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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?

EDIT

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.

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Another insightful reference for the semantics of dual-intuitionistic logics is Yaroslav Shramko's paper on the "logic of scientific research" (Studia Logica 2005).

Just to add a brief comment on @TMF's answer, for the benefit of the author of the original question, it is worth noting that Priest's "Da Costa Logic" was indeed proposed as a fragment of Rauszer's "Brouwer-Heyting logic", extending the conjunction-disjunction fragment of classical/intuitionistic logic by adding an intuitionistic implication and a co-intuitionistic negation, but not the duals of the latter two connectives.

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Most of the "dualized" intuitionistic logics in the literature, e.g., Priest-da Costa and anti-intuitionistic, are fragments of Cecylia Rauszer's Heyting-Brouwer logic, in which all connectives---not merely negation---are given duals. It's probably worth your time to review Rauszer's 1974 "Semi-Boolean Algebras and Their Application to Intuitionistic Logic with Dual Operations" and 1977 "Applications of Kripke Models to Heyting-Brouwer Logic."

The use of the prefix "co-" is a function of the use of the term "dual" and the latter is used in a variety of ways. Still, I suspect that whatever you intend by "co-Heyting algebra," a variety of pretenders to that title can be found implicitly in Rauszer.

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The paper H.P. Sankappanavar, "Heyting algebras with dual pseudocomplementation", published in Pacific journal of Mathematics 117 (1985), 405–415, I believe, provides an algebraic semantics for what Priest calls as "Da Costa Logic".

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