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  1. Klaus Aehlig (2005). Induction and Inductive Definitions in Fragments of Second Order Arithmetic. Journal of Symbolic Logic 70 (4):1087 - 1107.
    A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae (...)
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  2. Katie Atkinson & Trevor J. M. Bench-Capon (2007). Practical Reasoning as Presumptive Argumentation Using Action-Based Alternating Transition Systems. Artificial Intelligence 171 (10-15):855-874.
    In this paper we describe an approach to practical reasoning, reasoning about what it is best for a particular agent to do in a given situation, based on presumptive justifications of action through the instantiation of an argument scheme, which is then subject to examination through a series of critical questions. We identify three particular aspects of practical reasoning which distinguish it from theoretical reasoning. We next provide an argument scheme and an associated set of critical questions which is able (...)
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  3. Arnold Beckmann (2002). Proving Consistency of Equational Theories in Bounded Arithmetic. Journal of Symbolic Logic 67 (1):279-296.
    We consider equational theories for functions defined via recursion involving equations between closed terms with natural rules based on recursive definitions of the function symbols. We show that consistency of such equational theories can be proved in the weak fragment of arithmetic S 1 2 . In particular this solves an open problem formulated by TAKEUTI (c.f. [5, p.5 problem 9.]).
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  4. John L. Bell (2004). Whole and Part in Mathematics. Axiomathes 14 (4):285-294.
    The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.
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  5. T. J. M. Bench-Capon (2012). Representing Popov V Hayashi with Dimensions and Factors. Artificial Intelligence and Law 20 (1):15-35.
    Modelling reasoning with legal cases has been a central concern of AI and Law since the 1980s. The approach which represents cases as factors and dimensions has been a central part of that work. In this paper I consider how several varieties of the approach can be applied to the interesting case of Popov v Hayashi. After briefly reviewing some of the key landmarks of the approach, the case is represented in terms of factors and dimensions, and further explored using (...)
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  6. Francesco Berto (2009). The Gödel Paradox and Wittgenstein's Reasons. Philosophia Mathematica 17 (2):208-219.
    An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...)
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  7. G. S. Ceĭtin (ed.) (1971). Five Papers on Logic and Foundations. Providence, R.I.,American Mathematical Society.
    Markov, A. A. On constructive mathematics.--Ceĭtin, G. S. Mean value theorems in constructive analysis.--Zaslavskiĭ, I. D. and Ceĭtlin, G. S. On singular coverings and properties of constructive functions connected with them.--Maslov, S. Ju. Certain properties of E. L. Post's apparatus of canonical calculi.--Zaslavskiĭ, I. D. Graph schemes with memory.
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  8. Snehashish Chakraverty (ed.) (2010). Proceedings of International Conference on Challenges and Applications of Mathematics in Science and Technology: Camist, January 11-13, 2010. [REVIEW] Macmillan Publishers India.
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  9. Rodney Coleman (1979). An Introduction to Mathematical Stereology. University of Aarhus, Dept. Of Theoretical Statistics, Institute of Mathematics.
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  10. D. Dantzig (1959). Review of G. Pólya, Mathematics and Plausible Reasoning, Vols. I and II. Synthese 11 (4):353-358.
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  11. N. G. de Bruijn (1995). On the Roles of Types in Mathematics. In Philippe De Groote (ed.), The Curry-Howard Isomorphism. Academia. 27-54.
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  12. Rafael de Clercq (2012). On Some Putative Graph-Theoretic Counterexamples to the Principle of the Identity of Indiscernibles. Synthese 187 (2):661-672.
    Recently, several authors have claimed to have found graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles. In this paper, I argue that their counterexamples presuppose a certain view of what unlabeled graphs are, and that this view is optional at best.
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  13. S. S. Demidov (1988). On an Early History of the Moscow School of Theory of Functions. Philosophia Mathematica (1):29-35.
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  14. A. P. Ershov & Donald Ervin Knuth (eds.) (1981). Algorithms in Modern Mathematics and Computer Science: Proceedings, Urgench, Uzbek Ssr, September 16-22, 1979. Springer-Verlag.
  15. James Franklin (2006). Review of N. Wildberger, Divine Proportions: Rational Trigonometry to Universal. [REVIEW] Mathematical Intelligencer 28 (3):73-74.
    Reviews Wildberger's account of his rational trigonometry project, which argues for a simpler way of doing trigonometry that avoids irrationals.
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  16. Alexandre Guay & Brian Hepburn (2009). Symmetry and its Formalisms: Mathematical Aspects. Philosophy of Science 76 (2):160-178.
    This article explores the relation between the concept of symmetry and its formalisms. The standard view among philosophers and physicists is that symmetry is completely formalized by mathematical groups. For some mathematicians however, the groupoid is a competing and more general formalism. An analysis of symmetry that justifies this extension has not been adequately spelled out. After a brief explication of how groups, equivalence, and symmetries classes are related, we show that, while it’s true in some instances that groups are (...)
