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It seems to me that the state of things is that something can be true, not true, or undecidable, in mainstream mathematics. So, morally, we really should be using three valued logic.

I looked at the Wikipedia article and saw that interpretation of the third value as undecidable is done, but saw no reference. Is there anything to gain by doing this? Can we use this to treat undecidable propositions as having that third truth value in a useful way?

Matt Samuel
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    Morally? That's not a mathematical reason. Pragmatically? understandable. How about "many-valued logic" from the Stanford Encyclopedia of Philosophy. Also search for "multi-valued logic(s)". – amWhy Dec 15 '18 at 23:27
  • @Derek Tried to be more specific. – Matt Samuel Dec 15 '18 at 23:58
  • To the downvoters, any comments on how I can improve would be helpful. – Matt Samuel Dec 16 '18 at 00:28
  • Not as an answer to your last comment, but why stop at three truth values? Why not ask about four or more: e.g. "true, not true, might be true (depends), don't know if it's true, might be true in the future, is true in some worlds but not in others, ..." – amWhy Dec 16 '18 at 00:33
  • @amWhy I thought I was clear about that. Three truth values accurately describes the possibilities in mainstream mathematics, like ZFC. – Matt Samuel Dec 16 '18 at 00:34
  • I'm concerned about introducing a truth value meaning $\varphi$ and $\lnot \varphi$ are both unprovable in the current system that I'm using, right now. It seems like there's a paradox lurking in there. – Greg Nisbet Dec 16 '18 at 00:51
  • @Greg Probably, I guess this would likely be a meta theory. – Matt Samuel Dec 16 '18 at 01:06
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    There is certainly an easy construction here. Given a theory - say, ZFC - we get a corresponding "three-ary valuation" for sentences in the language: ZFC-provable, ZFC-disprovable, and ZFC-independent. A serious drawback, however, occurs if we try to view these as truth values, since the Boolean operations are no longer truth-functional: if $P$ and $Q$ are independent, what is the "truth value" of $P\vee Q$? Sometimes it will be independent (e.g. $P=Q$) and sometimes it will be provable (e.g. $P=\neg Q$). – Noah Schweber Dec 16 '18 at 01:17
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    What we're seeing here is really the fact that provability is better understood as a modal operator, with - for example - the truth value of "$\neg\Box\neg(P\vee Q)$" not being deducible from just knowing the truth values of $P$ and $Q$ (here "$\Box$" means "is provable from [theory]"). Personally, I think that one of the crucial aspects of truth values is that they have a rich set of truth functionals, so the provability picture doesn't seem adequate to me for this. – Noah Schweber Dec 16 '18 at 01:19
  • If you'd like, I can add some details and turn this into an answer (along with a bit more info on other many-valued logics), but I'm not sure this is exactly what you're asking about. – Noah Schweber Dec 16 '18 at 01:20
  • @Noah Sure. It's basically a negative answer to my question. If it can't be done, then it is what it is. Thanks. – Matt Samuel Dec 16 '18 at 01:21
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    See Kleene three-value logic where the third value is "unknown". – Mauro ALLEGRANZA Dec 16 '18 at 11:48
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    And see Quantum Logic and Probability Theory. – Mauro ALLEGRANZA Dec 16 '18 at 11:49
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    And see Graham Priest, An Introduction to Non-Classical Logic, Cambridge UP (2nd ed 2008), Ch.7 Many-valued Logics. – Mauro ALLEGRANZA Dec 16 '18 at 11:51
  • @Mauro Thanks, I'll take a look, though I think Noah's eventual answer is the one I am looking for, when he gets a chance. The specific application of multivalued logic I'm interested in would treat undecidable propositions as having the third truth value, and he seems to be saying that this doesn't work well with logical operations. – Matt Samuel Dec 16 '18 at 12:13
  • @MattSamuel I've written a first stab at an answer. Let me know if you'd like me to go into more detail on some point. In particular, I haven't written much about non-truth-functional operators and in particular the view of provability as a modality. – Noah Schweber Dec 16 '18 at 17:27
  • @Noah Looks like it's something I'll need to find some quiet time to read, which is something that occurs only at certain times of day with two young children. I will definitely let you know. Thanks. – Matt Samuel Dec 16 '18 at 17:31
  • I think the wikipedia https://en.wikipedia.org/wiki/Three-valued_logic is not the right one.

