Many-valued logics. A mathematical and computational introduction. (English) Zbl 06934362

Studies in Logic (London) 67. London: College Publications (ISBN 978-1-84890-250-3/pbk). xiii, 325 p. (2017).

This book cannot be described better than the publisher’s description we recall hereafter. However this description mostly concentrates on “Many-valued logics”, whereas “Part I” that takes up almost half of the book, presents “an extensive introduction to logical “\textit{things}”, i.e. logical languages, systems, and decisions.” Important notions such as logical consequence, adequateness of a logical system are rigorously defined. Notions regarding decision procedures in classical logic are introduced, namely satisfiability and validity, and they are put into relation with the several proof systems available. It makes the book a self-contained instrument for the reader who wishes to do something with many-valued logics.

Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic – as well as other non-classical logics – is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome.

The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize – which also means automate – decisions (i.e. reasoning) in many-valued logics. This, in turn, requires a mathematical foundation for these logics. This book provides both this mathematical foundation and this practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem(s) and automated deduction.

The main text is complemented with a large selection of exercises, a plus for the reader wishing not only to learn about, but also to do something with, many-valued logics.

Reviewer: Albert Hoogewijs (Gent)

MSC:

 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations Many-valued logic Logic in computer science

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