[boris_m]

The bool category is not involved in this. True and False are the initial and terminal object (related to limit and colimit, but not the same thing) of all logical categories as I call them, of which there are many (one for each set of axioms that you might construct).

1. Ordering is not required for a category to be a category, the necessary requirements are just the ones listed in the beginning of the book. It is just that orders can be seen as categories.

2. You can express "A or not A" in intuitionistic logic it is just that it is not necessarily true. Also, not sure how would you express that or any logical relation in set theory.