Friday, August 05, 2005

Disappearing Diamonds and Disappearing Disjunctions

(I'm using '-->' to represent the material conditional in this post.)

Kvanvig argues (here) that it is surprising (in some deep way) when Fitch's paradox shows us that 'p --> possibly Kp' is equivalent to 'p --> Kp'. It's surprising, he says, because it looks like we're collapsing a distinction between its being possible that Kp and (what is strictly stronger) its actually being the case that Kp.

But if that's the sole source of the puzzlement, why aren't we similarly surprised that '¬A --> (AvB)' is equivalent to '¬A --> B'? Here it looks like we're collapsing a distinction between the truth of a disjunction and (what is strictly stronger) the truth of one of its disjuncts. (Or at least, it looks as much like we're doing that as it looks like we're collapsing a modal distinction in the original case).

I suspect that, if there is something deeply surprising about the Fitch proof that stands in need of explanation, there must be more to it than this.

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