This evenhanded treatment addresses the decades-old dispute among probability theorists, asserting that both statistical and inductive probabilities may be treated as sentence-theoretic measurements, and that the latter qualify as estimates of the former. Beginning with a survey of the essentials of sentence theory and of set theory, the author examines statistical probabilities, showing that statistical probabilities may be passed on to sentences, and thereby qualify as truth-values. An exploration of inductive probabilities follows, demonstrating their reinterpretation as estimates of truth-values. Each (...) chapter is preceded by a summary of its contents. Illustrations and footnotes elucidate definitions, theorems, and technicalities. 1962 ed. (shrink)

The more traditional approaches to the history and philosophy of science and technology continue as well, and probably will continue as long as there are skillful practitioners such as Carl Hempel, Ernest Nagel, and th~ir students. Finally, there are still other approaches that address some of the technical problems arising when we try to provide an account of belief and of rational choice. - These include efforts to provide logical frameworks within which we can make sense of these notions. This (...) series will attempt to bring together work from all of these approaches to the history and philosophy of science and technology in the belief that each has something to add to our understanding. The volumes of this series have emerged either from lectures given by authors while they served as honorary visiting professors at the City College of New York or from conferences sponsored by that institution. The City College Program in the History and Philosophy of Science and Technology oversees and directs these lectures and conferences with the financial aid of the Association for Philosophy of Science, Psychotheraphy, and Ethics. MARTIN TAMNY RAPHAEL STERN PREFACE The papers in this collection stem largely from the conference 'Foun dations: Logic, Language, and Mathematics' held at the Graduate Center of the City University of New York on 14-15 November 1980. (shrink)

DEALING INITIALLY WITH QC, THE STANDARD QUANTIFICATIONAL CALCULUS OF ORDER ONE, THE AUTHOR COMMENTS ON A SHORTCOMING, REPORTED IN 1956 BY MONTAGUE AND HENKIN, IN CHURCH'S ACCOUNT OF A PROOF FROM HYPOTHESES, AND SKETCHES THREE WAYS OF RIGHTING THINGS. THE THIRD, WHICH EXPLOITS A TRICK OF FITCH'S, IS THE SIMPLEST OF THE THREE. THE AUTHOR INVESTIGATES IT SOME, SUPPLYING FRESH PROOF OF UGT, THE UNIVERSAL GENERALIZATION THEOREM. THE PROOF HOLDS GOOD AS ONE PASSES FROM QC TO QC asterisk , THE (...) PRESUPPOSITION-FREE VARIANT OF QC. TURNING NEXT TO QC subscript = , THE STANDARD QUANTIFICATIONAL CALCULUS OF ORDER ONE WITH '=', AND TO THE PRESUPPOSITION-FREE VARIANT QC subscript = OF QC subscript = , THE AUTHOR NEXT ESTABLISHES THE LEMMAS NEEDED THERE TO PROVE UGT. THAT, GIVEN FITCH'S ACCOUNT OF A PROOF FROM HYPOTHESES, UGT HOLDS FOR QC subscript = WAS ARGUED IN LEBLANC'S "TRUTH-VALUE SEMANTICS", BUT THE PROOF WAS IN ERROR. (shrink)

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability functions, in particular, an exhaustive account of (...) probability functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels. (shrink)

Provability, Computability and Reflection.