Fitch's paradox of knowability

Fitch's paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the "knowability thesis", which states that any truth is, in principle, knowable. The paradox is that this assumption implies the "omniscience principle", which asserts that any truth is, in actual fact, known. Essentially, Fitch's paradox asserts that the existence of an unknown truth is unknowable. So if all truths were knowable, it would follow that all truths are in fact known.

The paradox is of concern for verificationist or anti-realist accounts of truth, for which the "knowability thesis" is very plausible, but the omniscience principle is very implausible.

The paradox appeared as a minor theorem in a 1963 paper by Frederic Fitch, "A Logical Analysis of Some Value Concepts". Other than the knowability thesis, his proof makes only modest assumptions on the modal nature of knowledge and of possibility. He also generalised the proof to different modalities. It resurfaced in 1979 when W.D. Hart wrote that Fitch's proof was an "unjustly neglected logical gem".

Proof

Suppose "p" is a sentence which is an "unknown truth"; that is, the sentence "p" is true, but it is not known that "p" is true. Then the sentence "the sentence "p" is an unknown truth" is true; and, since all truths are knowable, it would be possible to know that "p" is an unknown truth. But this isn't possible. If it were known that "p" is a truth, then it wouldn't be an "unknown" truth. Therefore, there are no unknown truths; equivalently (in classical logic), "all truths are known".

This can be formalised with modal logic. K and L will stand for "known" and "possible", respectively. Thus LK means "possibly known", in other words, "knowable". The modality rules used are:

This time the proof proceeds:

The last line matches line 6 in the previous proof, and the remainder goes as before. So if any true sentence could possibly be believed by a rational person, then that person does believe all true sentences.

Some anti-realists advocate the use of intuitionistic logic; however, except for the very last line which moves from "there are no unknowable truths" to "all truths are known", the proof is, in fact, intuitionistically valid.

The knowability thesis

Rule (C) is generally held to be at fault rather than any of the other logical principles employed. It may be contended that this rule does not faithfully translate the idea that all truths are knowable, and that rule (C) should not apply unrestrictedly. Kvanvig contends that this represents an illicit substitution into a modal context.

See also

* Moore's paradox

External links

* [http://plato.stanford.edu/entries/fitch-paradox/ Fitch's Paradox of Knowability] . Article at the Stanford Encyclopedia of Philosophy, by Berit Brogaard and Joe Salerno.
* [http://consequently.org/writing/notevery/ Not Every Truth Can Be Known: at least, not all at once] . Discussion page on an article of the same name by Greg Restall to appear in Salerno's book

References

* Frederick Fitch, " [http://www.jstor.org/pss/2271594 A Logical Analysis of Some Value Concepts] ". Journal of Symbolic Logic Vol. 28, No. 2 (Jun., 1963), pp. 135-142
* W. D. Hart. "The Epistemology of Abstract Objects", Proceedings of the Aristotelian Society, suppl. vol. 53, 1979, pp. 153--65.
* Johnathan Kvanvig. [http://books.google.ca/books?id=nhRZqgREEQMC The Knowability Paradox] . Oxford University Press, 2006.
* Joe Salerno, ed. [http://knowability.googlepages.com/home New essays on the knowability paradox] . Oxford University Press, to appear.