Approaching the square of opposition in terms of the f-indexes of inclusion and contradiction

We continue our research line on the analysis of the properties of the f-indexes of inclusion and contradiction; in this paper, specifically, we show that both notions can be related by means of the, conveniently reformulated, Aristotelian square of opposition. We firstly show that the extreme cases of the f-indexes of inclusion and contradiction coincide with the vertexes of the Aristotelian square of opposition in the crisp case; then, we allocate the rest of f-indexes in the diagonals of the extreme cases and we prove that the Contradiction, Contrariety, Subcontrariety, Subalternation and Superalternation relations also hold between the f-indexes of inclusion and contradiction.Funding for open Access charge: Universidad de Málaga / CBUA. Partially supported by the Ministry of Science, Innovation, and Universities (MCIU), the State Agency of Research (AEI) and the European Social Fund (FEDER) through the research projects PGC2018-095869-B-I00 (MCIU/AEI/FEDER, UE) and VALID (PID2022-140630NB-I00 MCIN/AEI/10.13039/501100011033), and by Junta de Andalucía, Universidad de Málaga and the European Social Fund (FEDER) through the research project UMA2018-FEDERJA-001

Graded cubes of opposition and possibility theory with fuzzy events

International audienceThe paper discusses graded extensions of the cube of opposition, a structure that naturally emerges from the square of opposition in philosophical logic. These extensions of the cube of opposition agree with possibility theory and its four set functions. This extended cube then provides a synthetic and unified view of possibility theory. This is an opportunity to revisit basic notions of possibility theory, in particular regarding the handling of fuzzy events. It turns out that in possibility theory, two extensions of the four basic set functions to fuzzy events exist, which are needed for serving different purposes. The expressions of these extensions involve many-valued conjunction and implication operators that are related either via semi-duality or via residuation

Fuzzy Natural Logic in IFSA-EUSFLAT 2021

The present book contains five papers accepted and published in the Special Issue, “Fuzzy Natural Logic in IFSA-EUSFLAT 2021”, of the journal Mathematics (MDPI). These papers are extended versions of the contributions presented in the conference “The 19th World Congress of the International Fuzzy Systems Association and the 12th Conference of the European Society for Fuzzy Logic and Technology jointly with the AGOP, IJCRS, and FQAS conferences”, which took place in Bratislava (Slovakia) from September 19 to September 24, 2021. Fuzzy Natural Logic (FNL) is a system of mathematical fuzzy logic theories that enables us to model natural language terms and rules while accounting for their inherent vagueness and allows us to reason and argue using the tools developed in them. FNL includes, among others, the theory of evaluative linguistic expressions (e.g., small, very large, etc.), the theory of fuzzy and intermediate quantifiers (e.g., most, few, many, etc.), and the theory of fuzzy/linguistic IF–THEN rules and logical inference. The papers in this Special Issue use the various aspects and concepts of FNL mentioned above and apply them to a wide range of problems both theoretically and practically oriented. This book will be of interest for researchers working in the areas of fuzzy logic, applied linguistics, generalized quantifiers, and their applications