TITLE: Is there a category whose internal logic is paraconsistent?
QUESTION [19 upvotes]: The internal language of topoi is higher-order typed intuitionistic logic. Now according to wikipedia, the dual of intuitionistic logic, in some sense is paraconsistent. They say

Intuitionistic logic allows $A ∨ ¬A$ not to be equivalent to true, while paraconsistent logic allows $A ∧ ¬A$ not to be equivalent to false. Thus it seems natural to regard paraconsistent logic as the "dual" of intuitionistic logic.

they go on to say:

A specific paraconsistent logic is dual-intuitionistic logic or paracomplete logic, this duality can be best seen in sequent calculus framework, where
Both $\vdash A \vee \neg A$  and $ \neg \neg A \vdash A$ are not derivable in intuitionistic logic, whereas
Both $ A ∧ \neg A \vdash$  and $   A \vdash \neg \neg A$ are not derivable in paraconsistent logic.

Given duality has a strong presence in Category theory, given that the internal language of toposes are intuitionistic, are there categories whose natural interpetation as a logic is dual-intuitionistic, or paraconsistent in some other way?

REPLY [2 votes]: If $a:A\rightarrow X$ and $b:B\rightarrow X$ are two monomorphisms in a category $\mathcal C$ such that $a\leq b$ in $\text{Sub}(X)$, shouldn't we have that $b^\text{op} \leq a^\text{op}$ in $\mathcal C^\text{op}$ ? Then if $\mathcal C$ is a topos, $\mathcal C^\text{op}$ should have quotient objects forming co-Heyting algebras.