Experiment #136 · Thought experiment

Curry's Paradox

"If this sentence is true, then everything is true." Prove anything you like.

Haskell Curry · 1942 · Logic

First published: H. B. Curry, "The Inconsistency of Certain Formal Logics", *J. Symbolic Logic* 7 (1942): 115–117.

Let C = "If C is true, then arithmetic is inconsistent." Trivial reasoning derives the conclusion. The paradox is harder than the Liar.

Curry's paradox: consider the sentence C = "If C is true, then ψ" for an arbitrary ψ. By naive principles (unrestricted truth, conditional introduction, modus ponens), one can derive ψ — *any* sentence whatever, including "1 = 0." Unlike the Liar, Curry doesn't require negation, only conditional reasoning. The paradox motivates restrictions on naive truth that go further than standard responses to the Liar, and bears directly on paraconsistent and substructural logics. It is the cleanest reason to deny *some* of the basic logical principles classical logic takes for granted.

Formulation

C = "If C is true, then ψ" (any ψ). 1. Suppose C is true. 2. Then by truth-schema, "If C is true, then ψ" is true. 3. Hence by MP, ψ. 4. Discharging assumption (→I): C → ψ ≡ C. 5. Hence C is true. 6. By MP again, ψ. Conclusion: ψ for any ψ. Naive truth + classical conditional → triviality.

Compare this experiment

Dimensions Engaged

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Information

Bears on Information · Granularity: even more strongly than the Liar, naive truth principles produce informational collapse.

Responses — How Schools Engage

Affirms / takes the bait 3

Logical Positivism

Self-referential truth-talk must be regimented by Tarskian hierarchies or similar restrictions; ordinary-language pretensions to unrestricted truth are formally untenable.

Structuralism

A clean motivation for substructural logics: classical structural rules (e.g., contraction) are too strong; restricting them blocks Curry without sacrificing useful inference.

Postmodernism

A formal vindication of postmodern suspicion of totalising truth-claims: the very logic that supports them is self-undermining.

Reframes the question 2

Jainism / Anekantavada

Jain logic's seven-valued treatment of contradiction is congenial to substructural responses; Curry is a Western rediscovery that absolute truth-talk must be qualified.

Pragmatism

Truth-talk is for useful inquiry; Curry shows that pretending we have a complete, unrestricted truth predicate is a mistake — practical truth is local and contextual.

Holds it inconclusive 1

Analytic Metaphysics / Logical Atomism

A live constraint on theories of truth: any responsible treatment must address Curry as well as the Liar. Standard responses block one or another of the principles invoked.