# Duality between metric and logical viewpoints

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).
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