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Is there more than one form of logic in mathematics? I would be inclined that mathematics only cover one type of formal logic, but I would be interested to know if there are variants thereof or completely different logics than what we're taught in university. In philosophy, there are several logics, but in mathematics it seems there's only one, or at least only one that's prevalent.

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    Hard to say what you mean by "form of logic", but there are parts of mathematics that are based on non-classical logics, constructive mathematics and paraconsistent mathematics are the most prominent.
    – Conifold
    Dec 15 '20 at 5:22
  • There is the syntactic and semantic.
    – Elliot
    Dec 15 '20 at 5:45
  • Mathematics is not limited to any kind of logical system since it doesn't describe directly any phenomena. Scientist uses mathematical tools to interpret and describe the phenomena they're studying. In fact, with math, you can conceive any kind of logic! Richard Feynman described this nicely in his lectures on the character of physical law. Dec 15 '20 at 6:47
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Yes, on a fundamental level, much simpler than fleshed out arithmetic, we have classical, non-classical, dialetheistic logic, just to name a few.

You can choose either one of those systems and we can build complex mathematical structures, like fields, etc, using it - though it would be tedious, for example, if you don't have proof by contradiction at your disposal. So, yes, we most definitely have more than one form of logic in mathematics.

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