fuzzy intermediate quantifiersPeterson's rules graded Peterson's square of opposition

In real world, interpretation, understanding and deriving information from natural data using natural language is very common. One of the theories that deals with this modern topic is the theory of syllogistic reasoning. There are several methods to verify the validity and invalidity of classical Aristotle’s syllogisms as well as fuzzy syllogisms. In previous publications, we have focused on the mathematical construction of syntactic proofs of fuzzy Peterson's syllogisms. The main goal of this publication is to explain Peterson's rules of distributivy, quality and quantity for verifying the validity of syllogisms and to demonstrate them on selected examples of syllogisms.