# SV: SV: On logic and fuzziness

Narayan S. Raja raja at IFA.HAWAII.EDU
Mon Nov 9 21:31:36 UTC 1998

```On Mon, 9 Nov 1998, Narayan S. Raja wrote:

> On Mon, 9 Nov 1998, H.M.Hubey wrote:
>
> > Definition of distance is precise and well known.
> >
> >         d(x,z) <= d(x,y) + d(y,z)
> >
> > It means simply that if I take a detour (to go to z from x) via y then
> > the
> > sum of the two distances which comprise the detour (i.e. d(x,y)+d(y,z))
> > must be greater than or equal to the direct distance d(x,z). Any kind
> > of a function that satisfies this can be called a distance metric.
>
>
>              Y
>             /|
>            / |
>           /  |
>          /   |
>         /    |
>        X-----Z
>
> Here is a simple case where d(x,z) is not <= d(x,y) + d(y,z)

Oops, forgot to mention that x,z are
on the surface of a sphere and y is the
center of the sphere.  (The point is that
adding an extra dimension, i.e., allowing
tunneling thru the sphere, invalidates the
previous definition of "shortest distance
between two points".

> A precise definition of distance presupposes
> a precise definition of the "space" (to put
> it simplistically, at least how many dimensions
> are involved)!  Is there a precise and uncontroversial
> definition of "linguistic space", or even of
> "evolutionary phonological space"?  I doubt it.

Raja.

```