Fitch's Knowability Paradox
If all truths are knowable, all truths are known
First published: F. Fitch, "A Logical Analysis of Some Value Concepts", *J. Symbolic Logic* 28 (1963): 135–142.
Anti-realists hold that all truths are knowable. Fitch shows this entails that all truths are actually known — provided there are any unknown truths, contradiction.
Fitch's proof: assume the knowability principle (KP) — for every truth p, possibly someone knows p — and assume there exists at least one unknown truth p (a non-omniscience assumption). Consider q = "p is true and no one knows p." By KP, q is knowable: someone could know q. But knowing q entails knowing both conjuncts, including "no one knows p" — which contradicts the very knowing required. Hence either no truths are unknown, or KP is false. The result is a serious technical pressure on semantic anti-realism (Dummett, Wright) and has spawned decades of replies — restricted KPs, intuitionist responses, and structural reformulations.
Formulation
KP: ∀p (p → ◇K p). Non-omniscience: ∃p (p ∧ ¬Kp). Let q = (p ∧ ¬Kp). KP applied to q: ◇K(p ∧ ¬Kp). But K distributes over conjunction and is factive: K(p ∧ ¬Kp) → Kp ∧ K¬Kp → Kp ∧ ¬Kp, contradiction. Hence ¬◇K(p ∧ ¬Kp), contradicting KP.
Dimensions Engaged
Observer
Observer · Knowledge Extent: the structural limits of what can be known under modest principles.
Information
A formal result about which propositions are *informationally* available to any knower.
Responses — How Schools Engage
Affirms / takes the bait 2
Fitch corroborates classical realism: there are truths that, even if knowable in principle, are not actually known. The anti-realist conflation of truth and knowability fails.
Sharpens the skeptical conclusion: even modest epistemic principles fail under formal scrutiny. Caution about epistemic claims is structurally well-founded.
Reframes the question 1
Intuitionist constructivism handles Fitch by rejecting classical disjunctive reasoning at the relevant step; the proof goes through only on classical assumptions the constructivist already rejects.
Holds it inconclusive 2
A canonical pressure-test for anti-realism. Restricted-KP responses (Tennant), intuitionist responses, and Salerno's extensive analyses keep the debate active.
A technical headache for verificationist accounts of meaning: the connection between truth and verifiability is more subtle than early positivists assumed.
Related Experiments
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Further reading
- Fitch (1963), op. cit.
- Salerno (ed.), *New Essays on the Knowability Paradox* (2009)
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