In this article, we will show how relations forming the graded Peterson's square of opposition (contrary, contradictory, sub-contrary and sub-alterns) are connected with intermediate logical syllogisms, i.e., syllogisms that contain intermediate quantifiers. Our results are related to the theory of non-monotonic reasoning.

In this article, we will form a graded hexagon of opposition as an extension of the graded Peterson's square, incorporating new variations of intermediate quantifiers. We will start with the classical Aristotle's square of opposition and its extension to hexagon. Then we introduce the graded square of opposition that was suggested by Peterson and fully formalized by the authors. We also introduce quantifiers A few ( A little ) and Several and include them in the latter. Then we introduce new forms of quantifiers and study relations of contrary, sub-contrary, contradictory and sub-(sup-)altern among them. On the basis of the knowledge of the relations among all the intermediate quantifiers we construct the graded Peterson's hexagon of opposition.

This paper delves into logical syllogisms featuring intermediate quantifiers. In our previous works, we established the validity of logical syllogisms involving fundamental intermediate quantifiers “Almost all”, “Most”, and “Many”. In this paper, we focus on syllogisms incorporating also the quantifiers “Several” and “A few (A little)”.

We recently introduced special fuzzy quantifiers named quantifier-based operators to form extended fuzzy concept lattices and to construct graded squares, hexagons, octagons and decagons of oppositions. This article aims to extend our previous works by organizing quantifier-based operators in more general structures of oppositions: the so-called graded cubes of opposition and 5-graded cubes of opposition.

In real world, interpretation of natural data using a natural language is very popular. In our previous papers, we introduced mathematical definitions of intermediate quantifiers, which were used for an analysis of natural data using fuzzy association rules. The main objective of this paper is to show how we can apply the theory of syllogistic reasoning for a derivation of new information which is not included in data. An idea of this paper is to use generalized structure of syllogisms with more premises.

This paper is a contribution to the study of a special kind of syllogisms with intermediate quantifiers. We stem from our previous papers where a formal theory of the intermediate quantifiers was introduced. Besides other results, we syntactically proved validity of 105 basic syllogisms with them. We also demonstrated how our theory works in the semantic interpretation. In this paper, we will address some special kinds of syllogisms that are non-trivial in the sense that both premises as well as conclusion contain general intermediate quantifiers.