After my previous post on the paradoxes, Hud
Hudson reminded me that there are some much simpler paradoxes involving
validity rather than soundness. These don’t appeal to anything as controversial
as excluded middle, so they are a little more forceful.

Argument
A

A1.    Argument A is valid

AC.    Santa Claus is coming to town.

The argument that this leads to paradox is
as follows.

(1)      If
Argument A is valid, then if Argument A is valid, then Santa Claus is coming to
town.
          (By the definition of validity)

(2)      If
Argument A is valid, then Santa Claus is coming to town.
          (Contraction from (1))

(3)      Necessarily,
if Argument A is valid, then Santa Claus is coming to town.
          (By (2), since all the premises
we used to get there are necessary truths.)

(4)      Argument
A is valid
          (From (3), and the definition of
validity)

(5)      Santa
Claus is coming to town.
          (Modus Ponens on (2) and (4).)

Well, that was a little easy. Anyway, here’s
the serious problem. The obvious way to block the paradox here is to say that
(2) doesn’t really follow from (1). And Hartry
Field has been arguing just that for a while now, with I think some
success. But it seems to me that it’s much more plausible to say that p ® (p ® q)
doesn’t entail p ® q when the arrow denotes ordinary ‘if…then’ than when it denotes
entailment. When ®
means ‘entails’, contraction seems much more plausible. (I could be wrong about
the relative plausibility of the claims here – I think contraction is still
quite plausible for ordinary conditionals, but I think it is if anything stronger
for entailment conditionals.) Anyway, then the argument to paradox goes as
follows. (Using ® for
the entailment conditional.)

(6)      (Argument
A is valid) ® ((Argument A is valid) ® Santa Claus is coming to town))
          Assuming here that an analysandum entails its analysans

(7)      (Argument
A is valid) ®
Santa Claus is coming to town)
          Contraction on (6)

(8)      Argument
A is valid
          (7), connection between validity
and entailment

(9)      Santa
Claus is coming to town.
          Modus Ponens on (7), (8)

Of course, none of this is original to moi. I only mention it here because I had forgotten there were proof-theoretical paradoxes, and it is not entirely obvious that the solutions that work for other paradoxes also work for these.