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).