Is there a category whose internal logic is paraconsistent?

Is there a category whose internal logic is paraconsistent?

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 \vee \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?