Logical proofs depend on premises you choose to deduce an argument and the means of such premises (perception, inference, experience etc.). If the chosen premises already have biases then one cannot expect the conclusion to be a fair and un-biased deduction. As for choosing premises fairly, ordinary languages only help to certain extent because we constantly run into mis-interpretation. Thus, a formal set is used within logical structures (with its own rules, like propositions) that allow us to deduce any statement, event with some degree of confidence. However, when we use logic in our daily lives, we necessarily do not examine the truths of each and every premise, since those premises are born out of various means (like perception, inference) which we for granted take to be conclusive truths already (like, I saw such and such so it must be true!). So, to answer your question after all this mumbo jumbo, yes, there is enough room for logic, but just depends on how cleanly one uses it to reach a conclusion.