Thought I might jump in here with some minor comments. First, I am not by any means a master of "Logic" and I do not claim to be. Nevertheless, it appears to me that "Logic" is a formal field with formal rules and a specific "playing field". The whole issue is directed towards clear communications in order to make the existence of erroneous thoughts difficult to overlook.
Now, from that perspective, an "axiom" is something taken to be true in that no arguments against the statement are allowed: i.e., a foundation from which the arguments under examination may proceed. Whether or not the "axiom" can be false is an issue outside the chain of logic under examination.
However, I believe it is a bit presumptuous to hold that there exist no logical arguments as to the "need" of an axiom. One of the issues forgotten by many who use the word "need" is that the concept "need" is intimately connected to the concept of "purpose". If one says something is "needed" the immediate question is "why" which clearly asks for a purpose.
Now, when it comes to a mental model of anything, the purpose of any mental model is to answer questions. If different (equally logical) paths of deduction consistent with some model yield different answers to the same question, then the model fails its purpose (it fails to provide an answer). This phenomena is called inconsistency. Thus it is entirely logical that one should hold that any mental model of anything needs to obey the "Law of Non-Contradiction".
What I understand, what I believe Dick is talking about, what I think Alan is starting to grasp, if he hasn't already, is why the axiom cannot possibly be false.
I never said the axiom cannot possibly be false. Essentially, what I said was that anyone interested in case where it was false certainly wasn't interested in valid explanations of anything. If everything which exists is in fact no more than an explanation of reality then one could certainly hold the logical position that the "Law of Non-Contradiction" could not possibly be false! But, each to their own; far be it from me to tell people how to think.
No matter what you think, you have to begin somewhere.
Have fun-- Dick