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I am looking for some alternative logic where a sentence p could not only be true or false but also absurd, which is different of false in such a logic.

I see that there is some content about three valued logic : https://en.wikipedia.org/wiki/Three-valued_logic. However mine would be different because absurd would be an absorbing element for any operation.

A formal system of this logic would be complete if for any true sentence there is a proof that the sentence is true.

A formal system of this logic would be consistent if you can't prove 0=1.

For sure there is no excluded middle in such a logic since a proposition can be neither true nor false, but absurd.

The goal of this logic would be to get a formalization for arithmetic both complete and consistent. In classical logic Godel theorem forbid it but in such a logic the proof of Godel theorem is no longer possible because Godel assumes at some point that either P(G(P)) or P(G(P)) is false, but in this logic P(G(P)) could also be absurd.

Such a logic could also formalize the "not well defined sentence", like division by 0 and so on.

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    In your "logic" the law of non-contradiction, p ∧ ¬p = F, fails too. There are paraconsistent logics where this is allowed, but they are not truth-functional - truth values of connectives are not determined by truth values of their arguments. Truth-functional many valued logics are generally dismissed as models of logic these days exactly because meaningful generalizations of classical logic are not truth-functional, see Urquhart, p. 43, they are studied for other reasons.
    – Conifold
    Commented Jul 19, 2022 at 18:00
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    Do you assume an analogue of the law of the excluded middle where every statement must be either true, false, or absurd? Note that if you do you will still get an analogue of the liar paradox with something like "this statement is either false or absurd".
    – Hypnosifl
    Commented Jul 19, 2022 at 23:09
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    By your own rules, Absurd ∧ ¬Absurd = Absurd ∧ Absurd = Absurd ≠ F.
    – Conifold
    Commented Jul 20, 2022 at 8:44
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    I believe your rough core idea here is to use the Absurd truth value to account for those non-halting results of the non-recursive Provable() property, but how are you gonna constructively assign it through some algo/formula?
    – Double Knot
    Commented Jul 21, 2022 at 6:43
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    Pls note computer scientists usually won't say such "result" is Absurd which is really a synonym of Falsity, they'll usually employ an idea called Turing jump as some kind of infinite logic forcing pi_1 sentences such as G or Con(PA) lower in the arithmetic hierarchy to halt, and this can go on for anything "Absurd" in your term in another level. It's a famous result that the set of encodings of true formulas in PA with a predicate for any decision problem X definable in PA always halt from the Turing-reducible jumped problem X^(ω)...
    – Double Knot
    Commented Jul 24, 2022 at 19:49

3 Answers 3

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The goal of this logic would be to get a formalization for arithmetic both complete and consistent.

There is no such thing known. If you did discover it, you'd get a guaranteed Fields prize (or equivalent).

There are also some negative results in this area (Pudlák).

Furthermore, the issue with trying ternary logic for this is that you'd first have to (re)define "complete", because the usual definition is that

A theory is called complete (see wikpedia:complete theory) if for every sentence either it or its negation is provable in the theory.

So you'd have to carefully consider what negation does in your logic and how you define "complete". I suspect you'd simply end up with a trivial embedding of classical/binary logic if you follow that def.

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Two background pointers:

  1. Although you exclude the law of excluded middle, you have the option of the excluded fourth(?), or so on down that line. That's a neat tool to have at your logician's disposal.
  2. The really first step, though, is gonna be defining absurd as a truth value. So in physics, there's this idea of imaginary time, or time's signature being not just another real number, but a function of imaginary numbers also/instead (to put it loosely...). Given what absurdity is,??? let's say that in this highly exotic logic, the third truth value is i, the imaginary unit.
  1. This opens the door to the unit sphere of the quaternions allowing the possibility of an infinite-valued absurdity-theoretic logic, where it is precisely the Absurd truth values that are infinite in number, whereas True and False are just those two themselves. This due to the absurdity of infinity, perchance, then.

???Because it would be absurd for i to be a truth value, wouldn't it? Yet there it is. Actually, almost the whole of transalethism seems unexplored. By this I mean the esoteric possibility of using numbers not in the [0, 1] interval as truth values, e.g. i here, but also -1 (for an alternative conception of falsity), or 2, or whatever. So supertruth/hypertruth/ultratruth, if you will. Maybe I just haven't found much about it yet (there are some references I'm familiar with, which thematically resonate with the transalethist problematique), but at any rate, the possibility of using even more "bizarre" truth values (like π, for example) is captivating for me, so know that I offer my advice in the spirit of commending that bizarre form of absurdity for which I am arguing.

Modern abstract objects typically don't have to exemplify themselves, but in a weird twist, the Form of Absurdity (such as it is) is absurd Itself, too; in fact, the Absurd-in-Itself is bizarre/weird/surreal to boot. It is absurd (and bizarre and weird) that there is absurdity, no less. It's hard for me to imagine, off the top of my head, how having i as a truth value could actually work. For now, I just hope that it can.

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Here is an inchoate idea: perhaps others more knowledgable than me can modify.

If you wish to talk about completeness, consistency, etc. in your logic, you might try working out the syntactic parallel to "absurd". It seems that the closest is "ill formed" or "ill-defined". These also fit in with your "absorb" notion- concatentating ill-formed and ill-defined statements results in further ill-formed statements. But, Gödel’s operations are well-defined for a very large class of formal systems, namely those with a reasonable proof and arithmetic system.

Presumably- although I haven't worked it out myself- you could construct a very similar diagonal lemma. Certainly you could get the Gödel numbering to go thru and you could probably define most of the relations and properties in a similar manner. In particular, at no point is there anything that looks ill-formed, or ill-defined, or absurd.

However, if you do go thru, you may wish to look at Post's false operator, which sends everything to a truth value. It acts as the bottom element of a lattice and obeys the absorption laws, so perhaps there is something there for you.

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  • Thank you, yes may be ill formed or meaningless is better. I am not a native english speaker, so it seems to me that it is the same than absurd. May be it is also a philosophical point of view, I believe that there is no semantical absurd, thus absurd is always refering to some syntax.
    – François
    Commented Jul 21, 2022 at 11:23
  • If you don't believe that there is a semantic absurd, why would you attempt a valuation? this amounts to giving a formal semantics to it.
    – emesupap
    Commented Jul 21, 2022 at 14:09
  • Yes, good point, I think the goal of the third valuation is to make explicit the mistake (in my opinion) of classical logic.
    – François
    Commented Jul 22, 2022 at 9:36
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    you do realize that his first incompleteness theorem goes through in intuitionistic logic as well, right?
    – emesupap
    Commented Jul 22, 2022 at 17:34

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