Logical Paradoxes in Non-Classical Logic Systems

It is shown that well-known logical paradoxes such as Barber paradox can be interpreted differently in non-classical logic systems such as multi-valued, continuous and quantum logic with possibility of solutions of the paradox. The results of this research can have applications in investigations of completeness of logic systems.

SELF-REFERENCE, THEORY OF TYPES, AND CATEGORIZATION IN WITTGENSTEIN’S PICTURE THEORY OF STATEMENTS

Рассматривается источник логических парадоксов, выявленных Б. Расселом в системе обоснования математики, предложенной Г. Фреге. Самореферентность выражений, предложенная Б. Расселом как объяснение возникновения парадоксов, рассматривается с точки зрения разработанной им простой и разветвленной теории типов. Обосновывается, что теория типов, предложенная Б. Расселом, основана на онтологических предпосылках. Онтологические предпосылки зависят от предпочтения семантическому перед синтаксическим подходом, который принимается Б. Расселом. Рассмотрены синтаксические подходы к логическому символизму, которые позволяют устранить парадоксы с точки зрения языка современной символической логики. Анализируется подход к решению парадоксов Л. Витгенштейна, который основан на синтаксическом подходе. Показано, что этот подход отличается от способов построения языка, принятых в современной логике. The article analyzes the source of logical paradoxes Bertrand Russell identified in the foundations of mathematics proposed by Gottlob Frege. Russell proposed self-reference of expressions as the source of paradoxes. To solve paradoxes, he developed the simple and ramified theory of types. Ontological presuppositions are well substantiated for his theory; they depend on semantic, but not syntactic, preference. Contemporary approaches in symbolical logic prefer syntactic methods. But Wittgenstein’s approach in his Tractatus Logico-Philosophicus is more interesting, especially from the perspective of his picture theory of statements.

Paradoxes and Inconsistent Mathematics

Logical paradoxes – like the Liar, Russell's, and the Sorites – are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses “dialetheic paraconsistency” – a formal framework where some contradictions can be true without absurdity – as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly addresses a longstanding open question: how much standard mathematics can paraconsistency capture? The guiding focus is on a more basic question, of why there are paradoxes. Details underscore a simple philosophical claim: that paradoxes are found in the ordinary, and that is what makes them so extraordinary.

A History of Philosophy and Psychology in Finland, 1809-1917

Includes:PART ONEAN OUTLINE OF FINNISH PHILOSOPHY BEFORE 18091.1. Prehistory1.2. Christianity Arrives1.3. The Academy of TurkuPART TWOFROM IDEALISM TO NATIONALISTIC AND LIBERAL CONSTRUCTIONS OF THE STATE2. 1. Varieties of ldealism: Franzén, Lagus, and Hartman2.2. Romanticism in Turku: Bergbom, Ottelin, Arwidsson, and Hwasser2.3. J. J. Tengström's Teaching of Hegel's Philosophy of Right2.4. J. V. Snellman's Career: Philosopher, Journalist, Senator2.5. Academic Freedom and Bildung2.6. Personality, Spirit, and Nation2.7. Snellman’s Doctrine of the State2.8. Fight for Enlightenment: Wilhelm Bolin2.9. Ways out of HegelPART THREEENTANGLEMENT OF PHILOSOPHY AND EMPIRICAL RESEARCH3.1. From Philosophical Psychology to Experimental Psychology3.2. Logic and Psychology in Lotze's Spirit: Thiodolf Rein3.3. Psychological Laboratory: Hjalmar Neiglick3.4. Philosophy of History: Arvi Grotenfelt3.5. Psychological Ethics and Social Anthropology: Edward 'Westermarck3.6. A Promethean Philosopher and Psychologist: Rolf Lagerborg3.7. Logical Paradoxes: Hjalmar Magnus Eklund3.8. Modern Philosophical Currents: Eino Kaila3.9. Debates on Mach3.10. New Trends in Psychology in the 1910s3.11. Philosophy and Public Affairs: The Philosophical Society

Paradoxes of Nature

Parmenides’s insinuation of an unchanging universe is assertively supported by Zeno with various logical paradoxes that question the very nature of plurality, space, time, and the reality of apparent motion. The dichotomy is his most famous paradox. To begin a trip, say, from here to the door, a traveler must travel the first half of it, but before she does that she must travel half of the first half, and in fact half of that, ad infinitum. Since there will always exist a smaller first half to be traveled first, Zeno questions whether a traveler can ever even start a trip. Zeno’s analysis is logical; on the other hand, things everywhere appear to be moving. Hence, either Zeno’s reasoning is wrong or appearances are deceptive. Empowered by the uncertainty principle of quantum mechanics, it will be argued that, at best, the phenomenon of motion is experimentally unverifiable!

Paradoxes in scientific cognition and nonclassical logics

The subject of this research is the scientific paradoxes and such means for its resolution as nonclassical logics. The author defends a thesis that paradoxes often stimulate the scientific development. It is demonstrated that most vividly the problem of paradoxes manifested in crises in the fundamentals of mathematics; the attempts for its resolution in many ways contributes to the emergence of nonclassical logics. It is substantiated that nonclassical logics helped to resolve and explain the paraded occurring in scientific cognition. Comparative analysis is conducted on the capabilities of  three-valued “quantum logics” of Garrett Birkhoff and John von Neumann and “logics of complementarity” of Hans Reichenbach. Potential of the three-valued logics of D. Bochvar and nonclassical systems of A. Zinoviev in resolution and explanation of logical paradoxes, as well as importance of temporary logics of G. H. Wright for the philosophy of science is revealed. Special attention is paid  to the paraconsistent logics. The author determines two points of view in understanding of their essence and value for science and philosophy, which juxtaposition shows that none of them fully complies with the actual state of affairs. The main conclusion consists in the statement that paradoxes of scientific cognition should not be assessed just negatively; they also carry a positive meaning: detection of paradoxes in the theory testifies to the need for their elimination, more detailed research and stricter approach towards development of the theory, which in solution of this task can be accomplished by nonclassical logics.