Dick,
A few thoughts:
***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.***
I agree.
***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.***
My understanding is that axioms don't have to be anything else but able to create consistent or para-consistent rules that acceptable from the standpoint of the formal system. In mathematics, the criteria for an axiom seems to be a little more restricted since some axioms seem acceptable by some while rejected by others (e.g., axiom of choice, or AC). The AC is often rejected by some mathematicians because of the implications. The implications are curiosities that some mathematicians have no problem with, while others squirm and some even shout.
The issue cited by Alan, on the other hand, seems to be that LNC (or "LNC" - ha) cannot be done without in order for any formal system to be workable (i.e., it is needfully required). What some logicians are saying is that doing without LNC opens up more avenues than it closes. Therefore, it is not needfully required by any stretch. Even logicians who oppose a logical system that denies LNC will admit that there is no obvious needful requirement.
***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".***
I agree in principle. In terms of modelling the real world we can't cut ourselves off from classical logic. On the other hand, if we are modelling reality we have to be careful that we don't enforce our notion of logic onto reality. As I said before, our notion of logic may only be an approximation.
Warm regards, Harv |