If we reject logic we reject all meaning. So we cannot prove logic without assuming logic - indeed the whole concept of proof is meaningless if we do not. Indeed, as Chiroptera says proofs in the fullest sense are logical proofs, that depend on axioms and rules which cannot themselves be proven - including the laws of logic.
However I would add that in my view it is possible to justify the laws of logic. To explain how and why they work. They are not so much facts taken on faith as linguistic and intellectual tools of proven value. Thus there is justification, but not proof.
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