There is a principle in things, about which we cannot be deceived, but must always, on the contrary, recognize the truth – viz. that the same thing cannot at one and the same time be and not be": with these words of the Metaphysics, Aristotle introduced the Law of Non-Contradiction, which was to become the most authoritative principle in the history of Western thought. However, things have recently changed, and nowadays various philosophers, called dialetheists, claim that this Law does not (...) hold unrestrictedly – that in peculiar circumstances the same thing may at the same time be and not be, and contradictions may obtain in the world. This book opens with an examination of the famous logical paradoxes that appear to speak on behalf of contradictions (e.g., the Liar paradox, the set-theoretic paradoxes such as Cantor’s and Russell’s), and of the reasons for the failure of the standard attempts to solve them. It provides, then, an introduction to paraconsistent logics – non-classical logics in which the admission of contradictions does not lead to logical chaos –, and their astonishing applications, going from inconsistent data base management to contradictory arithmetics capable of circumventing Gödel’s celebrated Incompleteness Theorem. The final part of the book discusses the philosophical motivations and difficulties of dialetheism, and shows how to extract from Aristotle’s ancient words a possible reply to the dialetheic challenge. How to Sell a Contradiction will appeal to anyone interested in non-classical logics, analytic metaphysics, and philosophy of mathematics, and especially to those who consider challenging our most entrenched beliefs the main duty of philosophical inquiry. (shrink)

It hath been an old remark, that Geometry is an excellent Logic.

: Nagarjuna seems willing to embrace contradictions while at the same time making use of classic reductio arguments. He asserts that he rejects all philosophical views including his own-that he asserts nothing-and appears to mean it. It is argued here that he, like many philosophers in the West and, indeed, like many of his Buddhist colleagues, discovers and explores true contradictions arising at the limits of thought. For those who share a dialetheist's comfort with the possibility of true contradictions commanding (...) rational assent, for Nagarjuna to endorse such contradictions would not undermine but instead confirm the impression that he is indeed a highly rational thinker. It is argued that the contradictions he discovers are structurally analogous to many discovered by Western philosophers and mathematicians. (shrink)

The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established.

This paper defends that the both the Buddhist tradition and the Aristotelian allow us to think of the distinction between to reason with contradictions and to accept them, understanding ' accept a contradiction ' by taking it as consistent. From this viewpoint, none of two would disagree with most contemporary paraconsistent views. The conclusions are, thus, that, firstly, there is no compelling reason to endorse any kind of metaphysical dialetheism, and, second, that a coherent form of reasoning with contradictory statements (...) is neither opposed to the Aristotelian conception of logic, nor to certain texts in the Buddhist tradition, as the Madhyamikas. (shrink)

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested (...) in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body. (shrink)

Certain Buddhist texts contain statements that are prima facie contradictions. The scholarly consensus has been that such statements are meant to serve a rhetorical function that depends on the apparent contradictions being resolvable. But recently it has been claimed that such statements are meant to be taken literally: their authors assert as true statements that are of the form ‘p and not p’. This claim has ramifications for our understanding of the role played by the principle of non-contradiction in Buddhist (...) argumentation. I argue that these make the claim less plausible. (shrink)

Can God create a stone too heavy for him to lift? Can time have a beginning? Which came first, the chicken or the egg? Riddles, paradoxes, conundrums--for millennia the human mind has found such knotty logical problems both perplexing and irresistible. Now Roy Sorensen offers the first narrative history of paradoxes, a fascinating and eye-opening account that extends from the ancient Greeks, through the Middle Ages, the Enlightenment, and into the twentieth century. When Augustine asked what God was doing before (...) He made the world, he was told: "Preparing hell for people who ask questions like that." A Brief History of the Paradox takes a close look at "questions like that" and the philosophers who have asked them, beginning with the folk riddles that inspired Anaximander to erect the first metaphysical system and ending with such thinkers as Lewis Carroll, Ludwig Wittgenstein, and W.V. Quine. Organized chronologically, the book is divided into twenty-four chapters, each of which pairs a philosopher with a major paradox, allowing for extended consideration and putting a human face on the strategies that have been taken toward these puzzles. Readers get to follow the minds of Zeno, Socrates, Aquinas, Ockham, Pascal, Kant, Hegel, and many other major philosophers deep inside the tangles of paradox, looking for, and sometimes finding, a way out. Filled with illuminating anecdotes and vividly written, A Brief History of the Paradox will appeal to anyone who finds trying to answer unanswerable questions a paradoxically pleasant endeavor. (shrink)