Paraconsistent logic which modifies classical logic by rejecting the principle of explosion or ex contradictione sequitur quodlibet can be seen purely formally. The question remains as to how one interprets the logic - that is what is its semantics.
One broad class of paraconsistent logics are dual-intuitionistic logics. Now the standard semantics of intuitionisitic logic uses Kripke frames. Does one dualise here to get semantics for dual-intuitionistic logics?
Priest, in his paper Dualising intuitionistic negation shows that dualising the truth condition for negation in the Kripke frame gives a paraconsistent logic which he calls a De Costa logic (it is stronger than Da Costa own logic - C-Omega). So, the semantics of De Costa logic has a Kripke possible world semantics but with negation dualised.
This implies that there is more than one way to dualise intuitionistic logic. I'm primarily interested in viewing intuitionistic logic as a heyting algebra in category theory, and standardly dualising that to get co-heyting algebras.