YOu raise a good point. The basic concepts of logic are built into language - although they are usually not rigorously applied. (By the basic concepts I mean negation, conjunction and disjunction - as well as the concept of "truth").
On the other hand logic can be reagarded as a formalisation of these basic principles of language.
IMHO the latter view is more accurate. The basic logical principles we find in language are there because they are useful - they give the language a good deal of power that it would not otherwise have. Logic, then, is derived from formalising and rigourously applying the principles adopted for more pragmatic reasons.
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