An Analysis of Ruspini Partitions in Gödel Logic

By a Ruspini partition we mean a finite family of fuzzy sets Formula Not Shown, such that Formula Not Shown for all Formula Not Shown, where [0, 1] denotes the real unit interval. We analyze such partitions in the language of Gödel logic. Our first main result identifies the precise degree to which the Ruspini condition is expressible in this language, and yields inter alia a constructive procedure to axiomatize a given Ruspini partition by a theory in Gödel logic. Our second main result extends this analysis to Ruspini partitions fulfilling the natural additional condition that each Formula Not Shown has at most one left and one right neighbour, meaning that Formula Not Shown holds for Formula Not Shown .