Areas of Mathematics

Edited by Nemi Boris Pelgrom (Ludwig Maximilians Universität, München)
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  1. 1 — Consistency and faithful interpretations.S. Feferman, G. Kreisel & S. Orey - 1962 - Archive for Mathematical Logic 6 (1-2):52-63.
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  2. Free Choice Sequences: A Temporal Interpretation Compatible with Acceptance of Classical Mathematics.Saul Kripke - 2019 - Indagationes Mathematicae 30 (3):492-499.
    This paper sketches a way of supplementing classical mathematics with a motivation for a Brouwerian theory of free choice sequences. The idea is that time is unending, i.e. that one can never come to an end of it, but also indeterminate, so that in a branching time model only one branch represents the ‘actual’ one. The branching can be random or subject to various restrictions imposed by the creating subject. The fact that the underlying mathematics is classical makes such perhaps (...)
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  3. Retreat from Non-Being: Graham Priest, Towards Non-Being: The Logic and Metaphysics of Intentionality, Oxford: Clarendon Press, 2005, pp. xv + 190, £30. [REVIEW]Terry Horgan - 2006 - Australasian Journal of Philosophy 84 (4):615-627.
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  4. The Unseen Déjà-Vu: From Erkki Huhtamo’s Topoi to Ken Jacobs’ Remakes: Commentary to Edwin Carels “Revisiting Tom Tom: Performative anamnesis and autonomous vision in Ken Jacobs’ appropriations of Tom Tom the Piper’s Son”.Wanda Strauven - 2018 - Foundations of Science 23 (2):231-236.
    This commentary on Edwin Carels’ essay “Revisiting Tom Tom: Performative anamnesis and autonomous vision in Ken Jacobs’ appropriations of Tom Tom the Piper’s Son” broadens up the media-archaeological framework in which Carels places his text. Notions such as Huhtamo’s topos and Zielinski’s “deep time” are brought into the discussion in order to point out the difficulty to see what there is to see and to question the position of the viewer in front of experimental films like Tom Tom the Piper’s (...)
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  5. Reflections on Relativism: From Momentous Tautology to Seductive Contradiction.Susan Haack - 1996 - Philosophical Perspectives 10:297-315.
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  6. Critique de la Mesure. G. Beneze.Karol Meisels - 1939 - Philosophy of Science 6 (2):258-259.
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  7. Ontology and the Vicious-Circle Principle. Charles S. Chihara. [REVIEW]Fred Wilson - 1975 - Philosophy of Science 42 (3):339-341.
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  8. Experience and the Analytic. Alan Pasch. [REVIEW]Harry G. Frankfurt - 1960 - Philosophy of Science 27 (2):222-223.
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  9. Oneirics and Psychosomatics. Rolf Loehrich. McHenry, Ill.: The Compass Press, 1953. Pp. xiv, 157. $6.00.Calvin S. Hall - 1955 - Philosophy of Science 22 (1):69-69.
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  10. From Euclid to Eddington. E. Whittaker. [REVIEW]D. J. Struik - 1951 - Philosophy of Science 18 (1):88-91.
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  11. Logical Consequence for Nominalists.Marcus Rossberg & Daniel Cohnitz - 2009 - Theoria 24 (2):147-168.
    It is often claimed that nominalistic programmes to reconstruct mathematics fail, since they will at some point involve the notion of logical consequence which is unavailable to the nominalist. In this paper we use an idea of Goodman and Quine to develop a nominalistically acceptable explication of logical consequence.
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  12. Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. Providence, RI: American Mathematical Society, 2014. Pp. xi + 441. ISBN 978-1-4704-1493-1. £79.95. [REVIEW]Michael J. Barany - 2016 - British Journal for the History of Science 49 (1):140-141.
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  13. The classification of $${\mathbb {Z}}p$$ Z p -modules with partial decomposition bases in $$L{\infty \omega }$$ L ∞ ω.Carol Jacoby & Peter Loth - 2016 - Archive for Mathematical Logic 55 (7-8):939-954.
    Ulm’s Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to L∞ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\infty \omega }$$\end{document}-equivalence. In this paper, we extend this classification to a class of mixed Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_p$$\end{document}-modules which includes all Warfield modules and is closed under L∞ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...)
