Aristotle's square for mining fuzzy concepts

Aristotle's Square also known as Square of Opposition, is a mathematical diagram dating back to Greek philosophy and exhibiting the connection between four logical propositions in a simple graphical form. Fuzzy Relational Concept Analysis (FRCA) is a technique for extracting special clusters called fuzzy concepts from a Fuzzy Relational Context Family (FRCF), which is a dataset organized as multiple fuzzy object-attribute and object-object relations. The primary FRCA tools to obtain information from data are special fuzzy quantifiers viewed as interpretations in a model of formulas of the formal theory of the intermediate generalized quantifiers. This work focuses on the issue of generating a collection of fuzzy concepts from a certain FRCF, by choosing one of four particular FRCA quantifiers: the positive universal quantifier $S_1$, the negative universal quantifier $S_1^{-}$, the positive existential quantifier $S_{\exists }$, and the negative existential quantifier $S_{\exists }^{-}$. Certainly, the selection of the quantifier is crucial in the FRCA procedure since it affects the final concept classification: diverse fuzzy concepts arise from varying quantifiers. As the initial objective, this article introduces the logical relations involving $S_1,S_1^{-}$, $S_{\exists }$, and $S_{\exists }^{-}$, in order to arrange them in a graded version of the Aristotelian square. The second goal of this study is to examine the connections among fuzzy concepts produced by distinct quantifiers in $\{S_1,S_1^{-},S_{\exists },S_{\exists }^{-}\}$. Therefore, our findings provide a twofold contribution to the advancement of Aristotle's square. Indeed, they reveal a novel interpretation of the square of opposition within the framework of Fuzzy Relational Concept Analysis, emphasizing its potential as a valuable tool for the analysis of data.