  In this paper we split every basic propositional connective into two
versions, one is called extensional and the other one intensional. The
extensional connectives are semantically characterized by standard truth
conditions that are relative to possible worlds. Assertibility and deniability
of sentences built out of atomic sentences by extensional connectives are
defined in terms of the notion of truth. The intensional connectives are
characterized directly by assertibility and deniability conditions without the
notion of truth. We pay special attention to the deniability condition for
intensional implication. We characterize the logic of this mixed language by a
system of natural deduction that sheds some light on the inferential behaviour
of these two kinds of connectives and on the way they can interact.

Logic in Computer Science