In article <351d0bd7-fe7d-4005-a10a-42b63a9037e5@q16g2000yqg.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes: > On 26 Mai, 04:07, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > In article <15af9b47-eac6-4d22-8720-0d6e7ebba...@e21g2000yqb.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes: > > > The problem boils down to the following: > > > > > > En Am: m =< n <==> Am En m =< n [*] > > > En Am: m =< n ==> Am En m =< n [**] > > > > > > You know: Classical logic was obtained from finite sets ... > > > Show me a finite set that obeys [**] but not [*]. > > > > The above is not contested: [**] implies > > [*]. > > I said: For complete linear sets [*] is true.
Not in the article to which I responded.
> You said [*] is not true, but [**] is true.
That is not what I said. I said that for the case involved you have to *prove* that it is true, because it is not generally true.
> Weyl said: Classical logic was obtained from finite sets.
Right.
> Therefore I asked you: Show me a linear complete finite set, that > makes your claim [**] right and my claim [*] wrong.
But now you include the word "linear". Where did "Weyl" include the word "linear"?
> This is so simple that it should be understandable even for someone > who is not "very deep in logic". > > > What is contested is that: > > En Am: m =< n <== Am En m =< n [***] > > implies [*]. And *that* is the form you do use. > > No. I do not use the implication only, I use the full equivalence. Of > course the equivalence includes the implication [***] as well as the > implication [**]
Because the implication [**] is always true, the only part of the equivalence that is new is the implication [***].
> > There is a trivial finite > > counter-example. Take three dice where on each of the sides one of the > > numbers one to nine is printed (some of them repeated). Say the set is > > {d1, d2, d3}. Define di < dj when the probability to throw a higher > > number with dj than with di is larger than to 1/2. (I would submit that > > all this is quite physical.) There is a set of three dice such that > > d1 < d2, d2 < d3 and d3 < d1. > > > > And so we have: > > Am En d_m < d_n (d1 < d2 < d3 < d1) > > but not > > En Am d_m < d_n (there is no best die). > > Of course there is no best die. Therefore this set is not linear. > Every finite set of natural numbers has a "best" number. > > Why do you bother with such nonsense examples? > But the answer is easy: Because you have no other examples.
Because they are answers to what you actually ask. It proves, in general, that [***] does *not* imply [*], even not in finite cases. So in order to use such implication you have to *prove* it.
Now you state that classical logic is derived from the finite case. But in the finite case that implication is not generally available, so it is not part of classical logic, I would say. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
