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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?...

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    Yes; see e.g. Intuitionism and Constructive Mathematics, as well as Inconsistent Mathematics. Jan 22, 2018 at 12:31
  • In addition to Mauro's answer, by "a different arithmetic" do you mean something like wheel theory?
    – Alexis
    Jan 22, 2018 at 20:08
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    Not only were they imagined but even worked out in detail for centuries. See What are the differences between philosophies presupposing one Logic versus many logics? and set-theoretic pluralism.
    – Conifold
    Jan 22, 2018 at 21:54
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    Yes. In some sense, mathematics is just a set of strings. Imagine a computer writing out every possible string, and a second computer determining which strings are "correct". Depending on how you define "correct", you can have any mathematics you want.
    – user935
    Jan 23, 2018 at 16:27
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    A lot hinges on what "completely different" means. As the above comments, yes. But arguably there are features of mind that narrow the scope of what can be imagined and eliminate incomprehensible possibilities (incomprehensible because our minds cannot follow them at all).
    – virmaior
    Jan 31, 2018 at 14:22

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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

http://costaleite.info/logics-and-their-galaxies.pdf

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  • hey, while this looks like a good and neat answer, whoever asked the question, like me, will probably struggle to understand it due to the how it's notated. why not write it out in natural language??
    – user38026
    Apr 12, 2019 at 1:49

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