On keeping blue swans and unknowable facts at bay : a case study on Fitch's paradox

In Joe Salerno (ed.), New Essays on the Knowability Paradox. Oxford University Press (2009)
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Abstract

(T5) ϕ → ◊Kϕ |-- ϕ → Kϕ where ◊ is possibility, and ‘Kϕ’ is to be read as ϕ is known by someone at some time. Let us call the premise the knowability principle and the conclusion near-omniscience.2 Here is a way of formulating Fitch’s proof of (T5). Suppose the knowability principle is true. Then the following instance of it is true: (p & ~Kp) → ◊K(p & ~Kp). But the consequent is false, it is not possible to know p & ~Kp. That is because the supposition that it is known is provably inconsistent.3 The inconsistency requires us to deny the possibility of the supposition, yielding ~◊K(p & ~Kp). This, together with the above instance of the knowability principle, entails ~(p & ~Kp), which is (classically) equivalent to p → Kp. Since p occurs in none of our undischarged assumptions, we may generalize to get near-omniscience, ϕ → Kϕ. QED.

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Berit Brogaard
University of Miami

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References found in this work

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What is a Theory of Meaning? (II).Michael Dummett - 1976 - In Gareth Evans & John McDowell (eds.), Truth and Meaning: Essays in Semantics. Oxford: Clarendon Press.
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A logical analysis of some value concepts.Frederic Fitch - 1963 - Journal of Symbolic Logic 28 (2):135-142.
The paradox of knowability.Dorothy Edgington - 1985 - Mind 94 (376):557-568.

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