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  14. An induction principle over real numbers.Assia Mahboubi - 2017 - Archive for Mathematical Logic 56 (1-2):43-49.
    We give a constructive proof of the open induction principle on real numbers, using bar induction and enumerative open sets. We comment the algorithmic content of this result.
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  15. Univalent foundations as structuralist foundations.Dimitris Tsementzis - 2017 - Synthese 194 (9):3583-3617.
    The Univalent Foundations of Mathematics provide not only an entirely non-Cantorian conception of the basic objects of mathematics but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a pragmatist reading of the structuralist thesis in the philosophy of mathematics, which I will use to define a criterion that a formal (...)
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  16. X*—On The Logic Of Relations.Stephan Körner - 1977 - Proceedings of the Aristotelian Society 77 (1):149-164.
    Stephan Körner; X*—On The Logic Of Relations, Proceedings of the Aristotelian Society, Volume 77, Issue 1, 1 June 1977, Pages 149–164, https://doi.org/10.1093/a.
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  17. A Fixed Point, A Point of Interruption.Mark Hewson - 2006 - Cosmos and History : The Journal of Natural and Social Philosophy 2 (1-2):376-379.
    A review of Alain Badiou, emInfinite Thought: Truth and the Return to Philosophy/em, ed. and trans. Justin Clemens and Oliver Feltham, New Edition, London, Continuum, 2005. ISBN: 0826479294.br /.
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  18. On Axiom Systems of Słupecki for the Functionally Complete Three-Valued Logic.Mateusz Radzki - 2017 - Axiomathes 27 (4):403-415.
    The article concerns two axiom systems of Słupecki for the functionally complete three-valued propositional logic: W1–W6 and A1–A9. The article proves that both of them are inadequate—W1–W6 is semantically incomplete, on the other hand, A1–A9 governs a functionally incomplete calculus, and thus, it cannot be a semantically complete axiom system for the functionally complete three-valued logic.
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  19. Exploring the tractability border in epistemic tasks.Cédric Dégremont, Lena Kurzen & Jakub Szymanik - 2014 - Synthese 191 (3):371-408.
    We analyse the computational complexity of comparing informational structures. Intuitively, we study the complexity of deciding queries such as the following: Is Alice’s epistemic information strictly coarser than Bob’s? Do Alice and Bob have the same knowledge about each other’s knowledge? Is it possible to manipulate Alice in a way that she will have the same beliefs as Bob? The results show that these problems lie on both sides of the border between tractability (P) and intractability (NP-hard). In particular, we (...)
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  20. Infinite lotteries, large and small sets.Luc Lauwers - 2017 - Synthese 194 (6):2203-2209.
    One result of this note is about the nonconstructivity of countably infinite lotteries: even if we impose very weak conditions on the assignment of probabilities to subsets of natural numbers we cannot prove the existence of such assignments constructively, i.e., without something such as the axiom of choice. This is a corollary to a more general theorem about large-small filters, a concept that extends the concept of free ultrafilters. The main theorem is that proving the existence of large-small filters requires (...)
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  21. On the Use of Hilbert's ε-Operator in Scientific Theories.Rudolf Carnap - 1961 - In Yehoshua Bar-Hillel, E. Poznanski, M. Rabin & Abraham Robinson (eds.), Essays on the Foundations of Mathematics: Dedicated to A. A. Fraenkel on His 70th Anniversary. Amsterdam: North-Holland Publishing Company. pp. 156--164.
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  22. Set Partitions and the Meaning of the Same.R. Zuber - 2017 - Journal of Logic, Language and Information 26 (1):1-20.
    It is shown that the notion of the partition of a set can be used to describe in a uniform way the meaning of the expression the same, in its basic uses in transitive and ditransitive sentences. Some formal properties of the function denoted by the same, which follow from such a description are indicated. These properties indicate similarities and differences between functions denoted by the same and generalised quantifiers.
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  23. Can the Law of Contradiction be Stated?Slbajiban Slbajiban - 1960 - Analysis 21 (5):101.
    Slbajiban; Can the Law of Contradiction be Stated?, Analysis, Volume 21, Issue 5, 1 April 1961, Pages 101–105, https://doi.org/10.1093/analys/21.5.101.
