Say your preferred account of the a
priori is that S is a priori iff it knowable just on the basis of
ones understanding of S, as Stephen Yablo suggests in his paper in the
Gendler and Hawthorne volume. And say that you also think, not unreasonably,
that understanding the logical connectives just means finding their elimination
and introduction rules primitively compelling. And say that you also think (as
Im not sure that I do) that the relevant rules are those for a Gentzen style
single conclusion natural deduction system. So the introduction and elimination
rules for ® are
given by the equivalence of the following two sequents.
(1) S,
A: B
(2) S:
A ® B
I use S as a variable over pluralities
of propositions, A and B as variables over propositions, and :
for the consequence relation. Nothing above is utterly unreasonable, and indeed
everything is close to majority opinion amongst the relevant theorists, though this
being philosophy nothing in uncontroversial. Now it is well known that these
rules do not let us prove (3)
(3) ((p ® q) ® p) ® p
(3) can be proven using the standard rules
for the connectives, but you have to use the rules for negation, not just the
rules for ®.
If you accept all the assumptions in the
first paragraph, then you have a dilemma. Either (3) is not a logical truth,
despite being a truth-functional tautology, or some logical truths are not a
priori. Dummett accepts all of the initial assumptions (I think) and
concludes that (3) is not a logical truth. Yablo I think is committed to (3)
not being a priori, despite being a logical truth. This isnt an unintelligible
position, but it might be grounds for rejecting his account of what it is to be
a priori.