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Logic is supposed to be an objective foundation of all knowledge. But if that's the case then why are there multiple systems of logic? For example there's 'dialetheism', which allows for true contradictions, and 'fuzzy logic' in which the law of excluded middle does not apply. If people can just re-write the rules to create their own system of logic, then doesn't that make logic subjective and arbitrary? It doesn't seem like arguments would have much weight if I could simply just choose whichever system best supports the conclusion I want.

Logic is supposed to be an objective foundation of all knowledge. But if that's the case then why are there multiple systems of logic? For example there's 'dialetheism', which allows for true contradictions, and 'fuzzy logic' in which the law of excluded middle does not apply. If people can just re-write the rules to create their own system of logic, then doesn't that make logic subjective and arbitrary? It doesn't seem like arguments would have much weight if I could simply just choose whichever system best supports the conclusion I want.

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You've asked a very good question, and your final sentence makes a good point. Those who defend one or another non-classical system of logic (paraconsistent, dialetheistic, intuitionistic, fuzzy, quantum, etc.) insist that they're not simply choosing a system of logic on a whim or merely out of convenience. Instead, they say, we're forced to accept non-classical logic because (a) it's an objective fact that arbitrary contradictions don't imply every proposition; because (b) some propositions are objectively both true and false; because (c) some propositions are objectively neither true nor false; because (d) some tautologies aren't completely true and some contradictions aren't completely false; because (e) the data gleaned from reliable experiments don't obey the classical laws of distribution, etc.

Having looked into them, I find none of their arguments for (a)-(e) persuasive. But what's most interesting, as various philosophers have observed, is that the defenders of non-classical logic sooner or later all rely on classical logic in arguing for their preferred systems. Their reliance on classical logic to argue for non-classical logic suggests to me that classical logic is the logic they trust when the chips are down.