🚀 go-pugleaf

This question is perhaps trivial but...

I am first going to give the definition I am using, since many people
use the same words for different things: 

Consistency: A system is consistent if, for any wff F, we
can not have at the same time (F is a theorem) and (not F is a
theorem). 

Categoricity: a system is categoric if, for any wff F, we have either
(F is theorem) or (not F is a theorem).

Strong completeness: a system based on a set S of axioms is strongly
complete if for any wff F, we have either (F is a theorem) or (F added
to S is an inconsistent system).


Now the questions:

1) It seems quite clear that in any system which is a superset of
propositional calculus Categoricity and Strong completeness are
equivalent. However a short and clear demonstration would be very much
welcome.

2) Does the property hold in the general case (I especially think of
intuitionist logic) ?

If the question is too trivial, a reference to a book giving the
answers would be also welcome.