We have to assume that logic is true because the preponderance of the evidence supports that conclusion, but we can only attempt to prove logic is true by using logic.
Generally I read "assume" as "assume without evidence" or "baseless assumption". Perhaps we mean the same thing then.
Also, I wouldn't use "prove". For me, what we are demonstrating is the utility of the tool. Experience shows that we are "doing it right" or not.
The question "is logic true" is almost meaningless. The question, I think, is does our tool gives us useful results.
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