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What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end
BERNAYS AND SET THEORY
"... Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Göd ..."
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Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the twovolume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of firstorder logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent reevaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higherorder reflection principles, and produced a stream of
This is a preliminary version of a review which will appear in Research in the History of Economic Thought and Methodology, Volume 24A (2006), edited by
"... formalist revolution or what happened to orthodox economics after World War II?”, the specter of mathematician David Hilbert has haunted economists’s discussions of formalization and axiomatization. Briefly, if one looks upon formalized economics, or formalism, with a loathing built on fear (or a fe ..."
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formalist revolution or what happened to orthodox economics after World War II?”, the specter of mathematician David Hilbert has haunted economists’s discussions of formalization and axiomatization. Briefly, if one looks upon formalized economics, or formalism, with a loathing built on fear (or a fear based on loathing), one demonizes Hilbert since the philosophical notion of formalism, in the history of metamathematics, is usually associated with Hilbert. That is, in the history of philosophy of mathematics, there appears to be a distinction between formalists and empiricists, on the nature of mathematical objects. The ontological reflections in that arcane literature has Hilbert holding the position that mathematics is simply a formal system, and its symbols are simply marks on paper. It is an easy step then to look for traces of David Hilbert in the development of mathematical economics in the 20 th century. Seek, and ye shall find, and critics of mainstream economics have found Hilbertian connections in Vienna with Menger’s seminar. As a result Hilbert gets caught up in the origin stories of general equilibrium theory which lead all the way to von Neumann and the development of game theory. From Vienna and general equilibrium theory it is a short step, though a false step, to have Hilbert as the spiritual advisor to the Cowles Commission and thence to ArrowDebreu. From there of course one can launch tirades about the formalist revolution in economics and have Hilbert bearing some of the blame for a misguided economics.