    In my opinion mathematics should be formalized not as an open system that you can extend indefinitly but as a close, finite, complete thing. That's why, if the goal is to use this logic for the foundation of mathematics, I would suggest as a third value A for Absurd.

    A would be an absorbing element for any operation. A OR something is A, A AND something is A, not A is A.

     – François Jul 19 '22 at 11:46
  • I wonder also if we could speak about the false only as a negation of the true and thus remove the false of this logic. It would have only 2 syntaxical values : true/absurd and only one semantical value : true. The aburd syntax would not have a semantical interpretation because absurd only belong to syntax world and not to semantical world, which is single-valued in my idiosyncrasy. – François Jul 19 '22 at 11:52

2 Answers2

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The OP writes:

...the state of things is that something can be true, not true, or undecidable, in mainstream mathematics.

To my way of thinking, this statement appears to be fundamentally flawed, conflating truth with provability. We do not identify truth in set theory or mathematics with provability in any particular theory, and indeed, the distinction between truth and provability has been one of the most important advances in mathematical logic in the twentieth century.

It seems to me that the relevant trichotomy to which the OP refers should be: provable, refutable, independent. But these are not the same as truth values, and it would seem wrong to me to present the trichotomy as a system of truth values, for this would be to confuse truth with provability.

In any model of set theory (or universe of set theory, if you will), every statement is either true or false; there is no third possibility for truth or truth-in-a-model.

The pervasive independence phenomenon in set theory, in contrast, is about provability rather than truth. Statements like the continuum hypothesis are neither provable nor refutable in ZFC, and there are hundreds of natural examples, fundamental questions in set theory, which turn out not to be decided by the commonly considered axioms. But meanwhile, set theory is full of stronger theories, which do settle these statements. For example, the proper forcing axiom PFA implies the negation of CH over ZFC.

Meanwhile, a more sophisticated analysis of the independence phenomenon leads one naturally to the hierarchy of consistency strength. We have an enormous hierarchy of large cardinal notions and other natural mathematical statements, which have a relative degree of independence, in the sense that they increase in consistency strength. In this sense, it could make sense on these grounds to say that one statement is more or less independent than another. This would lead one naturally to more than three "values" in your system. But a more precise account is given simply by the claims of the consistency-strength hierarchy itself.

JDH
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  • By definition, we maintain consistency no matter what truth value we assign something undecidable, provided we have consistency in the first place. Arguably there could be reasons for one over the other, and like you said they may actually have a knowable truth value independent of the provability, but why not assign them a third truth value? – Matt Samuel Dec 16 '18 at 15:20
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    Because that's about provability in a theory, not truth. Independence assertions are always relative to a theory. For example, while CH is independent of ZFC, it is not independent of other set theories, such as ZFC+PFA or ZFC+V=L. So I wouldn't want to say that CH, by itself, has the truth value of "independent". Rather, in some set-theoretic universes, it is true, and in others, it is false. – JDH Dec 16 '18 at 15:41
  • While mathematically I'm quite mainstream, freely using infinities and all that, to me it's largely a game, and I doubt whether there is such a thing as an infinite collection of objects. So, to me, there's no reason for the continuum hypothesis to be true or false, because it might just be a question about something that isn't real anyway. I'm not a set theorist and the study of these axiomatic systems is basically irrelevant to my research. I understand the truth value may be important to some people, but perhaps there should be a theory where it's explicitly neither true nor false. – Matt Samuel Dec 16 '18 at 16:03
  • It could be an axiom. $Unknown(CH)$ – Matt Samuel Dec 16 '18 at 16:24
  • Basically, unlike a finitist, I accept the axioms of ZFC and therefore will tolerate the consequences. But it seems arbitrary to then add $\neg$CH, because even if you assert the existence of such a set, it's an empty assertion, because you can't get your hands on it. If you could, then it would simply be true with the original axioms. – Matt Samuel Dec 16 '18 at 16:36
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    I view the proposal as something like this. Some integers are even and others are odd; but also, if we say that $n$ is an integer, and we haven't yet decided which one, then perhaps it is not yet determined whether it is even or odd. So we really should have three parities: even, odd, and undetermined. – JDH Dec 16 '18 at 17:16
  • I can imagine, from my own experience with research, how someone's intuition for their own field might make them see things that way. For some things I agree. If Goldbach's conjecture is undecidable, then it undoubtedly must be true, otherwise we could find a counterexample. For other things, like what we're discussing, it's not so clear to me. – Matt Samuel Dec 16 '18 at 17:27
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    @MattSamuel Note that the distinction you draw here, insofar as it describes any degree of "absolute meaningfulness" to Goldbach, already suggests that "provable/disprovable/independent" isn't the end of the story. – Noah Schweber Dec 16 '18 at 17:33
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Tl;dr: there are drawbacks of what you propose, both algebraic and foundational, which I consider serious enough to indicate that the "valuation" in question really be a secondary rather than fundamental object. However, there are definitely more intricate approaches along the lines of what you suggest that are much more satisfactory, and even find application in classically-construed mathematics.

