Logical inference (and proof) is simply the use of the transformations built into the structure of a language (its grammar, etc.) to simplify a set of utterances. There is nothing magical about it.
Logicians will tell you about a “rule of logic” called modus ponens, which simply says that if:
-
If A then B
-
A
Then you can safely conclude that B.
Of course, it is gratuitous* to call this a “rule of logic” when it is really just a rule of grammar, or just part of the definition or grammatical function of the word “if” that we are all familiar with. Loosely put, the word “if” in the construction “If X then Y” just means that once you learn that X is true you can safely conclude that Y is also true. This is, by convention, what an if-then structure does in English.
When you logically deduce, for example, that “If A implies B and B implies C, then A implies C,” you are merely processing English grammar and the meanings of the words “if” and “imply”. You could mistrust your ability to process English grammar or to correctly apply the transformations built into English, or you could doubt the premises, or you could find ambiguities in the terms used, but there seems to be no other way to coherently “mistrust” logic.
The confusion arises from the defining logic in a vague way, as “careful reasoning” or whatever. The essence of logical deduction is merely the cleaning up of sets of statements using the grammatical and semantic transformations built into a language.
Lastly, regarding the question of whether logic (grammar!) can be refuted by emperical evidence, the question answers itself. If a scientist says that “A is not A” or “A is never B, but A is sometimes B,” he is simply speaking incoherently. If we take this incoherence on authority as some kind of paradoxical Deep WisdomTM, rather than demanding clarification, that is our loss.
*This is not to say that English grammar is always unambiguous and propositional logic adds nothing; indeed propositional logic makes the grammar unambiguous. Still, there is no reason to call it propositional logic; it could just as well be called “unambiguous grammar” and logical deduction could just as well be called “grammatical transforms.”