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One resolution of the Liar Paradox is that the Liar sentence is neither true nor false because no sentence L which is either true or false could possibly satisfy L ⇔ ¬L.

I couldn't attribute this resolution to any known academic author. Does anyone have any ideas?

Thank you for any scholarly reference.

EDIT

Given the comments so far, I maybe need to explain that I am aware that the neither-true-nor-false resolution of the Liar is certainly not new. Apparently, several logicians in the Middle-Ages already were of this view, although I don't know who they were and what their arguments were. So my question is not about this, but about the argument that no sentence L (which is either true or false) could possibly satisfy L ⇔ ¬L.

I suspect that the idea is of recent origin because it is reminiscent of Tarski's definition of the Liar as a sentence λ satisfying λ ⇔ ¬T⟨λ⟩.

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    Wikipedia on liar has refs: Kripke... If a statement's truth value is ultimately tied up in some evaluable fact about the world... If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value... Andrew Irvine has argued in favor of a non-cognitivist solution to the paradox, suggesting that some apparently well-formed sentences will turn out to be neither true nor false and that "formal criteria alone will inevitably prove insufficient" for resolving the paradox...
    – Double Knot
    Commented Jul 6 at 6:05
  • There exists another much more radical view called aletoric nihilism discussed in a post in this site which asserts truth shouldn't be used at all even for regular groundable well formed propositions...
    – Double Knot
    Commented Jul 6 at 6:36
  • @DoubleKnot, I'd be really surprised if Kripke was the first. Łukasiewicz, the Polish mathematician was doing multi-valued logics back in the 1920s, and I believe (from very faint memory) that the Liar was one of his motivations.
    – David Gudeman
    Commented Jul 6 at 7:59
  • @DavidGudeman I'm only commenting OP's body "I couldn't attribute this resolution to any known academic author. Any idea?", and didn't claim Kripke is the first to propose such, but Łukasiewicz doesn't appear in the wikipedia ref as contributor to 'resolution' of liar. One possible personal explanation is Łukasiewicz's three-valued logic is really about semantically-closed logic same as the bivalent logics, while per Tarski liar sentence if sensible and non self-referential belongs to the open natural language, thus compared to Kripe's and Irvine's, Łukasiewicz's doesn't really apply... –
    – Double Knot
    Commented Jul 6 at 8:31
  • Regarding Łukasiewicz , see Jan Wolenski, Łukasiewicz on the Liar Paradox (ML, 1994)
    – Mauro ALLEGRANZA
    Commented Jul 8 at 7:24

2 Answers 2

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I think that the earliest attested statement which has the form you've given is Russell's explanation to Frege that their axioms admitted an inconsistency. Quoting Russell's letter to Frege on the 16th of June, 1902:

There is just one point where I have encountered a difficulty. You state … that a function too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate.

That is, Russell concludes that w ⇔ ¬w by case analysis, and therefore that w must not have been eligible for truth- or falsehood. Frege replied a week later, writing on the 22nd of June, 1902:

Your discovery of the contradiction has surprised me beyond words and, I should almost like to say, left me thunderstruck, because it has rocked the ground on which I meant to build arithmetic. … Your discovery is at any rate a very remarkable one, and it may perhaps lead to a great advance in logic, undesirable as it may seem at first sight.

This reply shows that Frege had not had anything like Russell's approach in mind, and suggests that the mathematicians of the era had not yet considered the severity and precision of liars' sentences as counterexamples.

I cannot find verbatim freely-readable sources for this letter; this particular translation is supposedly from Kaal 1980 (ISBN-10 0226261972). I will approve edits that can give better sourcing or translations.

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  • I get the idea but it doesn't mean that Russell would have wanted to translate the Liar as L ⇔ ¬L. But this may have inspired Tarski's λ ⇔ ¬T⟨λ⟩, which is nearly λ ⇔ ¬λ.
    – Speakpigeon
    Commented Jul 7 at 10:13
  • @Speakpigeon - correct; Russell's proposed solution to the Liar was through the Type theory.
    – Mauro ALLEGRANZA
    Commented Jul 7 at 12:59
  • @MauroALLEGRANZA Yes, and his theory takes the sentence "This sentence is in English" as being ill-formed. I guess this falsifies his theory.
    – Speakpigeon
    Commented Jul 7 at 15:50
  • Note that Frege himself, in the quoted letter, suggests that one way around Russell's problem is to stratify predicates into multiple tiers. Russell would eventually develop Frege's suggestion over the next few years.
    – Corbin
    Commented Jul 7 at 19:02
  • @Speakpigeon: Ah, the first person to talk of the diagonal lemma was likely Carnap in Carnap 1934. I can't read German or find a good translation to verify this, but folklore is that Gödel's 1936 notes mention Carnap's work. Note that the diagonal lemma isn't the result you've described; indeed, it looks quite backwards from Russell's argument!
    – Corbin
    Commented Jul 7 at 19:08
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You could look into paraconsistent logic (and perhaps many-valued logics more generally). The basic idea is that some statements or sentences might not have (or it might be impossible to obtain for them) a determinate truth value ('true' or 'false'), so they need to be considered as 'undefined' or 'ungrounded'. In particular, the Liar would be an example of an 'ungrounded' sentence. A very good paper is Saul Kripke's Outline of a Theory of Truth (1975) in which he explicitly deals with the Liar paradox (and it's variations) by sketching an alternative to Tarski's "orthodox" approach. The "orthodox" approach avoids paradox (sort of salvages "truth") by constructing a hierarchy of languages or language levels:

Let Lo be a formal language, built up by the usual operations of the first- order predicate calculus from a stock of ... primitive predicates, and adequate to discuss its own syntax (perhaps using arithmetization). ... Such a language cannot contain its own truth predicate, so a metalanguage L1 contains a truth (really satisfaction) predicate T1(x) for Lo.

Philosophers have not been completely satisfied with such a construction, since it doesn't seem to do justice to our intuitions. Kripke gives a pretty elaborate overview of why he agrees with that and then gives a sketch of a possible alternative theory which avoids having to construct 'truth' as predicate in a meta-language. So, this alternative theory allows 'truth' as predicate in the base language, but then has to pay the price that only 'grounded' sentences can have a definite truth value. (The simplest example of a 'grounded' empirical statement is for instance "Snow is white". It doesn't refer to other statements and doesn't assert it's own or other sentences' truth or falsehood. Once the language starts using truth as predicate and also starts referring to sentences in the language, potential trouble arises.)

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