Reflections on an old problem: That of preserving the logical forms and symmetry

This paper is devoted to analyze and discuss some of the new aspects about the preservation of the logical forms appearing in ordinary reasoning and fuzzy logic. Considering a Basic Fuzzy Algebra (BFA) as the basic formal model for representing imprecise predicates, several classical properties of logical forms which can be not valid depending on the representation of the logical connectives in a BFA are studied. Special emphasis is put on the general symmetry property, or its absence in some logical forms that, in some particular cases and in some sense, can become symmetrical as they are, for instance, the linguistic negation, the law of perfect repartition, or the additive law of measures. In any case, the difference between what is supposed in a mathematical model and what can be checked in the reality cannot be forgotten since any reality, and the study of the ordinary reasoning in natural language is a clear example of that, is but an ‘observed reality’. Anyway, if what is in a mathematical fuzzy model is not always symmetrical, and since these models represent linguistic statements affected by imprecision and uncertainty, it can be concluded that, in general, neither plain language, nor commonsense or ordinary reasoning, show the same level of symmetry exhibited by the specialized language and the deductive reasoning allowing mathematical proofs.