As all logical arguments must make the assumption that the rules of logic work, is there any way to derive the laws of logic?

As you suggest, all logical arguments (and hence all derivations) depend at least implicitly on laws of logic. So I can't see any way of deriving any law of logic without relying on other laws of logic. Nevertheless, we can derive every law of logic, provided we're allowed to use other laws of logic in our derivation. We needn't fret about our inability to derive a law of logic while relying on no laws of logic, because the demand that we do so is simply incoherent.

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