This monograph shows that, through a recourse to the concepts and methods of abstract algebraic logic, the algebraic theory of regular varieties and the concept of analyticity in formal logic can profitably interact. By extending the technique of Plonka sums from algebras to logical matrices, the authors investigate the different classes of models for logics of variable inclusion and they shed new light into their formal properties.
The book opens with the historical origins of logics of variable inclusion and on their philosophical motivations. It includes the basics of the algebraic theory of regular varieties and the construction of Plonka sums over semilattice direct systems of algebra. The core of the book is devoted to an abstract definition of logics of left and right variable inclusion, respectively, and the authors study their semantics using the construction of Plonka sums of matrix models. The authors also cover Paraconsistent Weak Kleene logic and survey its abstract algebraic logical properties. This book is of interest to scholars of formal logic.
Chapter 1. Analyticity, Consequence, and Meaninglessness.- Chapter 2. Plonka Sums and Regular Varieties.- Chapter 3. Dualities for Regular Varieties.- Chapter 4. Logics of Left Variable Inclusion.- Chapter 5. Logics of Right Variable Inclusion.- Chapter 6. Paraconsistent Weak Kleene Logic.
Stefano Bonzio is assistant professor of Mathematical Logic at the University of Cagliari, where he teaches Algebra and Discrete Mathematics. He is author of more than 20 publications in international peer-reviewed journals. His research interests include non-classical logics (in particular Kleene logics), algebraic logic and universal algebra, the foundation of probability.
Francesco Paoli is a full professor of Logic at the University of Cagliari. He published, among other things, the book Substructural Logics: A Primer (Kluwer, 2002) and over 60 papers in international peer-reviewed journals. His research interests include nonclassical (substructural, quantum, many-valued) logics, universal algebra, and the foundations of physics.
Michele Pra Baldi is a post-doctoral researcher at the Artificial Intelligence Research Institute, Spanish National Council of Research (Bellaterra, Barcelona, Spain). He obtained his PhD at the University of Padua in 2019. His research interests include abstract algebraic logic, universal algebra, non-classical logics and their applications to formal epistemology and philosophical logic.
Report
"Logics of Variable Inclusion is a valid, rigorous, and comprehensive study in algebraic logic of the employment of Plonka sums in formal semantics, an essential reference for the future developments in the field." (Nicolò Zamperini, Studia Logica, Vol. 111 (3), 2023)
