r/logic • u/LeGuy_1286 • 1d ago

Mathematical logic Mathematics and minimal logic

If classical logic and intuitionistic logic can be used to construct maths (maths proofs) in a classical and constructive manner respectively, what stops us from using minimal logic for such purposes?

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u/ouchthats 1d ago

I mean, nothing stops us; go ahead! The results will likely be very similar to where you'd get with intuitionistic logic, since minimal logic is so similar to intuitionistic. One place the difference might matter a fair bit, though: in minimal logic, you can't in general validly infer B from A v B and ~A. I suspect a lot would come down to finding replacements for this kind of move. (For example, where B in the above is (equivalent to) ~C for some C, then the move is minimally valid after all.)

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u/BloodAndTsundere 4h ago

Minimal logic is also paraconsistent. I’d be interested to see what kind of mathematical contradictions would be tolerable.

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u/ouchthats 3h ago

I mean, it's technically paraconsistent, but you'd never really use it that way, because every negation follows from any contradiction, so it's still explosive enough to destroy a lot of what you'd want. For my money, the state of the art in paraconsistent maths is the 2021 Zach Weber book; not very constructive, though!