Nam Nguyen wrote: > > On 26/02/2013 1:16 AM, Virgil wrote: > > In article <m%XWs.20125$mC2.392@newsfe29.iad>, > > Nam Nguyen <namducnguyen@shaw.ca> wrote: > > > >> On 25/02/2013 10:25 PM, Virgil wrote: > >>> In article <SDWWs.99982$Hq1.27823@newsfe23.iad>, > > > >>> Since I said "not always", any such situation shows I am right. > >> > >> I think you misunderstood my point: > >> > >> In the context of language structure truth verification, > >> your original statement would _always_ fail: because for > >> Ex[P(x)] to be true, P(x0) must be true for some _example_ x0. > > > > To know that something must be true for some x0, it need not be known > > for which x0 it is true, only that it is true for SOME x0. Which was my > > original point! > > Then, can you construct a _language structure_ that would illustrate > your point?
Here is an example of a non-constructive existence proof.
Thm: There are solutions of x^y = z with x and y irrational and z rational.
Prf: sqrt 2 is irrational and (sqrt 2)^{sqrt 2} is either rational or irrational. Put x = sqrt 2, y = sqrt 2 so that x = (sqrt 2)^{sqrt 2}, which by hypothesis, is rational. If, on the other hand, (sqrt 2)^{sqrt 2} is irrational, put x = (sqrt 2)^{sqrt 2} and y = sqrt 2, so that z = ((sqrt 2)^{sqrt 2})^{sqrt 2} = (sqrt 2)^2 = 2, which is certainly rational. Thus in ether case a solution exists.
That is classically valid, but intuitionistically not.
Dummett once attributed the example to Fred C Benenson, my philosophy tutor at the time, who now seems to be a realtor[2] (is that the word?). Now [1], he attributes it to Peter Rogosinski and Roger Hindley.
[1] Dummett, Elements of Intuitionism, second edition, OUP, 2000. The misattribution was in the first, 1977, edition. [2] http://www.benensoncapital.com/
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
