[bitdiddle]

I think the general program of categorical logic, the work of Lambek and Scott, and J. Bell on topos theory and local set theory really make clear the relationship between category theory and logic, as well as lambda calculus.

A topos is essentially a cartesian closed category with a subject classifier. In Set this is the two element set of 1/0 which is a Boolean algebra and thus the internal logic of the category Set is classical.

In general though the subobject classifier is a heyting algebra which expresses the semantics of intuitionistic logic.

There is also a very good, but introductory, book by Goldblatt on Topoi that covers this logical aspect

So in terms of logics the category of Sets is the exception.

By internal logic I mean that for every topos one builds up a theory using it's objects and function between them. An equivalence theorem (see J. Bell) states that a given topos is essentially equal to the category generated by this internal theory.

This program began with Lawvere who noticed that conjunction and implication were really adjoints, the same one as between the product and hom functors in a cartesian closed category.