Tuesday, December 4, 2007

Epistemology of logical equivalences

Chris had an objection to Williamson brought up last class. It goes something like this:
Williamson argues:
1) Every metaphysically modal statement is logically equivalent to a counterfactual statement
2) For any two statements, if one is logically equivalent to another then when one is in an epistemic position to know the first, one is necessarily in an epistemic position to know the second
3) (1&2) -> 4
4) If we're in an epistemic position to know counter-factual claims, we're in an epistemic position to know metaphysically modal claims

Chris denies (2). After some thought, I think this is obviously right. For instance, consider a tautology too complex for the human mind to grasp. Then take ~(P&~P). The latter is equivalent to the former, yet we're in no epistemic position to know the former. Alternately, consider someone who's convinced by McGee's argument that modus ponens is invalid. Suppose further this person has (given his evidence) every reason to believe McGee and no reason not to. Arguably, this person could know P&(P -> Q), and have excellent independant evidence for ~Q. Not only would he be in no epistemic position to know Q, he'd be in a very good epistemic position to justifiably doubt Q.

Williamson is a smart guy, and it's hard to believe that he simply missed this. So let's see what he says:
"Given that quivalences (17) and (18) are logically true, metaphysically modal thinking is logically equivalent to a special case of counterfactual thinking, and the epistemology of the former is tantamount to a special case of the epistemology of the latter. Whoever has what it takes to understand the counterfactual conditional and the elementary logical auxiliaries ~ and (contradiction) has what it takes to understand the possibility and necessity operators"
He goes on to consider something like Chris's objection a bit later:
"Indeed, we have no sufficient reason to regard any of the equivalences as strict synonymies. That detracts little from their philosophical significance, for failure of strict synonymy does not imply failure of logical equivalence. The main philosophical concerns about possibility and necessity apply equally to anything logically equivalent to possibility or necessity. A non-modal analogy: ~A is logically equivalent to A->(contradiction), but presumably they are not stictly synonymous; nevertheless, once we have established that a creature can handle -> and (contradiction), we have established that it can handle something logically equivalent to negation, which answers the most interesting questions about its ability to handle negation"

I'm not sure exactly what to make of this, but it seems like Williamson is talking about a specific sort of logical equivalence. Not logical equivalence between statements, rather equivalence between logical operations. A new argument may be set up as follows:

1) If one is in a cognitive position to fluently use logical operator P, and applying P to a formula(s) is logically equivalent to applying Q to a formula(s), then one is in a cognitive position to fluently use logical operator Q.
2) [] is logically equivalent to C, where CA is defined as (~A []-> contradiction).
3) We're in a cognitive position to fluently use C
4) (1&2&3) -> (5)
5) We're in a cognitive position to fluently use []

I see this as no better off than the original argument. For instance, C' could be defined as (~A[]-> (negation of incredibly complex tautology)), and an analogous argument would say we're in a cognitive position to use C'. However, we're not in a cognitive position to use C'.
One could respond that we ARE in a cognitive position to use C', we're just not aware of the fact. This seems weak.
Furthermore, even if this revised argument is sound, it misses the point. We want to know how we know metaphysically modal truths, not how we've come into a position to use them (or something logically equivalent to them). If I want to know how you found out that all renates are alive, I won't be satisfied with the answer "I found out that all cordates are alive". More is needed.
I found this all quite vexing. any thoughts anyone?

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