On a strong negation-based representation of modalities. (English) Zbl 1464.03021
Summary: In this study, the notion of a dual pair of modal operators is interpreted according to the algebraic criteria for necessity and possibility operators on De Morgan lattices presented by G. Cattaneo et al. [Inf. Sci. 181, No. 19, 4075–4100 (2011; Zbl 1242.03088)]. Here, a representation theorem is introduced which demonstrates that, in this algebraic model, a dual pair of modal operators can be represented by compositions of two strong negations, where one of them is stricter than the other. Then, the Pliant negation operator is utilized to derive dual modal operators. It is demonstrated that using the generator function of Dombi operators, the composition of two Pliant negations results in modal operators that have simple forms and easy-to-use characteristics. Next, we examine how the proposed modal operators are connected with the drastic necessity and possibility operators. Also, the necessary and sufficient condition for the distributivity of modal operators induced by compositions of strong negations over strict t-norms and strict t-conorms is presented. Lastly, a connection between the modal operators and hedges is highlighted.
Citations:
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