Epistemic BL-algebras: An algebraic characterization for the fuzzy logic of belief KD45

PENÉLOPE CORDERO

Niterói

Conferencia; Conference on Mathematical Logic, satellite event of the ICM 2018; 2018

Instituto de Computaçao, Universidade Federal Fluminense

In \cite{Hajeketal1995}, the authors define a modal logic to reason about possibility and necessity degrees of many-valued propositions. The fuzzy modal belief logic, that they call $KD45(\mathcal{C})$, is defined as the set of valid formulas in the class of $\mathcal{C}$-possibilistic frames. This logic is a generalization of the so-called \textit{Possibilistic Logic} \cite{Duboisetal1,Duboisetal2} a well-known uncertainty logic to reasoning with graded belief on classical propositions by means of necessity and possibility measures. In his book (Chapter 8 of \cite{HajekBook98}), H\'{a}jek leaves as an open problem to find an elegant axiomatization for $KD45(\mathcal{C})$ with models over BL-algebras $\mathcal{C}$. %Some attempts to solve the open problem have been done, considering that the most natural semantics of fuzzy $KD45$ is a fuzzy version of possibilistic frames (closely related to the Kripke-style semantics), see for example \cite{BoEsGoRo11} and the reference therein.Our aim is to give a more general characterization of the logical system by an algebraic approach. In this sense, we introduce the class of Epistemic BL-algebras (EBL-algebras), as BL-algebras with two unary operators that behave generalizing the modal operators of $KD45$.\\Algebraic models for KD45 have already been studied for some particular logics. In \cite{Bez2002}, Bezhanishvili introduces the class of pseudomonadic algebras as algebraic semantics for the KD45 system over classical logic. In this sense, we show that the class of pseudomonadic algebras coincide with the class of EBL-algebras whose reduct is a Boolean algebra. In a similar way, we prove that the EBL-algebras whose reducts are G\"{o}del algebras coincide with the variety of serial, euclidean and transitive bi-modal G\"odel algebras defined by Caicedo et al. in \cite{CaiRod2015}.\\To establish a connection with possibilistic BL-frames described in \cite{HajekBook98}, we introduce a special class of epistemic BL-algebras, which we call c-EBL-algebras and we prove that each possibilistic BL-frame determines a unique c-EBL-algebra and we give necessary conditions for a BL-algebra of functions to be the reduct of a c-EBL-algebra corresponding to a possibilistic frame.%References\begin{thebibliography}{999}\bibitem{Bez2002}Bezhanishvili N (2002) Pseudomonadic Algebras as Algebraic Models of Doxastic Modal Logic. Mathematical Logic Quartely 48:624-636.\bibitem{CaiRod2015}Caicedo X, Rodriguez R (2015) Bi-modal G\"{o}del logic over [0,1]-valued Kripke frames. Journal of Logic and Computation 25:37-55.\bibitem{Duboisetal2} Dubois D, Land J, Prade H (1994) Possibilistic Logic. In Gabbay et al. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programing, Non-monotonic Reasoning and Uncertain Reasoning. Vol.3 Oxford UP. 439-513.\bibitem{Duboisetal1} Dubois D, Prade H (2004) Possibilistic Logic: a retrospective and perspective view. Fuzzy Sets and Systems 144:3-23.\bibitem{Hajeketal1995} H\'{a}jek P, Harmancov\'a D, Verbrugge R (1995) A Qualitative Fuzzy Probabilistic Logic. Journal of Approximate Reasoning 12:1-19.\bibitem{HajekBook98} H\'{a}jek P (1998) Metamathematics of Fuzzy Logic. Trends in Logic, Kluwer.%book