Since we seem to be pretty much in a "he said/she said" mode of posting, I'm gonna cut right to the chase to comment specifically on the issue of substance below. If you consider further comment is necessary on omitted material you're welcome to restore it. - LZ
On 31 Jul 2006 01:53:08 -0700, "Rupert" <email@example.com> wrote:
[. . .]
>> >> > You can ask for some sort of >> >> >justification. I haven't provided one. But we have given you reference >> >> >where you can find a discussion of the issue. >> >> >> >> I see. I can ask for "some sort of justification". I just can't ask >> >> for any true justification. >> > >> >What would count as "true justification" for you? >> >> Aha! You're just now beginning to raise pertinent issues. I imagine it >> would run along the same lines to which mathematicians hold the >> demonstration of theorems. >> > >Mathematicians demonstrate theorems using the axiomatic method. That >means some things are accepted without demonstration. It's the only >way. You will have to accept *some* things without demonstration.
Certainly not on your say so. You can't possibly know what can't be done when you can't even demonstrate what has been done.
Normally theorems are demonstrated true with respect to axioms by showing that alternatives must be false with respect to those axioms. Now the problem with demonstrating the truth of axioms themselves is that we have nothing to regress them to and thus no way to show that alternatives are necessarily false with respect to other axioms. Thus we are left to consider how to show that axiomatic alternatives must be false and an axiom itself true.
This doesn't necessarily mean the problem can't be solved but it does mean obviously we only have the axiomatic assumption itself to work with. So let's say we have axiomatic assumption A and we ask how to demonstrate alternatives to A must be false.
To do this we address alternatives to A in the form of "not A" and ask how it is possible to establish that "not A" is necessarily false? We don't exactly know what A is at this point only that we need to establish that "not A" must be false and can never be true.
Now there is one and only one way I can think of to establish this and that is to make "not A" self contradictory and in so doing establish A must necessarily be true to the extent that "A, not A" exhausts all possibilities for truth.
Thus if "not A" is to be considered necessarily self contradictory we must find some A which contradicts "not". In other words we cannot make A "apples" for example because "not apples" does not contradict itself.
Therefore we find that only by making our initial axiomatic assumption A "not" do we find alternatives to A in the form of "not not" are self contradictory and necessarily so such that A itself must perforce be universally true and cannot be otherwise.
In effect all we're saying is that "contradiction" must be universally true of everything because the "contradiction of contradiction" must necessarily be self contradictory and hence false. Or we can argue that "differences" must necessarily be universally true of everything because "different from differences" is itself self contradictory.
There are different ways to express the universal truth of axiomatic assumptions but there is only one way to demonstrate the universal truth of whatever axiomatic assumptions we make. And the truth of other axiomatic assumptions depends on their demonstrability in terms of that one universally true axiomatic assumption.