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In article <45476ebd@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
> cbrown@xxxxxxxxxxxxxxxxx wrote:
> > Tony Orlow wrote:
> >> cbrown@xxxxxxxxxxxxxxxxx wrote:
> >>> Tony Orlow wrote:
> >>>
> >>> <snip>
> >>>
> >>>> t=-1/n ^ t=0 -> -1/n=0. T v F?
> >>> T.
> >>>
> >>> However, in a mathematical argument it is equally true that (assuming t
> >>> is a real and n a natural):
> >>>
> >>> t=-1/n ^ t=0 -> -1/n > 7
> >>>
> >>> Cheers - Chas
> >>>
> >> That's debatable. Intuitionistic logicians reject that a false premise
> >> implies anything.
> >
> > That is not my understanding; what they instead reject from the usual
> > classical logic is only the law of the excluded middle i.e., that we
> > cannot assume that (A v ~A) is true. It does /not/ say that we cannot
> > assume that (A ^ ~A) is false.
> >
> > See, for example:
> >
> > http://en.wikipedia.org/wiki/Intuitionistic_logic
> >
> > The rule NOT-2 states:
> >
> > A -> (~A->B)
> >
> > which is also true in classical logic and can be described as saying
> > "anything follows from a falsehood".
> >
> > For example, we can use the above to state your example: if (the moon
> > is not a balloon), then it follows that "if (the moon is a balloon)
> > then (Santa Claus is gay)" is a true statement.
> >
> > Since we all agree that the moon is not a balloon, therefore
> >
> > "If the moon is a balloon, then Santa Claus is gay".
> >
> > is a /logically true/ statement.
> >
> > I agree that intuitionists as well as classical logicians both say that
> > we cannot /deduce/ anything from a logiclly true implication with a
> > logically false premise; i.e., that when A is false, we cannot /deduce/
> > from (A->B) that "therefore B", nor can we /deduce/ that "therefore
> > ~B".
> >
> > That's why we refer to statements of the form "(something false) ->
> > (anything)" as being "vacuously true": they're technically true
> > statements; but who cares? We can't deduce anything at all from them.
> >
> >> If the moon's a balloon, Santa Claus is gay. Is that a
> >> valid logical statement? You say it's true. I say it's nonsense.
> >
> > I say it's a /logically/ true statement; and I say its /premise/ is
> > logically false. That's why it's useless to us in an argument: nothing
> > can be /deduced/ from it; i.e., it is nonsense to claim that you have
> > deduced something from it.
>
> Then why bring it up?
>
> >
> >> You
> >> claim my premise cannot be true because it's nonsense?
> >
> > No, I claim your premise cannot be true because it is demonstrably
> > false; and even intuitionists maintain that ~(~A and A).
> >
>
> Not if n is infinite. Prove that n cannot be infinite, if not so restricted.
>
> >> It's not nonsense
> >> if "t=0" means that 0-t is infinitesimal...
> >
> > This is another example of where it seems that you disagree with
> > premise (1): when we speak of a time t, we mean a real number t.
>
> Infinitesimals lie on the real line.
As TO uses them, they certainly lie, every time.
TO conflates different number systems without warning.
> Of course. -1/n is only an infinitesimal for infinite n, not for any
> finite natural n.
And outside of TOdreamland, no such non-finite natural n can exist.
> Where did I say n was a standard natural? Clearly 1/n>0 for any standard
> natural.
Keep your nightmares out of our number systems, TO.
> No, you've not shown the premise false except by assuming something
> incompatible with it.
The premise that there are infinite naturals in ZF or NBG is delusional.
The premise that TO's number systems are actually systems is delusional.
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