aleph_1$-valency of fuzzy logic possibly help map out something akin to a parameter space of possible axiomatizations wherein the truth/falsity of the continuum hypothesis is a parameter ranging in any of the infinitely many values between true and false?

 

[TeethWhitener]

Couple of things. First, "undecidable" usually refers to the non-existence of a generic algorithm to solve a given decision problem. It's more complexity theory than logic/set theory. The continuum hypothesis is more properly referred to as "independent" of the ZFC axioms. This means that including CH in ZFC doesn't produce contradictions, but neither does including the negation of CH. Analogously, you can consider ZF without the axiom of choice. In this case, it can be shown that the axiom of choice is independent of ZF (indeed, some researchers study the system ZF~C, Zermelo Fraenkel with the negation of the axiom of choice). In light of this, it's unclear what you would gain by using multivalent logic for terms that are independent of a given axiom system.

It may, however, prove useful to consider multivalent logic for the complexity/decision theory notion of "undecidable." I don't want to give a firm answer to this, but my intuitive reasoning is this: many of the tricky/counterintuitive situations we encounter in logic/set theory revolve around self-reference in some way (Liar paradox, incompleteness theorem, Tarski's truth undefinability, Russel's paradox, etc., etc.). Multivalent logics are sometimes used for handling self-reference in systems where this would usually be a problem (in fact, many moons ago on this site, I answered some questions about the Liar paradox, and how it's efficiently treated in trivalent logic). The undecidability of e.g., the halting problem relies on self-reference. So maybe there's something to trivalent logic there. Of course, the interpretation might end up being trivial and therefore not very useful (e.g., "Does this program halt?" "Yes/No/Undecidable").

 

[jk22]

This has few todo with the op but if we think in terms of quantum logic then the truth value would be in a plane, the axes being true false. An undecidable statement would not need a third dimension but it is the peculiar case of a null length vector, i.e. the origin.

 

[Auto-Didact]

This has few todo with the op but if we think in terms of quantum logic then the truth value would be in a plane, the axes being true false. An undecidable statement would not need a third dimension but it is the peculiar case of a null length vector, i.e. the origin.

Thanks for the post, and actually this has a direct bearing on the OP, seeing quantum logic is a noncommutative multivalent logic :)