The short version for me is that the map given by a "structure" (in whatever sense is meant) from sentences to truth values is the fundamental object in a given logic, and what you've described is really better considered a "derived" object.

First, let me push back against Joel's answer a bit. There is a perfectly meaningful semantics for your logic: a "structure" is just a set of sentences in the relevant language, and this fully determines a map from sentences into $\{P,D,I\}$ (= provable, disprovable, independent). While this is extremely far from the usual notion of first-order semantics, and countable structures (respectively computable, or etc., countable structures) aren't in my opinion any more objectinable than arbitrary sets of sentences (resp. computable, or etc., sets of sentences), and so this shift feels a bit unnatural to me, it's perfectly satisfactory. So let's go with it.

So what are the criticisms?

  • First problem: Theory and semantics are blurred. Self-explanatory. In my opinion, part of what makes logic interesting is the differentiation between semantics and syntax and their subsequent unification. Certainly some notions of semantics make philosophical commitments which are difficult to justify, but that doesn't mean that all semantices fall prey to this, and abandoning the whole semantic project seems inappropriate to me.

    • That said, for completeness (hehe) I need to mention Jean-Yves Girard, whose "antirealist" stance is both highly interesting and on face value strongly opposed to at least some of what I've written here. I have my own thoughts about his work, and the extent to which it actually pushes against what I've written here, but this isn't an appropriate forum for that; I mention him only because failing to mention a serious opposing viewpoint to what I'm saying would border on dishonesty.

  • Second problem: We lose truth-functionality of basic operations. The truth value of $p\wedge q$ is no longer determined by the truth values of $p$ and $q$: e.g. taking our "model" (in this sense) to be the theory ZFC, consider $(i)$ $p=CH, q=\neg CH$ versus $(ii)$ $p=CH, q=CH$.

    • It's worth noting that truth functionality is by no means a requirement of a logical system - non-truth-functional operations are extremely important, and in particular a modal interpretation of provability yields a rich and meaningful theory - but the degree to which we lose truth functionality here seems difficult to justify.

So while the map assigning $P,D$, or $I$ to a sentence depending on its status relative to a given theory is extremely important and interesting, in my opinion it has too big drawbacks to be considered the satisfactory fundamental object for a logical system. Rather, it's better thought of as a derived object, coming after the semantics has already been established. This semantics need not involve strong Platonist commitments, but does in my opinion need to be a bit richer.

However, all is not lost (in my opinion)!

In a comment to Joel's answer you write

perhaps there should be a theory where it's explicitly neither true nor false.

This is not the same as what you've proposed; it's a much less restrictive vision. And indeed, it's one which is realized by extremely interesting and useful approaches.

