The extension principle extends crisp/defined domains to fuzzy ones. A fuzzy set is different from a crisp one in that in a fuzzy set the elements have degrees of membership. The protoype of a fuzzy set is the most central to the set. The ostensive definition is using examples to indicate the defined term.
Essentially the difference between a crisp and fuzzy domain is that in a crisp domain one knows where the boundaries lay, as opposted to a fuzzy one in which the boundary is not exactly known.
Fuzzy logic is essentially how you can represent uncertainty mathematically. Probabilty achieves the same goal.