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Intuitionistic logic is a form of logic that doesn't have the law of the excluded middle, or LEM. The LEM says basically that a proposition that is not true is false, and a proposition that is not false is true. Classical logic has LEM but intuitionistic logic does not.

A 3-value logic is a logic where a proposition can be not merely True or False, but also something else, typically Unknown. It can be used to handle various propositions that 2-value logic cannot, propositions formed such that there is no consistent way to assign either true or false to them. Naturally, LEM also does not apply to a 3-value logic. I tend to view LEM as a device for defining what counts as a proposition. In a logic with LEM, for example, you can't allow a proposition like the Liar, "This sentence is false", so it just can't be formulated as a proposition, but in 3-valued logic it would be possible to allow it.

It occurs to me that Intuitionistic logic is a sort of stub 3-value logic, one where there is no way to deal with the third value, but where the possibility of a 3rd value is still allowed for. That is, without LEM, there is no way to prove that a proposition can only have one of two values.

Are there any developments of this idea in the literature? For example, is there any work where intuitionistic logic is used as a base for either classical logic or 3-value logic just by adding axioms or other features?

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    I expected otherwise, and so was surprised to find that intuitionistic logic is explicitly not three-valued. Feb 19, 2023 at 19:52
  • @KristianBerry And that's better - three-valued is more restrictive.
    – Frank
    Feb 19, 2023 at 22:37
  • @Frank, more restrictive in what sense? Feb 19, 2023 at 23:43
  • @DavidGudeman Saying that there are only 3 values seems more restrictive to me than removing the excluded middle?
    – Frank
    Feb 19, 2023 at 23:47
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    @Conifold, I don't see any rational point here, just personal distaste for the formalism. It works very well in practice. It has concrete semantics for giving the answers to questions, where the data is incomplete. Calling that a processing workaround is like calling zero an accounting workaround. Feb 22, 2023 at 21:11

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In answer to the last question: is there any work where intuitionistic logic is used as a base for either classical logic or 3-value logic just by adding axioms or other features?

It's not in the literature, but it may be possible to do it the other way around, that is, to construct a model that satisfies Heyting's axioms of intutionism ( as for instance https://www3.cs.stonybrook.edu/~cse541/15chapter11.pdf ) using suitable definitions based on Lukasiewicz 3-valued logic. This does not establish equivalence. It appears that intuitionism uses a "strong" negation instead of the standard, which in a way breaks the system to make it non-truth functional, but the idea that intuitionism may be an incomplete stub of a more complete and robust (as well as truth-functional) 3-valued logic seems likely.

This doesn't really solve the paradox of the liar, since L3 has an "excluded fourth" law in place of the excluded middle, but an incomplete logic that can consistently handle the unknown is already useful in some computer applications.

As a bonus, it also seems possible to do the same with the Lewis axioms for S5: that is, construct a truth-functional 3-valued model that satisfies the axioms. Again this is not an equivalence, because the definition of the strict conditional is different, and Lewis's definition breaks the system, but the 3-valued version appears to be more robust.

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A few points can be made here.

In the modern way that the Law of Excluded Middle is defined, it is a syntactic statement. It states that "φ ∨ ¬φ" is a theorem of logic. It is the Principle of Bivalence that states that "either φ is true or φ is false". Classical logic features both and intuitionistic logic lacks both. But they are not the same thing and some formal systems have one but not the other.

Intuitionistic logic is not three-valued. In fact, it is not n-valued for any finite n. This was proved by Kurt Gödel.

The term proposition is not usually limited to the two-valued case. For example, the SEP article on many-valued logic uses the term proposition and propositional logic to include the many-valued case.

Allowing three values does not satisfactorily resolve the liar paradox, since one can extend the paradox by weakening it, e.g. "This sentence either has no truth value or it is false."

You can treat superintuitionistic logic as a base for logics that are intermediate between intuitionistic and classical. There are several such logics.

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  • In the last sentence, did you intend to write superIL?
    – J D
    Feb 20, 2023 at 4:09
  • You seem to have misunderstood a couple of things in my question. First, I didn't say intuitionist logic is three-valued. I'm well aware that it is not. That's what I mean by calling it "a sort of stub 3-valued logic" where you can't reference a third value but also can't rule one out. Second, I was not saying that propositions are in general limited to two values; I was saying that in 2-valued logics they are limited to 2 values. I'm well aware of multi-value logics. Feb 20, 2023 at 5:43
  • Finally, the Law of the Excluded Middle harks back to classical Greece, and it is not a purely syntactic principle. Just because some microcosm of logicians uses the term in that way you describe doesn't make that the only correct way to use the term. Feb 20, 2023 at 5:46
  • Intuitionists typically do not accept that there is a third truth value. Sometimes this is expressed by distinguishing between a weak and strong version of bivalence. The weak version is that there are only two truth values; the strong version is that every proposition has one truth value or the other. Intuitionism is compatible with the weak version, but rejects the strong version.
    – Bumble
    Feb 20, 2023 at 8:37
  • If you accept that propositions can be multi-valued, then I'm not sure what to make of your statement that you view LEM as a device for defining what counts as a proposition. The distinction between LEM and bivalence is as standard as it gets in modern logic, and is not limited to a microcosm of logicians.
    – Bumble
    Feb 20, 2023 at 8:38
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In intuitionistic logic, we can't use LEM in order to prove propositions, but that doesn't mean that LEM is false. After all, the idea that "either LEM is true, or there's a proposition that is neither true nor false" is itself an LEM-like / classical piece of reasoning.

I mostly came into contact with intuitionistic logic via constructive logic, via proof assistants where you identify propositions with the set of their proofs and prove them by giving an explicit member of the set. In particular, when you're proving a disjunction "p or q", you need to pick which one of p or q you're going to produce a proof for, and then produce a proof for it, and on the other hand, when you have a proof for a disjunction, you're always able to inspect it and ask which of p or q it proves. In this way "p or q" means not only "p is true or q is true" but further "... and I can tell you which one".

Clearly we can't have LEM in such a system unless we're able to know not only that each statement is true or false, but decide which of those it is, which rules out any system that can encode basic integer arithmetic.

This doesn't mean that there's some secret third thing that propositions can be, that isn't either true or false. It just means that by restricting ourselves from exploiting the two-valued nature of our logic, we get proofs that can have more structure or content to them. These proofs are always also construction / decision procedures.

I intend this as an example of where an intuitionistic logical system may be useful despite conceptually having no need for a third truth value. I guess you could say "ok but this system doesn't track truth, it tracks provability, and there are three proof states, proven true, proven false, and unprovable", and that's kind of true? But it's not really central to what the system is for or how it works (and, as you suggested, the system doesn't really have a way of referring to the third state directly.)

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  • Everyone seems to be misunderstanding my point, thinking that I'm saying intuitionist logic has three truth values. If it had three truth values, then it wouldn't work as a base for classical logic. My point was that it is agnostic about additional truth values, not that it has a third truth value. Feb 28, 2023 at 22:53

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