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Mechanical theorem proving in geometries

The syllabi, the textbooks and the curriculum design in elementary logic, reasoning, argumentation and critical thinking are all currently ill-conceived and ill-executed. In a certain historical perspective, which is only slightly oversimplified, the same fate has befallen logic as an academic discipline has befallen ethics. Ethics began in the ancient Greece and more generally in the classical antiquity as a study of different forms of excellence. These types of excellence were not only moral in our sense, but included sundry forms of social, military and intellectual superiority. As Adkins (1960, pp. 32–33) sums up the Homeric sense of the word, to be a good or virtuous man, that is, 
to be agathos one must be brave, skillful, and successful in war and in peace; and one must possess the wealth and... the leisure which are at once the necessary conditions for the development of these skills and the natural reward of their successful employment.

Is Logic the Key to All Good Reasoning

Languages for the specification of software

Godel’s work from the beginning to his first substantive explorations in philosophy would to a significant extent be contextualized by, reactive to, and reflective of, Russell’s. Russell was the towering figure who set the stage for analytic philosophy in the first two decades of the 20 th Century; Godel insisted that his mathematical work was substantively motivated by his own philosophical outlook; and so it becomes especially pertinent to draw out and highlight the interconnections between the two. What follows is a narrative that focuses on the interplay of several arching motifs: Russell’s theory of types and Axiom of Reducibility; definability as analysis; Godel’s incompleteness theorems and constructible sets; and Russell’s and Godel’s respective construals of the nature of truth as a, if not the, philosophical problem. More specifi cally, Godel’s reflections on Russell’s so-called “multiple relation theory of judgment” (MRTJ), the theory of truth at work in the Principia Mathematica and later writings of Russell, especially his William James Lectures (1940), will be set forth and framed. This will allow us to draw out, in a provisional way, some of the philosophical signifi cance of Godel’s recently transcribed Max Phil notebooks. Russell’s MRTJ was an idiosyncratic version of the correspondence theory of truth, framed in terms of complex (non-dual) relations between judgers, the objects of their judgments, and their environment. Our view is that it was this specific theory of truth that preoccupied Godel by 1929, and again in 1942-1943, when he returned to reading Russell in earnest. We offer a narrative of influence, reaction, and confluence, of suggested thematic causalities discernible in Godel’s intellectual development. With the conviction that there is still much to be said about Godel vis-a-vis Russell, we forge a particular path through now well-known aspects of Godel’s work and thought, putting new weight on some lesser-known aspects. In particular, we give a nuanced reading of Godel’s mathematical platonism as emergent according to mathematical advances and, by his lights, increasingly indispensable. We also insist, especially given the evidence of the Max Phil notebooks, on the increasingly serious importance to Godel (as also evidently to Russell throughout) of philosophical questions of the most classical and fundamental kind concerning the relation of logic to grammar and to mathematics, of truth to perception, and these questions’ relevance to foundational questions about the very possibility of conceptual analysis. Others have weighed in here, in an increasingly informative and fine-grained way; our aim is primarily to broach the bearing of the

Gödel vis-à-vis Russell: Logic and set Theory to Philosophy

This article serves to present a large mathematical perspective and historical basis for the Axiom of Replacement as well as to affirm its importance as a central axiom of modern set theory.

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Wang Hao

Papers

Popular Lectures on Mathematical Logic.