# Can three valued logic serve as an adequate basis for a (nontraditional) modal logic?

I asked a similar question on the Math Stack Exchange, but the best answer so far provided was not entirely satisfactory.

I have examined much of the literature referenced in the SEP article on Many-valued logic, so I have at least encountered the most prominent versions. I have found that Lukasiewicz three-valued logic appears the most robust, especially when supplied with a strict conditional ( LCpq, in case anyone is interested)

There are important differences from the Lewis Systems S1-S5 and related systems. The Lewis systems accept the Law of the Excluded Middle, the three valued logic of course rejects it. The definition of the conditional and the strict conditional are different from both classical logic and from Lewis' definition. The required interpretations of the "necessity" and "possibility" operators differ significantly from common, accepted definitions in modal logic and Kripke-style semantics is inapplicable.

What would have to be settled or established for such a logic to be acceptable as some version of modal logic?

I don't see what abandoning bivalence is going to get you. Modality is about how a sentence is true or false--is it necessarily true, or just contingently true? I'm not saying that a trivalent modal logic can't be created, just that I don't see what the motivation for such a system would be. That's a first question you'd need to answer.

The next thing you'd need to do to create such a system would be to offer a semantic theory that fills in the ellipses here:

Box x is true iff ... Box x is false iff ... Box x is undefined iff ...

Using that semantic theory you would then need to prove which syntactic rules of inference were valid in your new system.

This is just the general procedure when introducing new modal systems. You can find examples of the procedure in any introductory textbook on modal logic.

• I seem to have gone about it backwards: first finding valid rules of inference, then finding/creating an interpretation consistent with them. Mar 4 '14 at 18:19
• I mean, you can do it that way too - - the point is that you want a system that is both syntactically and semantically correct. Usually it's easier to come up with the semantics first IMO, but your mileage may vary.
– user5172
Mar 4 '14 at 19:08
• My experience has been that the required semantics are different enough from established conventions that the system doesn't get more than a casual glance by experts, who then tend to dismiss it as uninteresting or unimportant, fail to investigate it, and never get to see for themselves how many long-standing controversies in logic it addresses and clarifies. Mar 4 '14 at 20:24
• Write a paper elaborating the semantics, proving the system consistent and complete, and showing how it solves an important problem. Then submit it to a journal. I would think the Journal of Philosophical Logic or the Notre Dame Journal of Formal Logic would be interested. If you can publish either of those places, people will take your ideas seriously.
– user5172
Mar 4 '14 at 21:00

Modal logics are generally (or formally) done by the introduction of new quantifiers like It is necessary that and It is possible that.

However one could interpret multi-valued logics as a fracturing of the bivalence of truth. So p is true meaning that p is necessary and p is false meaning that p is never possible.

So additional truth values will lie in between these two possibilities. In Lukasiewicz 3-valued logic one could intepret the third value as Unknown, or partly true and partly false, or neutral, or unknown. All these have different semantic ranges.

What might be remarkable, is if there is a mapping of some kind between these two conceptions of modal logics, ie a certain multi-valued logic maps onto a modal one. Certainly I haven't come across one - but I'm no expert.

But given that intuitionistic logic and modal logic are both interpreted by frames it seems a distinct possibility.

• Lukasiewicz did define such quantifiers for his logic. His attempt has been generally regarded as unsatisfactory, but that could be because his logic was incomplete and not quite fully functional. Mar 5 '14 at 1:21