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  24. On Wright’s Inductive Definition of Coherence Truth for Arithmetic.Jeffrey Ketland - 2003 - Analysis 63 (1):6-15.
    In “Truth – A Traditional Debate Reviewed”, Crispin Wright proposed an inductive definition of “coherence truth” for arithmetic relative to an arithmetic base theory B. Wright’s definition is in fact a notational variant of the usual Tarskian inductive definition, except for the basis clause for atomic sentences. This paper provides a model-theoretic characterization of the resulting sets of sentences "cohering" with a given base theory B. These sets are denoted WB. Roughly, if B satisfies a certain minimal condition, then WB (...)
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  25. Adequacy and consistency: A second reply to dr bar-Hillel.K. R. Popper - 1956 - British Journal for the Philosophy of Science 7 (27):249-256.
  26. The Foundations of Mathematics in the Theory of Sets. [REVIEW]Roy T. Cook - 2003 - British Journal for the Philosophy of Science 54 (2):347-352.
  27. La natura e il futuro della filosofia.Donald Gillies - 2003 - British Journal for the Philosophy of Science 54 (3):501-507.
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  28. What is a Higher Level Set?Dimitris Tsementzis - 2016 - Philosophia Mathematica:nkw032.
    Structuralist foundations of mathematics aim for an ‘invariant’ conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. I argue in favor of the former over the latter. First, I explain why to choose between them we need to ask the question of what is the correct ‘categorified’ version of a set. Second, I argue in favor of groupoids over categories as ‘categorified’ sets by introducing a pre-formal understanding of groupoids as (...)
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  29. Reply to Feferman, Koellner, Tait, and Sieg.Charles Parsons - 2016 - Journal of Philosophy 113 (5/6):286-307.
    I comment on Feferman’s views on set theory, in particular criticizing a priori arguments claiming that the continuum hypothesis has no determinate truth value and commenting on his responses to my paper on his skepticism about set theory. I respond to criticisms of his of the structuralism that I have advocated and comment on his view of proof theory. On Koellner’s paper, I register little disagreement but note a difference of sympathy about views such as constructivism. On Tait’s paper, I (...)
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  30. Frege Meets Aristotle: Points as Abstracts.Stewart Shapiro & Geoffrey Hellman - 2015 - Philosophia Mathematica:nkv021.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...)
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  31. Philosophy of the Matrix.A. C. Paseau - 2017 - Philosophia Mathematica 25 (2):246-267.
    A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.
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  32. If P, then Q: Conditionals and the Foundations of Reasoning.Vann Mcgee - 1992 - Philosophy and Phenomenological Research 53 (1):239-242.
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  33. The Logic Of Representation.Gabriella Ujlaki - 1993 - British Journal of Aesthetics 33 (2):121-131.
  34. Towards Non‐Being: The Logic and Metaphysics of Intentionality ‐ By Graham Priest.Brandon C. Look - 2007 - Philosophical Books 48 (1):83-84.
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  35. Die Rolle der Analogie in Bernays' Philosophie der Mathematik.Andrès R. Raggio - 1978 - Dialectica 32 (3-4):201-207.
    Bernays' idea of the power-set is neither univocal nor ambiguous. according to its domain of application it has a different mathematical meaning in spite of its identical logical definition. in his last axiomatisation of set-theory bernays' introduced a very powerful axiom stating that every property of the universe of all sets is mirrored by some set. this is a most peculiar application of the general--philosophical and theological--principle of analogy as a key principle in the foundations of mathematics.
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  36. Theorie der Konsequenzoperationen und Grundbegriffe der Logik.Christian Wallmann - 2011 - Kriterion - Journal of Philosophy 25 (1):64-77.
    We give an elementary introduction into the theory of consequence operations. We proof some elementary results concerning basic notions of logic like tautology, consistency, independence and completeness. We show in particular that every finite axiomatizable set is independent axiomatizable and that every consistent set has relative to a finitary consequence operation a maximal consistent extension. Finally we provide an abstract semantics for consequence operations.
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  37. Implication and Linear Inference.J. E. Creighton - 1921 - Journal of Philosophy 18 (2):52-54.
