Whether you ultimately agree with it or not, I feel that a study of modal logic, including David Lewis's (and others') notion of possible words is a valuable facet of a proper understanding of logic and metaphysics. However when I attempt to explain necessary truths (where one cannot imagine a world in which they would be false) I am often met with some form of the following criticism:
I can imagine universes in which the axioms of logic and mathematics don't hold, and who are you to tell me what I can and cannot imagine?
Of course, the premise of a necessary truth is that this isn't actually possible, and therefore the person making this statement is mistaken (assuming the example is a necessary truth). With enough time I am usually able to walk the person I'm talking to through why this is the case, but there is serious resistance on the part of many people being introduced to this concept. I understand where they are coming from, but I don't have a strong rebuttal to it other than "Oh really?" which is, of course, not satisfactory.
I most often see this criticism from people early on in their exposure to Philosophy as a formal discipline, so please target your responses for someone with limited knowledge of the discipline.