Duality between metric and logical viewpoints

[anhinga_anhinga]

The metric viewpoint: how far two objects are from each other. The logical viewpoint: to what degree two objects overlap.

Fuzzy mathematics is traditionally done from the logical viewpoint, so the first step in introducing fuzzy metrics is often the transformation f(x,y) = exp(-d(x,y)).

Then we have the following correspondences:

d(x,y) = 0 if and only if f(x,y) = 1.

d(x,y) is plus infinity if and only if f(x,y) = 0.

d(x1, y1) < d(x2, y2) if and only if f(x1, y1) > f(x2, y2).

The axiom d(x,x) = 0 becomes f(x,x) = 1.

The axiom d(x,z) < d(x,y) + d(y,z) becomes f(x,y) * f(y,z) < f(x,z).

Non-expansive maps become maps which respect overlap by not letting it decrease.

Etc..


However, this seems to be a rather superficial duality: basically two equivalent ways to write the same things using different notation.

The question is whether there is also a natural deeper duality here (of a contravariant nature, where function arrows would reverse direction when one switches between these two viewpoints).

Comments

[anhinga-anhinga.livejournal.com]

Sure, but there must be plenty of dualities within that category (just like we have dualities within the category of vector spaces), including the ones which are unknown, or considered unremarkable.

(One should note the number of various interesting categories of metric and metric-like spaces is quite high, because there are many attractive variants for the notion of morphism. So we really are talking about a family of equivalences, and in some cases the resulting notions of morphism are new for the metric viewpoint, or for the logical viewpoint.)

*************

You are certainly correct that any duality which might be found here would allow reformulation as a duality within metric viewpoint (although it might become less cute in the process).

I think the question whether interesting dualities can be obtained from considering the interplay between metric and logical viewpoints remains quite valid.