Pseudomonadic BL-algebras: an algebraic approach to possibilistic BL-logic
Fuzzy possibilistic logic is an important formalism for approximate reasoning. It extends the well-known basic propositional logic BL, introduced by Hájek, by offering the ability to reason about possibility and necessity of fuzzy propositions. We consider an algebraic approach to study this logic, introducing Pseudomonadic BL-algebras. These algebras turn to be a generalization of both Pseudomonadic algebras introduced by Bezhanishvili (Math Log Q 48:624–636, 2002) and serial, Euclidean and transitive Bimodal Gödel algebras proposed by Caicedo and Rodriguez (J Log Comput 25:37–55, 2015). We present the connection between this class of algebras and possibilistic BL-frames, as a first step to solve an open problem proposed by Hájek (Metamathematics of fuzzy logic. Trends in logic, Kluwer, Dordrecht, 1998, Chap. 8, Sect. 3).
KeywordsModal algebras Fuzzy possibilistic logic BL-algebras
The authors thank the referees for their comments that helped improve the paper. The authors were funded by the research project PIP 112-20150100412CO, CONICET, Desarrollo de Herramientas Algebraicas y Topológicas para el Estudio de Lógicas de la Incertidumbre y la Vaguedad. DHATELIV. M. Busaniche and Penélope Cordero were also funded by the project CAI+D PIC 50420150100090LI, UNL, Métodos algebraico-geométricos en la teoría de la información. Additionally, the author P. Cordero was supported by a CONICET grant during the preparation of the paper. R. O. Rodriguez was also funded by the projects UBA-CyT 20020150100002BA and PICT/O \(N^\circ \) 2016-0215. The authors have been funded by the Horizon 2020 project of the European Commission 689176–SYSMICS.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
- Bou F, Esteva F, Godo L, Rodriguez R (2016) Possibilistic semantics for a modal \(KD45\) extension of Gödel fuzzy logic. In: IPMU 2016, Eindhoven, The Netherlands, 20–24 June 2016, proceedings, part II. Communications in computer and information science 611, SpringerGoogle Scholar
- Busaniche M, Montagna F (2011) Hájek’s logic BL and BL-algebras. Handbook of mathematical fuzzy logic, volume 1 of studies in logic, mathematical logic and foundations, vol 1. College Publications, London, pp 355–447Google Scholar
- 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 University Press, Oxford, pp 439–513Google Scholar
- Hintikka J (1962) Knowledge and belief. An introduction to the logic of the two notions. Stud Log 16:119–122Google Scholar