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Fuzzy logic and mathematics : a historical perspective
The term “fuzzy logic” (FL) is a generic one, which stands for a broad variety of logical systems. Their common ground is the rejection of the most fundamental principle of classical logic—the principle of bivalence—according to which each declarative sentence has exactly two possible truth values—true and false. Each logical system subsumed under FL allows for additional, intermediary truth values, which are interpreted as degrees of truth. These systems are distinguished from one another by the set of truth degrees employed, its algebraic structure, truth functions chosen for logical connectives, and other properties. The book examines from the historical perspective two areas of research on fuzzy logic known as fuzzy logic in the narrow sense (FLN) and fuzzy logic in the broad sense (FLB), which have distinct research agendas. The agenda of FLN is the development of propositional, predicate, and other fuzzy logic calculi. The agenda of FLB is to emulate commonsense human reasoning in natural language and other unique capabilities of human beings. In addition to FL, the book also examines mathematics based on FL. One chapter in the book is devoted to overviewing successful applications of FL and the associated mathematics in various areas of human affairs. The principal aim of the book is to assess the significance of FL and especially its significance for mathematics. For this purpose, the notions of paradigms and paradigm shifts in science, mathematics, and other areas are introduced and employed as useful metaphors.
eBook
Pure inductive logic
\"Pure inductive logic is the study of rational probability treated as a branch of mathematical logic. This monograph, the first devoted to this approach, brings together the key results from the past seventy years plus the main contributions of the authors and their collaborators over the last decade to present a comprehensive account of the discipline within a single unified context.\" -- Provided by publisher.
Book
Souslin quasi-orders and bi-embeddability of uncountable structures
We provide analogues of the results from Friedman and Motto Ros (2011) and Camerlo, Marcone, and Motto Ros (2013) (which correspond
to the case
eBook
(L\\)-Modules
In this paper, considering \\(L\\)-algebras, which include a significant number of other algebraic structures, we present a definition of modules on \\(L\\)-algebras (\\(L\\)-modules). Then we provide some examples and obtain some results on \\(L\\)-modules. Also, we present definitions of prime ideals of \\(L\\)-algebras and \\(L\\)-submodules (prime \\(L\\)-submodules) of \\(L\\)-modules, and investigate the relationship between them. Finally, by proving a number of theorems, we provide some conditions for having prime \\(L\\)-submodules.
Journal Article
Tractatus logico-philosophicus
This new edition of Wittgenstein's book, strictly following the author's recommendations, allows a more immediate comprehension of the text and dissolves several false problems that had deceived readers and scholars for a century. The faithful interpretation of decimal numbers (which alone, according to Wittgenstein, \"give perspicuity and clarity to the book\") shows that the Tractatus stems from a home-page containing seven cardinal propositions and develops level by level, by perfectly coherent reading units. Indeed, \"the Tractatus must be read in accordance with the numbering system, and that demands that the reader follow the text after the manner of a logical tree, which is the way in which the book was composed and in which Wittgenstein arranged his philosophical remarks\" (Peter Hacker, The Philosophical Quarterly). Thence, the Tractatus is no longer an obstacle course, where critics and students were strenuously committed to decipher anacolutes, semantic jumps and bizarre combinations. On the contrary, it reveals to be, at long last, a book that every reader, from her own point of view, can enjoy. The actual form of Wittgenstein's work discloses the harmony and the aesthetic value of a philosophical text that is contemporary and is one of the most amazing masterpieces of world literature.
eBook
