Peterson's rules of distributivity; quality and quantity; Fuzzy intermediate syllogisms; Graded Peterson square of opposition

In our previous papers, we syntactically proved validity of 105 logical syllogisms with intermediate quantifiers. Peterson in his book suggested 6 rules using which it is possible to verify validity of such syllogisms in a simple way. The rules, however, are formulated in free language and so, it is not possible to check whether they are correct. In this paper, we focus on formalization of Peterson's rules. The main idea is based on the graded Peterson square of opposition, which has been formally analyzed in our previous publications.