Towards more realistic (e.g., non-associative) ?and?- and ?or?-operations in fuzzy logic

How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since AB and BA are the same, the corresponding and-operation should be commutative. Similarly, since AA means the same as A, we should expect that the and-operation should also satisfy this property, etc. It turns out to be impossible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if they contradict to the picked ones. This idea leads to a neat mathematical theory, but the analysis of real-life expert reasoning shows that all the requirements are only approximately satisfied. we should require all of these requirements to be satisfied to some extent. In this paper, we show the preliminary results of analyzing such operations. In particular, we show that non-associative operations explain the empirical 7±2 law in psychology according to which a person can normally distinguish between no more than 7 plus minus 2 classes.