I'll mention in particular Boolean-valued models. First, a purely syntactic comment. In some sense the truth functionality issue I highlighted above was only due to loss of information: if I kept track of the truth conditions of the sentences involved, and not just their ultimate proof-theoretic status, I wouldn't have run into trouble. This can be made precise by considering the Lindenbaum algebra of a theory: given a theory $T$, the set of sentences in the language of $T$ modulo $T$-provable equivalence forms a Boolean algebra, and this Boolean algebra - rather than the set $\{P,D,I\}$ - is a reasonable set of truth values. Semantically speaking, a Boolean-valued structure should be one where the atomic formulas are allowed to take on truth values in an arbitrary Boolean algebra, rather than just $\{\top,\perp\}$; so a Boolean-valued structure is probably going to be a pair consisting of the structure side and the "target" Boolean algebra. This winds up being a very useful generalization - in particular, it has serious value in the Boolean approach to forcing in set theory.

And we could also consider other algebraic structures for our sets of truth values, like Heyting algebras or even wilder things. Thinking along these lines also motivates the general study of algebraic logic, and for a perspective both general and syntactic see Czelakowski's book.

The moral of this line of thought is: mere provability status is not complete information, even if we adopt a radically anti-realist stance. It's this loss of information, not the absence of a philosophical commitment, which is intension with your suggestion.

One final note: another important loss-of-information occurring in what you propose comes from degrees of meaningfulness. You write in a comment to Joel's answer that you distinguish between Goldbach's conjecture and CH, at least partly because there is a strong sense in which "independence implies truth" for the former but not the latter, but once you accept this as a possible phenomenon to any degree the strict approach you've proposed seems even more inappropriate. I've written a bit about my own thoughts on relative meaningfulness here, and some grown-ups' thoughts on the matter can be found here.

Let me end with a bit of clarification. My stance above may come off as "semantics-first," which is pretty hypocritical: if my major foundational interest is the interaction between semantics and syntax, then I should be treating them as equally interesting.

My response is that I do think that it is perfectly valid (and often more interesting) to go the other way: first determine a purely syntactic logic, and then look for a semantics for it. Similarly, it's often extremely valuable to identify multiple equally good semantics for a given syntactically-defined logical system. However, I also think that we cannot really be satisfied until we have a semantics which is both reasonably natural and with respect to which our proof system is sound and complete. Moreover, my definition of "semantics" is sufficiently broad in my opinion that nothing is lost by my previous claim. I would say that the nature of my stance is aesthetic.

Noah Schweber
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  • You're right. There should be at least four values then, the Goldbach value, and the CH value. But then likely more gradations will pop up. – Matt Samuel Dec 16 '18 at 19:51
  • @MattSamuel Exactly (and actually even at first you'd want at least five, for the "anti-Goldbach value"). As we go up in "complexity" (e.g. as measured by quantifier rank - "independent implies true" is the same as "equivalent to $\Pi_1$"), we get less and less concrete - but we still have various degrees of concreteness. For example, if for some $n$ the statement "there are $(>n)$-many twin primes" is unprovable, then the twin prime conjecture is false, but here we have "mixed quantification." – Noah Schweber Dec 16 '18 at 19:56
  • And even at the level of CH we still have arguably-meaningful differences in concreteness - e.g. Shoenfield absoluteness separating $\Delta^1_3$ from non-$\Delta^1_3$, and even some separations between $\Sigma^2_1$ and non-$\Sigma^2_1$ (note that CH is $\Sigma^2_1$, so this is at the relevant level). – Noah Schweber Dec 16 '18 at 20:01
  • In my opinion mathematics should be formalized not as an open system that you can extend indefinitly but as a close, finite, complete thing. That's why, if the goal is to use this logic for the foundation of mathematics, I would suggest as a third value rather A for Absurd than I for Independent. If A is defined as an absorbing element for any operation I don't think there is the Goldbach value, the CH value and so on, all these values would collapse. – François Jul 19 '22 at 11:40