This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such as (...) a simple proof of the completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic. Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean partitions can be defined and computed in classical logic. Detailed bibliographic remarks at the end of each chapter and an extensive bibliography lead the reader on to further specialised topics. (shrink)

Pseudo MV-algebras are a non-commutative extension of MV-algebras introduced recently by Georgescu and Iorgulescu. We introduce states (finitely additive probability measures) on pseudo MV-algebras. We show that extremal states correspond to normal maximal ideals. We give an example in that, in contrast to classical MV-algebras introduced by Chang, states can fail on pseudo MV-algebras. We prove that representable and normal-valued pseudo MV-algebras admit at least one state.

Some algebraic structures of the set of all effects are investigated and summarized in the notion of a(weak) orthoalgebra. It is shown that these structures can be embedded in a natural way in lattices, via the so-calledMacNeille completion. These structures serve as a model ofparaconsistent quantum logic, orthologic, andorthomodular quantum logic.

We introduce the notion of quantum MV algebra (QMV algebra) as a generalization of MV algebras and we show that the class of all effects of any Hilbert space gives rise to an example of such a structure. We investigate some properties of QMV algebras and we prove that QMV algebras represent non-idempotent extensions of orthomodular lattices.