This volume celebrates the work of Petr Hájek on mathematical fuzzy logic and presents how his
efforts have influenced prominent logicians who are continuing his work. The book opens with a
discussion on Hájek's contribution to mathematical fuzzy logic and with a scientific biography
of him progresses to include two articles with a foundation flavour that demonstrate some
important aspects of Hájek's production namely a paper on the development of fuzzy sets and
another paper on some fuzzy versions of set theory and arithmetic. Articles in the volume also
focus on the treatment of vagueness building connections between Hájek's favorite fuzzy logic
and linguistic models of vagueness. Other articles introduce alternative notions of consequence
relation namely the preservation of truth degrees which is discussed in a general context
and the differential semantics. For the latter a surprisingly strong standard completeness
theorem is proved. Another contribution also looks at two principles valid in classical logic
and characterize the three main t-norm logics in terms of these principles. Other articles
with an algebraic flavour offer a summary of the applications of lattice ordered-groups to
many-valued logic and to quantum logic as well as an investigation of prelinearity in
varieties of pointed lattice ordered algebras that satisfy a weak form of distributivity and
have a very weak implication. The last part of the volume contains an article on possibilistic
modal logics defined over MTL chains a topic that Hájek discussed in his celebrated work
Metamathematics of Fuzzy Logic and another one where the authors besides offering unexpected
premises such as proposing to call Hájek's basic fuzzy logic HL instead of BL propose a very
weak system called SL as a candidate for the role of the really basic fuzzy logic. The paper
also provides a generalization of the prelinearity axiom which was investigated by Hájek in
the context of fuzzy logic.
