Quantum-logic/probability aspects of connectives in fuzzy logic
Prof. dr. Michal Baczynski
Prof. Dr. Hans De Meyer
The future of fuzzy set theory greatly depends on the development of systematic ways of generalizing classical results to a graded setting. Far too often, such a generalization is done on a case by case basis, reinventing the wheel over and over again. However, instead of looking for multiple theorems, one should rather aim for a single powerful meta-theorem.
In former work at KERMIT, an interesting example of such a meta-theorem has been uncovered. The generalization of classical similarity measures, abundant in practically all branches of science where set-based comparison is at stake, to a graded setting, essentially depends on a model for graded conjunction, often realized by means of a conjunctor, in particular a quasi-copula or t-norm. It turns out that the transitivity of such similarity measures is preserved when the conjunctor used satisfies a number of inequalities that show a striking resemblance with some Bell-inequalities from probability theory. A first such inequality was discovered by Pykacz and D’Hooghe. On the other hand, some connectives in fuzzy logic have been inspired by quantum logic operations.
It is clear that a more profound study would be welcome, revealing deeper connections between Bell-inequalities and fuzzy connectives, in particular those related to quantum logic in one sense or another. Furthermore, their role in the study of inequality-based properties of fuzzy relations should be further unveiled.