Curry’s paradox, not the liar, is the real test of a theory of truth, and most theories fail it

Tomasz Szewczak Fellow

October 16, 2026 · 4 replies

Fallacies & paradoxes Analytic Graham Priest J. C. Beall #contraction #curry-paradox #liar-paradox #paraconsistent-logic

Its coordinates: 4.2.1 Fallacies

Where it stands

For 3Against 1

Every proposed solution to the liar ("this sentence is false") is usually judged by what it says about negation. Some say the liar is neither true nor false; dialetheists say it is both. But Haskell Curry showed in 1942 that you can get trouble with no negation at all.

Take the sentence C: "If C is true, then the moon is made of cheese." Suppose C is true. Then, by what C says, if C is true the moon is cheese; so the moon is cheese. So we have proved: if C is true, the moon is cheese. But that is just what C says, so C is true. And so the moon is cheese. Any claim at all can replace the cheese.

The steps used are modest: modus ponens, conditional proof, and the rule called contraction, which lets you use an assumption twice.

Thesis: Curry's paradox is the real test. A theory that handles the liar by changing negation, whether with truth-value gaps or gluts, has not yet solved anything, because Curry needs no negation. Graham Priest's logic LP survives only because its conditional fails modus ponens, which is too high a price. J. C. Beall and others give up contraction instead, and I suspect that price is too high as well.

I write proofs that machines check for a living. Convince me there is a conditional worth having that survives Curry.

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4 replies

For the thesis 3

Oskar Nyberg 2 weeks ago

For Fellow

For. Ana Clara's resource picture is lovely for proofs and strange for belief. If I believe A and believe that A implies (A implies B), it is very hard to see what is wrong with my using A twice to get B. I didn't spend my belief in A the first time.

Curry exposes that the liar was never about negation. It's about self-reference plus a truth predicate that is too obliging. I'd rather restrict the truth predicate than the logic.
Derek Hollis 2 weeks ago

For Fellow

For, from someone who also writes machine-checked proofs. In Coq or Lean you can't even write Curry's sentence: the type system forbids the kind of self-reference it needs. People sometimes call that cheating, but it's exactly Oskar's option, a restriction on what can be said, built so deep that you never feel it. Most working mathematics has never missed what it rules out.
Tomasz Szewczak 2 weeks ago

For Fellow

Derek's point is the one I live with daily, and it is honest about its cost: the restriction is in the language, so we cannot state certain things at all. Ana Clara, the resource reading does move me. My remaining doubt is Oskar's: beliefs don't get used up. I'll keep my side but downgrade "too high" to "not yet paid for".

Against the thesis 1

Ana Clara Ferreira 2 weeks ago

Against Fellow

Against, or at least against "too high a price". Contraction is not as innocent as it looks. In linear logic, assumptions are resources, and using one twice is a real step that should be justified, not a free move. Once you see conditional reasoning as consuming premises, rejecting contraction looks like bookkeeping, not mutilation.

And notice what we keep: modus ponens, conditional proof, and a naive truth predicate with no hierarchy of languages. Compare Tarski's way out, which stops you saying "everything Tomasz says is true" in one language. I'd rather lose contraction than lose that.