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  17. Philip E. B. Jourdain (1915). The Purely Ordinal Conceptions of Mathematics and Their Significance for Mathematical Physics. The Monist 25 (1):140-144.
  18. Adam Kolany (1997). Consequence Operations Based on Hypergraph Satisfiability. Studia Logica 58 (2):261-272.
    Four consequence operators based on hypergraph satisfiability are defined. Their properties are explored and interconnections are displayed. Finally their relation to the case of the Classical Propositional Calculus is shown.
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  19. P. J. M. (1966). Colloquium on the Foundations of Mathematics, Mathematical Machines and Their Applications. Review of Metaphysics 19 (4):821-821.
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  20. V. McGee (2011). Francesco Berto. There's Something About Godel. Malden, Mass., And Oxford: Wiley-Blackwell, 2009. Isbn 978-1-4051-9766-3 (Hbk); 978-1-4051-9767-0 (Pbk). Pp. XX + 233. English Translation of Tutti Pazzi Per Godel! (Rome: Gius, Laterza & Figli, 2008). [REVIEW] Philosophia Mathematica 19 (3):367-369.
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  21. John Mighton (2004). Review of D. Aeheson, 1089 and All That. [REVIEW] Mathematical Intelligencer 26 (2):70.
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  22. Asutosh Mookerjee (2009). Mathematical Contributions of Sir Asutosh Mookerjee: Contemporaneity and Relevance. Jijnasa Pub. House.
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  23. Brunetto Piochi (1983). Logical Matrices and Non-Structural Consequence Operators. Studia Logica 42 (1):33 - 42.
    In the present paper, we study some properties of matrices for non-structural consequence operators. These matrices were introduced in a former work (see [3]). In sections 1. and 2., general definitions and theorems are recalled; in section 3. a correspondence is studied, among our matrices and Wójcicki's ones for structural operators. In section 4. a theorem is given about operators, induced by submatrices or epimorphic images, or quotient matrices of a given one.Such matrices are used to characterize lattices of non-structural (...)
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  24. Burkard Polster (2004). Q. Walker & Co..
    Q.E.D. presents some of the most famous mathematical proofs in a charming book that will appeal to nonmathematicians and math experts alike. Grasp in an instant why Pythagoras’s theorem must be correct. Follow the ancient Chinese proof of the volume formula for the frustrating frustum, and Archimedes’ method for finding the volume of a sphere. Discover the secrets of pi and why, contrary to popular belief, squaring the circle really is possible. Study the subtle art of mathematical domino tumbling, and (...)
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  25. Michael D. Potter (2000). Reason's Nearest Kin: Philosophies of Arithmetic From Kant to Carnap. Oxford University Press.
    This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.
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  26. David A. Rabson, John F. Huesman & Benji N. Fisher (2003). Cohomology for Anyone. Foundations of Physics 33 (12):1769-1796.
    Crystallography has proven a rich source of ideas over several centuries. Among the many ways of looking at space groups, N. David Mermin has pioneered the Fourier-space approach. Recently, we have supplemented this approach with methods borrowed from algebraic topology. We now show what topology, which studies global properties of manifolds, has to do with crystallography. No mathematics is assumed beyond what the typical physics or crystallography student will have seen of group theory; in particular, the reader need not have (...)
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  27. Ronald Rensink, A New Proof of the NP-Completeness of Visual Match.
    A new proof is presented of Tsotsos' result that the VISUAL MATCH problem is NP-complete when no (high-level) constraints are imposed on the search space. Like the proof given by Tsotsos, it is based on the polynomial reduction of the NP-complete problem KNAPSACK to VISUAL MATCH. Tsotsos' proof, however, involves limited-precision real numbers, which introduces an extra degree of complexity to his treatment. The reduction of KNAPSACK to VISUAL MATCH presented here makes no use of limited-precision numbers, leading to a (...)
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  28. Moses Richardson (1958). Fundamentals of Mathematics. New York, Macmillan.
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  29. Giovanni Sambin & Jan M. Smith (eds.) (1998). Twenty-Five Years of Constructive Type Theory: Proceedings of a Congress Held in Venice, October 1995. Oxford University Press.
    This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Lof over the last twenty-five years.
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  30. Dennis Sentilles (1975). A Bridge to Advanced Mathematics. Baltimore,Williams & Wilkins.
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  31. Stewart Shapiro (ed.) (1985). Intentional Mathematics. Sole Distributors for the U.S.A. And Canada, Elsevier Science Pub. Co..
    Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.
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  32. David Sherry (2011). Thermoscopes, Thermometers, and the Foundations of Measurement. Studies in History and Philosophy of Science Part A 42 (4):509-524.
