“Wrong”, not “false”. Logic can only establish whether an argument is valid, not whether it is sound. That is to say, it can only determine that, if an argument’s premisses are true, the conclusion must also be true; it cannot establish whether the premisses are true.
And even this holds only insofar as reality is logical–i.e., insofar as it corresponds to human reason.
You’re confusing two things here (perhaps not in your mind, but certainly in your words). A bacterium need not behave according to logic; it may just seem to behave that way, because we cannot think otherwise than “according to logic”–i.e., to human logic.
This is the profoundest insight with regard to logic. The rest is–no more than logical. To be sure, this insight requires the good use of logic. It is logical thought apprehending its own limits.
“We cease to think when we refuse to do so under the constraint of language [note: not necessarily of words]; we barely reach the doubt that sees this limitation as a limitation.
Rational thought is interpretation according to a scheme that we cannot throw off.” (Nietzsche, The Will to Power, section 522, Kaufmann translation.)
“We are unable to affirm and to deny one and the same thing: this is a subjective empirical law, not the expression of any ‘necessity’ but only of an inability.
[…] Either [the law of contradiction] asserts something about actuality, about being, as if one already knew this from another source; that is, as if opposite attributes could not be ascribed to it. Or the proposition means: opposite attributes should not be ascribed to it. In that case, logic would be an imperative, not to know the true, but to posit and arrange a world that shall be called true by us.” (op.cit., section 516.)
“The earthly kingdom of desires out of which logic grew: the herd instinct in the background. The assumption of similar [“identical”] cases presupposes ‘similar souls.’ For the purpose of mutual agreement and dominion.” (op.cit., section 509.)
Thus ethics (herd morality) comes before logic. See http://www.ilovephilosophy.com/viewtopic.php?f=1&t=187672.