Constructible falsity $\sim A$,
 also called strong negation,
is an alternative to Heyting's negation
$\lnot A$ $(\leftrightarrow (A\to\bot))$ in
intuitionistic logics.
In this paper we give the proofs for
Kripke completeness of the basic logic $N_\lnot$
and its five variations.
Among them the most novel results are
about the logics
with what we would like to call omniscience axiom,
$\lnot\lnot(A\lor\sim A)$.
We present two different proofs
based on tree-sequents:
one is by an embedding of classical logic, and the
other is by an extended version of tree-sequent.