Monday, November 13, 2006

Jottings on Fitch's Paradox I

None of this is to be taken as more than a few sketch thoughts.

Fitch's Paradox is a paradox that has occupied quite a few people in epistemology. Discovered by Fitch in 1963, the result is usually characterized along the lines of "If all truths are knowable, all truths are known." What makes it a paradox is that the knowability principle (KP) is usually considered to be very plausible:

(KP) p → ◊Kp If a claim is true, it is possible that it is known to be true.

But if KP is true, and Fitch's argument is right, then it follows that all truths are known. And it's difficult to determine what could be wrong with Fitch's argument. So here's the argument as portrayed in the SEP article linked to above, with a rough English translation to the right.

(1) p ∧ ¬Kp A particular claim is true, and it is not known to be true.
(2) (p ∧ ¬Kp) → ◊K(p ∧ ¬Kp) (1) implies that it is possible to know both that the particular claim is true and that it is not known to be true.
(3) ◊K(p ∧ ¬Kp) It is possible to know that both the particular claim is true and that it is not known to be true.

(1) is just a particular instance of the claim that there is at least one truth that is not known. (2) follows by KP from (1). (3) follows from (1) and (2). Now suppose that it is not only possible to know (p ∧ ¬Kp) but that this is actually known:

(4) K(p ∧ ¬Kp) It is known that both the particular claim is true and that it is not known to be true.
(5) Kp ∧ K¬Kp The particular claim is known to be true and it is known that it is not known to be true.
(6) Kp ∧ ¬Kp The particular claim is known to be true and it is not known to be true.

(4) is our supposition; (5) follws from (4) by the standard epistemic principle that a conjuction is known only if its conjucts are known; and (6) follows from the standard epistemic principle that if p is known, p is true. We have a contradiction here, of course. So (4)) is false.

(7) ¬K(p ∧ ¬Kp) It is not known both that the particular claim is true and that it is not known to be true.
(8) ¬K(p ∧ ¬Kp) (7) is necessary.
(9) ¬◊K(p ∧ ¬Kp) It is not possible that this is known: both that p is true and that it is not known to be true.
(10) ¬∃p(p ∧ ¬Kp) There is no particular claim for which it is true both that the claim is true and that it is not known to be true.
(11) ∀p(p → Kp) For any particular claim, if it is true, it is known to be true.

We know that (7) is true because we showed that (4) led to a contradiction. Because (4) led to a contradiction, however, (7) must be necessarily true, which gives us (8). (8) tells us that it is necessary that something is not true; and whenever it is necessary that something is not true, it is not possible for it to be true. So, given (8), we have (9). (9) contradicts (3). From this it follows that there is not some particular claim for which (p ∧ ¬Kp) is true. That's what (10) says; and from (10) we can get (11) directly. So if something is true, it is known to be true.

The general consensus is that this is very odd. I am, as I said, going to have a few sketchy thoughts on this; but I'll save those for another post. Right now, I want to make two brief points.

The first is that it is not at all surprising that (4) is necessarily false. Suppose that p = "The sky is blue". (4) then claims that this is true:

Both of these are known: The sky is blue and it is not known that the sky is blue.

It is not in the least surprising that that this leads to a contradiction. But this suggests that the whole argument from (4) to (11) is right, because if (4) is necessarily false, (7) is necessarily true; and the rest of the argument seems to follow in good order.

The second is a matter of translation. I already said that the result is usually put in terms of knowability: KP is usually read, "If p is true, it is knowable." I translated differently, as you can see. This is because I think "knowable" is a very bad translation of the double operator, ◊K. To see this, think about what we really mean when we say the following two things:

This is knowable: The sky is blue.
It is possible that this is known: The sky is blue.


The two are not equivalent, and for good reason. The English word 'knowable' in all but a very small handful of uses hides a third operator, a temporal operator -- an incipit, to be exact:

This can come to be known (can begin to be known): The sky is blue.

So when I say that some claim is knowable, I usually don't mean that it is possible that it is known; I mean that it is possible that it could come to be known. So I think there's reason to stay away from the added complications that are introduced by the word 'knowable'. Using 'knowable' makes it sound even more paradoxical; but (1) it doesn't need to be made to sound more paradoxical; and (2) it is misleading.

Both of these points will be seen again. More on Fitch's Paradox in a future post.

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