Includes bibliographical references and index.

In this book, it is argued that the notorious logical paradoxes-the Liar, Russell's, the Sorites-are only the noisiest of many. Contradictions arise in the everyday, from the smallest points, to the widest boundaries. Dialetheic paraconsistency-a formal framework where some contradictions can be true without absurdity-is used as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, this work directly addresses a longstanding open question of how much standard mathematics paraconsistency can capture. The guiding focus is on the question: why are there paradoxes? Details underscore a simple philosophical claim: that paradoxes are found in the ordinary-and that is what makes them so extraordinary. Argument: (1) There are true contradictions, both in the foundations of logic and mathematics, and in the everyday world. (2) If the world is inconsistent but not absurd, then the logic underlying our theory of the world ought to be paraconsistent. (3) Paraconsistent logic then must, and can, show that it supports some ordinary reasoning, including proving the motivating paradoxes in elementary mathematics. (4) The basic components of a non-classical picture come into view, and we are positioned to (re)address the question of why there are paradoxes.