Category: logic
Type: paradox
Associated thinker(s): Frederic Fitch
Original publication or approximate period: Fitch, “A Logical Analysis of Some Value Concepts,” The Journal of Symbolic Logic 28(2) (1963): 135–142; theorem 5
Difficulty: Advanced
Themes: knowability, verificationism, modal logic
Related experiments: The Unexpected Hanging / Surprise Test; The Liar Paradox
Suppose every true proposition is knowable in principle: if p is true, then it is possible that someone knows p. Some verificationist and anti-realist views have been argued to accept a version of this principle.
Now suppose that a true proposition p is not known by anyone at any time. Then the compound proposition “p is true and nobody knows p” is also true. If the knowability principle applies to every proposition, someone could know that compound. But knowing the conjunction would include knowing p and knowing that nobody knows p. Since knowledge is factive, that combination is impossible. So the unrestricted principle conflicts with there being an unknown truth.
Here Kp means “someone knows p at some time,” and possibility means that such a knowledge state could obtain. This informal sketch assumes that the knowability principle applies even to compound propositions, that knowledge is factive and distributes over conjunction, and that standard classical and modal principles are available. The formal proof and its assumptions are discussed in the cited SEP entry.
Does the principle that every truth is knowable commit us to the conclusion that every truth is known by someone at some time?
The result challenges views that accept an unrestricted knowability principle together with the relevant epistemic and modal logic. It does not by itself refute every view called verificationist or anti-realist: whether the principle accurately represents a particular theory is disputed.
It also shows how a claim about what could be known can have surprising consequences when it applies to complex propositions such as “p and nobody knows p.” The literature debates whether the result is best treated as a refutation, a paradox, or a reason to revise the principle or its logic.
One response rejects the unrestricted knowability principle. Others restrict it to selected kinds of propositions, revise the underlying logic, or change how the relevant modality or propositions are understood. The SEP distinguishes, among other approaches, intuitionistic and paraconsistent revisions from semantic and syntactic restrictions. These proposals have their own challenges and remain debated.
The proof does not show that people actually know everything by empirical means. It shows that an unrestricted knowability principle, together with its logical assumptions, is incompatible with there being a truth that no one ever knows. Fitch published the theorem in a paper about value concepts and credited its essential form to an anonymous referee in 1945. The SEP reports that the referee was later identified as Alonzo Church; Fitch also noted a related result in an earlier Anderson paper, attributed there to W. T. Parry. The priority history is not settled here.
Some proposals restrict the principle to selected propositions. For example, the SEP discusses “Cartesian” propositions, roughly those for which knowing p is not formally inconsistent. Other approaches revise the logic or the semantics. The SEP survey also explains why proposed restrictions face further Fitch-style challenges.
The proof, its formal assumptions, and the response literature were checked against Brogaard and Salerno’s SEP entry, especially §§1–5. Fitch’s 1963 article was checked through the publisher’s metadata and selected extract and footnotes; the full article was not accessed. The publisher extract supports the publication details and Fitch’s note crediting an anonymous referee, while the SEP supplies the full proof exposition and reports the later identification of the referee as Church. No claim is made that one priority account is uncontested or that the proof settles every form of anti-realism.
Research check: 2026-11-01. This is an original educational explanation, not a substitute for the cited sources. Additional comparisons are teaching prompts, not established historical variations.