Can you imagine a different logic and mathematics? For example, with a different arithmetic, or even a universe with no logic or mathematics and contradictions? A non consistent system?...
In order to consider the infinite number of logics and to make evident how different logics yield different ontological commitments, we make use of the notions of antilogic and counterlogic defined for a suitable logic. Here we will not consider any logic in particular, but rather logics in an abstract sense.
A logic L is a structure (F, ⊢) such that F is a set and ⊢ is a binary relation on ℘(F) × F without any restrictions. We use Γ ⊢ ϕ to indicate that (Γ, ϕ) ∈ ⊢, and we say that ϕ is a consequence of Γ in L. [...]
The antilogic L' of a given logic L = (F, ⊢) is a pair (F, ⊢') such that
Γ ⊢' ϕ if and only if it is not the case that Γ ⊢ ϕ