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In article <8712310840.AA03892@garnet.berkeley.edu> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes:
>And this reminds me of the infinite regression inside Lewis Carroll's
>Achilles and the tortoise story (-> binds tighter than &):
>
> Start              (q)  <--
>   |             [   p , p->q ]  <--
>   |          {      p , p->q , (p&p->q)->q }  <--
>   V       <         p , p->q , (p&p->q)->q , p&p->q&(p&p->q)->q >  <--
                                               ^^^^^^^^^^^^^^^^^^
(That should be: ((p&p->q)&((p&p->q)->q))->q.)
>  ... ( ...                                                           ... ) ...
>And so, by your reasoning, we should reject modus ponens.

No, we just note that this isn't a proof of it.

It seems to me that the fallacy in Carroll's story lies in confusing the
statement of modus ponens from modus ponens itself.  "Whenever p and p->q,
then we can conclude q" is a statement of modus ponens.  But when one
applies modus ponens, one does not refer to that statement at all.  One
merely notes that one has p and p->q, and concludes q.

"(p&p->q)->q" isn't modus ponens at all, although it has more than a passing
resemblence to it.
-- 

Frank Adams                           ihnp4!philabs!pwa-b!mmintl!franka
Ashton-Tate          52 Oakland Ave North         E. Hartford, CT 06108