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  38. Topics in Philosophical Logic.Marshall Swain - 1971 - Journal of Philosophy 68 (10):319-324.
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  39. A class of {Sigma {3}^{0}} modular lattices embeddable as principal filters in {mathcal{L}^{ast }(V{infty })}.Rumen Dimitrov - 2008 - Archive for Mathematical Logic 47 (2):111-132.
    Let I 0 be a a computable basis of the fully effective vector space V ∞ over the computable field F. Let I be a quasimaximal subset of I 0 that is the intersection of n maximal subsets of the same 1-degree up to *. We prove that the principal filter ${\mathcal{L}^{\ast}(V,\uparrow )}$ of V = cl(I) is isomorphic to the lattice ${\mathcal{L}(n, \overline{F})}$ of subspaces of an n-dimensional space over ${\overline{F}}$ , a ${\Sigma _{3}^{0}}$ extension of F. As a (...)
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  40. The principia: mathematical principles of natural philosophy.Ian G. Stewart - 2004 - Studies in History and Philosophy of Science Part A 35 (3):665-667.
  41. La logique de la contradiction.Fr Paulhan - 1910 - Revue Philosophique de la France Et de l'Etranger 69 (3):275-303.
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  42. The Philosophy of Rhetoric.George Campbell, William Creech, Thomas Cadell, W. Davies & George Ramsay and Company - 1808 - Printed by George Ramsay & Co. For William Creech, Edinburgh; and T. Cadell and W. Davies, London.
    The Philosophy of Rhetoric is widely regarded as the most important work of a theory of rhetoric produced in the 18th century. Campbell's work engages such themes in an attempt to formulate a universal theory of human communication. Campbell attempts to develop his theory by discovering deep principles in human nature that account for all instances and kinds of human communication. He seeks to derive all communication principles and processes empirically. In addition, all statements in discourse that have to do (...)
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  43. How I See Philosophy.Friedrich Waismann & Rom Harré - 1968 - New York: Palgrave-Macmillan.
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  44. Rhetoric Is a Counterpart of Dialectic (E retorike estin antistrophos te "dialektike".Brad McAdon - 2001 - Philosophy and Rhetoric 34 (2):113-150.
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  45. Are They Class-names?J. M. Hinton - 1982 - Philosophy 57 (219):27-50.
    We often, in effect, take it for granted that some word or phrase is what is called ‘the name of a class, be that class empty or non-empty’. We do so whenever in effect we either wonder about, or mean to be taking a view on, the number of members a certain suppositious class has, on the ultrasimple number scale: ‘None, more than none’.
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  46. Gregory Landini. Zermelo and Russell’s Paradox: Is There a Universal Set?: Correction Notice.Gregory Landini - 2014 - Philosophia Mathematica 22 (1):142-142.
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  47. What logic should we think with?R. M. Sainsbury - 2002 - Royal Institute of Philosophy Supplement 51:1-17.
    Logic ought to guide our thinking. It is better, more rational, more intelligent to think logically than to think illogically. Illogical thought leads to bad judgment and error. In any case, if logic had no role to play as a guide to thought, why should we bother with it?The somewhat naïve opinions of the previous paragraph are subject to attack from many sides. It may be objected that an activity does not count as thinking at all unless it is at (...)
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  48. Randomization, Persuasiveness and Rigor in Proofs.Catherine Womach & Matrin Farach - 2003 - Synthese 134 (1):71-84.
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  49. Paraconsistency And Dialogue Logic Critical Examination And Further Explorations.Jean Paul Van Bendegem - 2001 - Synthese 127 (1):35-55.
    The first part of this paper presents asympathetic and critical examination of the approachof Shahid Rahman and Walter Carnielli, as presented intheir paper The Dialogical Approach toParaconsistency. In the second part, possibleextensions are presented and evaluated: (a) top-downanalysis of a dialogue situation versus bottom-up, (b)the specific role of ambiguities and how to deal withthem, and (c) the problem of common knowledge andbackground knowledge in dialogues. In the third part,I claim that dialogue logic is the best-suitedinstrument to analyse paradoxes of the (...)
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  50. On elementary theories of some lattices of alpha-recursively enumerable sets.Mannel Lerman - 1978 - Annals of Mathematical Logic 14 (3):227.
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