A Resemblance Approach to Analogical Reasoning Functions

Fuzzy inference, as defined by Zadeh's Compositional Rule of Inference, may be viewed as a procedure that translates some analogy in the hypothesis space between the antecedent of a given rule and the hypothesis, into some other analogy, now in the thesis space, to obtain the thesis from the consequent of the rule. Starting from this fact, we analyze the so-called analogical reasoning functions, that is, functions that preserve some given resemblance in the hypothesis and thesis spaces respectively. The resemblance considered are supposed to be given by T-transitive fuzzy relations where T is a t-norm. From this standpoint, and in order to study the suitability of those functions to model reasoning procedures, we characterize the set of the analogies of a given set, that is, the set of images of the given set under such class of analogical reasoning functions.