Fitch’s Logical Analysis

Frederic B. Fitch  A Logical Analysis of Some Value Concepts The Journal of Symbolic Logic Vol. 28, No. 2 (Jun., 1963), pp 135-142,

as summarised by Wikipedia.

Fitch’s paradox of knowability … essentially … asserts that if all truths were knowable, it would follow that all truths were in fact known.

This is regarded as significant by some philosophers, e.g  Hart. The more recent opinion of the Stanford Encyclopaedia of Philosophy is that the apparent paradox is avoidable, as long as one doesn’t apply the logic to empirical reasoning.

The proof given supposes Qp ≡ p ^ ¬Kp (p is true but not known)  for some p. It then shows that KQp leads to a contradiction, which it interprets as meaning that Qp is not knowable. If we consider instead K’Qp, where K’ is some state of knowledge that may differ from K, there seems to be no contradiction: we may come to know something that we know was previously unknown. This may be because we have new empirical data or have invented a new line of reasoning, perhaps inventing a new construct (as so often happens in mathematics).

The apparent paradox seems to arise because K’Qp implies K’p, and one can’t have K’p^¬Kp^(K’=K).

Fitch’s original theorem (his ‘Theorem 5’) has been interpreted as ‘if p is true but not known, then Q(p) is true but not knowable.’ But in his formula K appears on both sides. To get the odd interpretation the first occurrence has to stand for some specific state of knowledge and the second has to stand for a possible state of knowledge. If, instead, both K’s stand for either a definite state of knowledge or a possible state of knowledge, the apparent difficulty disappears.

Dave Marsay


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