Handbook of mathematical fuzzy logic. Volume 1 (Q2871197)

Motivated by the need to define a good mathematical basis for modelling processes that include vagueness and imprecision, having a strong philosophical inspiration, fuzzy systems theory and fuzzy set theory are founded on fuzzy logics. Mathematical fuzzy logic can be considered a subfield of mathematical logic dealing with non-classical many-valued logics with a linearly ordered set of truth values. This handbook consists of two volumes (a third volume is in preparation).NEWLINENEWLINE Volume 1 provides an introductory overview of basic notions and results on the subject (``Introduction to mathematical fuzzy logic'' (pp. 1--101) by \textit{Libor Běhounek}, \textit{Petr Cintula} and \textit{Petr Hájek}) developing a parallelism with the classical counterparts of fuzzy logic, according to their complexity and metamathematical properties. Close connections between abstract algebraic logic and mathematical fuzzy logic (``A general framework for mathematical fuzzy logic'' (pp. 103--207) by \textit{Petr Cintula} and \textit{Carles Noguera}) open possibilities to develop particular technical notions corresponding to the intuition of fuzzy logics as the logics of chains. Two applications of proof theory are considered: a novel approach to tackling standard completeness, and extension of propositional fuzzy logics to a first-order language (``Proof theory for mathematical fuzzy logic'' (pp. 209--282) by \textit{George Metcalfe}). Development of algebraic semantics for fuzzy logics is also presented (``Algebraic semantics: semilinear FL-algebras'' (pp. 283--353) by \textit{Rostislav Horčík}), and particular attention is paid to Hájek's basic logic and its algebraic semantics (``Hájek's logic BL and BL-algebras'' (pp. 355--447) by \textit{Manuela Busaniche} and \textit{Franco Montagna}).NEWLINENEWLINE For Volume 2 of this handbook see [Zbl 1283.03002].