Polish Logic 1920 1939

Polish Logic 1920-1939

Many-valued logics were developed as an attempt to handle philosophical doubts about the "law of excluded middle" in classical logic. The first many-valued formal systems were developed by J. Lukasiewicz in Poland and E.Post in the U.S.A. in the 1920s, and since then the field has expanded dramatically as the applicability of the systems to other philosophical and semantic problems was recognized. Intuitionisticlogic, for example, arose from deep problems in the foundations of mathematics. Fuzzy logics, approximation logics, and probability logics all address questions that classical logic alone cannot answer. All these interpretations of many-valued calculi motivate specific formal systems thatallow detailed mathematical treatment. In this volume, the authors are concerned with finite-valued logics, and especially with three-valued logical calculi. Matrix constructions, axiomatizations of propositional and predicate calculi, syntax, semantic structures, and methodology are discussed. Separate chapters deal with intuitionistic logic, fuzzy logics, approximation logics, and probability logics. These systems all find application in practice, in automatic inference processes, which have been decisive for the intensive development of these logics. This volume acquaints the reader with theoretical fundamentals of many-valued logics. It is intended to be the first of a two-volume work. The second volume will deal with practical applications and methods of automated reasoning using many-valued logics.

Arguably the single most influential literary work of the European Middle Ages, the Roman de la Rose of Guillaume de Lorris and Jean de Meun has traditionally posed a number of difficulties to modern critics, who have viewed its many interruptions and philosophical discussions as signs of a lack of formal organization and a characteristically medieval predilection for encyclopedic summation. In Fortune's Faces, Daniel Heller-Roazen calls into question these assessments, offering a new and compelling interpretation of the romance as a carefully constructed and far-reaching exploration of the place of fortune, chance, and contingency in literary writing. Situating the Romance of the Rose at the intersection of medieval literature and philosophy, Heller-Roazen shows how the thirteenth-century work invokes and radicalizes two classical and medieval traditions of reflection on language and contingency: that of the Provençal, French, and Italian love poets, who sought to compose their "verses of pure nothing"in a language Dante defined as "without grammar," and that of Aristotle's discussion of "future contingents" as it was received and refined in the logic, physics, theology, and epistemology of Boethius, Abelard, Albert the Great, and Thomas Aquinas.Through a close analysis of the poetic text and a detailed reconstruction of the logical and metaphysical concept of contingency, Fortune's Faces charts the transformations that literary structures (such as subjectivity, autobiography, prosopopoeia, allegory, and self-reference) undergo in a work that defines itself as radically contingent. Considered in its full poetic and philosophical dimensions, the Romance of the Rose thus acquires an altogether new significance in the history of literature: it appears as a work that incessantly explores its own capacity to be other than it is.

Volume seven of a ten volume set which provides full and detailed coverage of all aspects of philosophy, including information on how philosophy is practiced in different countries, who the most influential philosophers were, and what the basic concepts are.

With the publication of the present volume, the Handbook of the History of Logic turns its attention to the rise of modern logic. The period covered is 1685-1900, with this volume carving out the territory from Leibniz to Frege. What is striking about this period is the earliness and persistence of what could be called 'the mathematical turn in logic'. Virtually every working logician is aware that, after a centuries-long run, the logic that originated in antiquity came to be displaced by a new approach with a dominantly mathematical character. It is, however, a substantial error to suppose that the mathematization of logic was, in all essentials, Frege's accomplishment or, if not his alone, a development ensuing from the second half of the nineteenth century. The mathematical turn in logic, although given considerable torque by events of the nineteenth century, can with assurance be dated from the final quarter of the seventeenth century in the impressively prescient work of Leibniz. It is true that, in the three hundred year run-up to the Begriffsschrift, one does not see a smoothly continuous evolution of the mathematical turn, but the idea that logic is mathematics, albeit perhaps only the most general part of mathematics, is one that attracted some degree of support throughout the entire period in question. Still, as Alfred North Whitehead once noted, the relationship between mathematics and symbolic logic has been an "uneasy" one, as is the present-day association of mathematics with computing. Some of this unease has a philosophical texture. For example, those who equate mathematics and logic sometimes disagree about the directionality of the purported identity. Frege and Russell made themselves famous by insisting (though for different reasons) that logic was the senior partner. Indeed logicism is the view that mathematics can be re-expressed without relevant loss in a suitably framed symbolic logic. But for a number of thinkers who took an algebraic approach to logic, the dependency relation was reversed, with mathematics in some form emerging as the senior partner. This was the precursor of the modern view that, in its four main precincts (set theory, proof theory, model theory and recursion theory), logic is indeed a branch of pure mathematics. It would be a mistake to leave the impression that the mathematization of logic (or the logicization of mathematics) was the sole concern of the history of logic between 1665 and 1900. There are, in this long interval, aspects of the modern unfolding of logic that bear no stamp of the imperial designs of mathematicians, as the chapters on Kant and Hegcl make clear. Of the two, Hcgel's influence on logic is arguably the greater, serving as a spur to the unfolding of an idealist tradition in logic - a development that will be covered in a further volume, British Logic in the Nineteenth Century.