Kitainik axioms do not characterize the class of inclusion measures based on contrapositive fuzzy implications

In this short communication, we refute the conjecture by Fodor and Yager from [5] that the class of inclusion measures proposed by Kitainik coincides with that of inclusion measures based on contrapositive fuzzy implications. In particular, we show that the conjecture only holds when the considered universe of discourse is finite.The research reported in this paper was conducted with the financial support of the Odysseus programme of the Research Foundation – Flanders (FWO) (grant number G0H9118N) and partially supported by the Spanish Ministry of Science, Innovation and Universities (MCIU), State Agency of Research (AEI), Junta de Andalucía (JA), Universidad de Málaga (UMA) and European Regional Development Fund (FEDER) through the projects PGC2018-095869-B- I00 (MCIU/AEI/FEDER) and UMA2018-FEDERJA-001 (JA/UMA/FEDER). Funding for open access charge: Universidad de Málaga / CBU

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