    Psychologists debate whether mental attributes can be quantified or whether they admit only qualitative comparisons of more and less. Their disagreement is not merely terminological, for it bears upon the permissibility of various statistical techniques. This article contributes to the discussion in two stages. First it explains how temperature, which was originally a qualitative concept, came to occupy its position as an unquestionably quantitative concept (§§1–4). Specifically, it lays out the circumstances in which thermometers, which register quantitative (or cardinal) differences, (...)
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  33. David Sherry (2009). Reason, Habit, and Applied Mathematics. Hume Studies 35 (1/2):57-85.
    Hume describes the sciences as "noble entertainments" that are "proper food and nourishment" for reasonable beings (EHU 1.5-6; SBN 8).1 But mathematics, in particular, is more than noble entertainment; for millennia, agriculture, building, commerce, and other sciences have depended upon applying mathematics.2 In simpler cases, applied mathematics consists in inferring one matter of fact from another, say, the area of a floor from its length and width. In more sophisticated cases, applied mathematics consists in giving scientific theory a mathematical form (...)
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  34. David Sherry (1985). A Concordance for Wittgenstein's Remarks on the Foundations of Mathematics. History and Philosophy of Logic 6 (1):211-213.
  35. Stephen G. Simpson (2010). The Gödel Hierarchy and Reverse Mathematics. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
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  36. Kim Solin (2012). Dual Choice and Iteration in an Abstract Algebra of Action. Studia Logica 100 (3):607-630.
    This paper presents an abstract-algebraic formulation of action facilitating reasoning about two opposing agents. Two dual nondeterministic choice operators are formulated abstract-algebraically: angelic (or user) choice and demonic (or system) choice. Iteration operators are also defined. As an application, Hoare-style correctness rules are established by means of the algebra. A negation operator is also discussed.
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  37. Edward Russell Stabler (1953). An Introduction to Mathematical Thought. Cambridge, Mass., Addison-Wesley.
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  38. Mark Steiner (1995). Review of S. Sternberg, Group Theory and Physics. [REVIEW] Philosophia Mathematica 3 (3):313-316.
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  39. Ian Stewart & David Tall (1977). The Foundations of Mathematics. Oxford University Press.
    The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books.
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  40. David S. G. Stirling (2009). Mathematical Analysis and Proof. Horwood Pub..
    This fundamental and straightforward text addresses a weakness observed among present-day students, namely a lack of familiarity with formal proof. Beginning with the idea of mathematical proof and the need for it, associated technical and logical skills are developed with care and then brought to bear on the core material of analysis in such a lucid presentation that the development reads naturally and in a straightforward progression. Retaining the core text, the second edition has additional worked examples which users have (...)
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  41. Claes Strannegård, Fredrik Engström, Abdul Rahim Nizamani & Lance Rips (2013). Reasoning About Truth in First-Order Logic. Journal of Logic, Language and Information 22 (1):115-137.
    First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof systems when (...)
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  42. Neil Tennant (1995). Review of K. Devlin, Logic and Information. [REVIEW] Philosophia Mathematica 3 (2).
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  43. George J. Tourlakis (2012). Theory of Computation. Wiley.
    In addition, this book contains tools that, in principle, can search a set of algorithms to see whether a problem is solvable, or more specifically, if it can be solved by an algorithm whose computations are efficient.
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  44. Raymond Turner (2009). Computable Models. Springer.
    Raymond Turner first provides a logical framework for specification and the design of specification languages, then uses this framework to introduce and study ...
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  45. Ervin E. Underwood (1970). Quantitative Stereology. Reading, Mass.,Addison-Wesley Pub. Co..
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  46. Jean Paul van Bendegem (1999). Review of C. Mortensen, Inconsistent Mathematics. [REVIEW] Philosophia Mathematica 7 (2):202-212.
  47. Michał Walicki (2012). Introduction to Mathematical Logic. World Scientific.
    A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness and (...)
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  48. Kai F. Wehmeier (1996). Classical and Intuitionistic Models of Arithmetic. Notre Dame Journal of Formal Logic 37 (3):452-461.
    Given a classical theory T, a Kripke model K for the language L of T is called T-normal or locally PA just in case the classical L-structure attached to each node of K is a classical model of T. Van Dalen, Mulder, Krabbe, and Visser showed that Kripke models of Heyting Arithmetic (HA) over finite frames are locally PA, and that Kripke models of HA over frames ordered like the natural numbers contain infinitely many PA-nodes. We show that Kripke models (...)
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  49. Mark J. West (2012). Basic Stereology for Biologists and Neuroscientists. Cold Spring Harbor, New York.
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  50. Andrew Wohlgemuth (1990/2011). Introduction to Proof in Abstract Mathematics. Dover Publications.
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