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  1. An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities.George Boole - 1854 - London, England: Dover Constable.
    Self-taught mathematician and father of Boolean algebra, George Boole (1815-1864) published An Investigation of the Laws of Thought in 1854. In this highly original investigation of the fundamental laws of human reasoning, a sequel to ideas he had explored in earlier writings, Boole uses the symbolic language of mathematics to establish a method to examine the nature of the human mind using logic and the theory of probabilities. Boole considers language not just as a mode of expression, but as a (...)
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  • Category Theory.Steve Awodey - 2006 - Oxford, England: Oxford University Press.
    A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems, as well as numerous examples and exercises.
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  • Philosophy of Mathematics and Natural Science.Hermann Weyl - 1949 - Princeton University Press.
    This is a book that no one but Weyl could have written--and, indeed, no one has written anything quite like it since.
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  • Modal Quantum Theory.Benjamin Schumacher & Michael D. Westmoreland - 2012 - Foundations of Physics 42 (7):918-925.
    We present a discrete model theory similar in structure to ordinary quantum mechanics, but based on a finite field instead of complex amplitudes. The interpretation of this theory involves only the “modal” concepts of possibility and necessity rather than quantitative probability measures. Despite its simplicity, our model theory includes entangled states and has versions of both Bell’s theorem and the no cloning theorem.
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  • Categories for the Working Mathematician.Saunders Maclane - 1971 - Springer.
    Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...)
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  • Quantum Computation and Quantum Information.Michael A. Nielsen, Isaac L. Chuang & Isaac L.. Chuang - 2000 - Cambridge University Press.
    First-ever comprehensive introduction to the major new subject of quantum computing and quantum information.
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  • An Introduction to Partition Logic.David Ellerman - 2014 - Logic Journal of the IGPL 22 (1):94-125.
    Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual (...)
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  • The Structure and Interpretation of Quantum Mechanics.R. I. G. Hughes - 1989 - Harvard University Press.
    R.I.G Hughes offers the first detailed and accessible analysis of the Hilbert-space models used in quantum theory and explains why they are so successful.
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  • Quantum Mechanics Over Sets: A Pedagogical Model with Non-Commutative Finite Probability Theory as its Quantum Probability Calculus.David Ellerman - 2017 - Synthese (12):4863-4896.
    This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...)
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  • The Logic of Partitions: Introduction to the Dual of the Logic of Subsets: The Logic of Partitions.David Ellerman - 2010 - Review of Symbolic Logic 3 (2):287-350.
    Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of (...)
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  • Quantum Mechanics Over Sets.David Ellerman - forthcoming - Synthese.
    This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...)
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  • An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities.Alonzo Church - 1951 - Journal of Symbolic Logic 16 (3):224-225.
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  • New Pathways in Science.Arthur Eddington - 1935 - Philosophy 10 (40):483-485.
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  • The Structure and Interpretation of Quantum Mechanics.R. I. G. Hughes - 1992 - Tijdschrift Voor Filosofie 54 (4):735-736.
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  • Category Theory.S. Awodey - 2007 - Bulletin of Symbolic Logic 13 (3):371-372.
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  • The Principles of Quantum Mechanics.P. A. M. Dirac - 1936 - Revue de Métaphysique et de Morale 43 (2):5-5.
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  • The Structure and Interpretation of Quantum Mechanics.R. I. G. Hughes, James T. Cushing & Ernan Mcmullin - 1991 - Synthese 86 (1):99-122.
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  • Philosophy of Mathematics and Natural Science.H. Weyl & Olaf Helmar - 1951 - Science and Society 15 (1):85-88.
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  • The Problem of Hidden Variables in Quantum Mechanics.Simon Kochen & E. P. Specker - 1967 - Journal of Mathematics and Mechanics 17:59--87.
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  • Theory of Equivalence Relations.Oystein Ore - 1943 - Journal of Symbolic Logic 8 (1):55-56.
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