Thursday, September 22, 2005

Time For Fitch

John Burgess has a paper here on a temporal analogue of Fitch's paradox: if whatever is true will be known, then whatever is true is known.

Using 'Kp' to mean 'It is known that p' and 'Fp' to mean 'it sometime will be that p', the analogue of Fitch's paradox is that from:
1: p --> FKp
together with the usual assumptions about knowledge, it appears we can derive 2: p --> Kp
We do this by noting that 1 implies
3: (p&¬Kp) --> FK(p&¬Kp)
but that the consequent of 3 is impossible, so that from 1 we can derive the negation of 3's antecedent, which is (classically) equivalent to 2.

Here, however, an analogue of Edgington's response to the vanilla Fitch paradox looks particularly easy to defend. Just as Edgington's anti-realist says that what's knowable are things of the form 'actually p', the defender of the future-knowledge principle should say that what's knowable are things of the form 'at t, p', where 't' is a name for whatever time it is now. This means that instead of 1 we have
1': p --> FK[p was true at t]
and instead of 3 we get
3': (p&¬Kp) --> FK[(p&¬Kp) was true at t]
the consequent of which is unproblematic.

Given, however, that Edgington's solution to vanilla Fitch is beset with objections, it's not clear how thinking about this temporal analogue can really throw any light on the original puzzle, as Burgess seems to hope it will.

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