In article <FImdnfw3_ePXa9HMnZ2dnUVZ_q-dnZ2d@giganews.com>, fom <fomJUNK@nyms.net> wrote:
> On 3/22/2013 6:40 PM, Virgil wrote: > > In article <1-KdnWUae4shRdHMnZ2dnUVZ_s-dnZ2d@giganews.com>, > > fom <fomJUNK@nyms.net> wrote: > > > >> On 3/22/2013 4:49 PM, WM wrote: > >>> > >>> Fools stay together. > >>> > >> > >> As observed before: > >> > >> Ex(phi(x)) -> Ax(phi(x)) > >> > >> is true in Wolkenmuekenheim > > > > And, far too often, so is > > > > Ax(phi(x)) -> Ex(phi(x)). > > > > > Is that one not always true? > > AxP(x) -> P(t)
Couldn't Ax(phi(x)) be true when Ex(phi(x)) isn't? E.G., All four sided triangles have less than three sides. and There is a four sided triangle having less than three sides.
> P(t) -> ExP(x) > > are both axiomatic. > > That is not your background, however. > > One of my objections involving the > failure to distinguish foundational > investigation from other types is > that the body of mathematical statements > used for practical application are > not obtained with free variables in > the premises and do not make assertions > having free variables in the conclusions. > > The "actuality" of any mathematical > object in set theory as an instantiated > object only occurs within the proper > interior of a proof since the language > has no individual constants. > > Hence, my unhealthy fascination concerning > the role of description theory. --