Search results for 'Goedel Theorem' (try it on Scholar)

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  1. I. Aimonetto (1993). Goedel Theorem of Incompleteness. Filosofia 44 (1):113-136.
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  2. I. Aimonetto (1988). The Foundations of the Goedel Theorem-From Peano to Frege and Russell. Filosofia 39 (3):231-249.
     
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  3.  13
    Douglas S. Robertson (2000). Goedel's Theorem, the Theory of Everything, and the Future of Science and Mathematics. Complexity 5 (5):22-27.
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  4.  6
    Robert A. Schultz (1980). What Could Self-Reflexiveness Be? Or Goedel’s Theorem Goes to Hollywood and Discovers That It’s All Done with Mirrors. Semiotica 30 (1-2).
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  5. Geoffrey Hellman (1981). How to Goedel a Frege-Russell: Goedel's Incompleteness Theorem. Noûs 15:451-68.
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  6. Michael Detlefsen (1976). The Importance of Goedel's Second Incompleteness Theorem for the Foundations of Mathematics. Dissertation, The Johns Hopkins University
     
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  7. U. Fidelman (1999). Goedel's Theorem and Models of the Brain: Possible Hemispheric Basis for Kant's Psychological Ideas. Journal of Mind and Behavior 20 (1):43-56.
    Penrose proved that a computational or formalizable theory of the brainís cognitive functioning is impossible, but suggested that a physical non-computational and non-formalizable one may be viable. Arguments as to why Penroseís program is unrealizable are presented. The main argument is that a non-formalizable theory should be verbal. However, verbal paradoxes based on Cantorís diagonal processes show the impossibility of a consistent verbal theory of the brain comprising its arithmetical cognition. It is suggested that comprehensive theories of the human brain (...)
     
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  8. F. RivettiBarbo (1996). A Surreptitious Change in the Designation of a Term: The Foundation of Goedel's Theorem of the Non-Demonstrability of Non-Contradictoriness-A New Metalinguistic Exposition and Philosophical Considerations. Rivista di Filosofia Neo-Scolastica 88 (1):95-128.
     
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  9.  69
    Taner Edis (1998). How Godel's Theorem Supports the Possibility of Machine Intelligence. Minds and Machines 8 (2):251-262.
    Gödel's Theorem is often used in arguments against machine intelligence, suggesting humans are not bound by the rules of any formal system. However, Gödelian arguments can be used to support AI, provided we extend our notion of computation to include devices incorporating random number generators. A complete description scheme can be given for integer functions, by which nonalgorithmic functions are shown to be partly random. Not being restricted to algorithms can be accounted for by the availability of an arbitrary (...)
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  10.  70
    Aaron Sloman (1992). The Emperor's Real Mind -- Review of Roger Penrose's The Emperor's New Mind: Concerning Computers Minds and the Laws of Physics. Artificial Intelligence 56 (2-3):355-396.
    "The Emperor's New Mind" by Roger Penrose has received a great deal of both praise and criticism. This review discusses philosophical aspects of the book that form an attack on the "strong" AI thesis. Eight different versions of this thesis are distinguished, and sources of ambiguity diagnosed, including different requirements for relationships between program and behaviour. Excessively strong versions attacked by Penrose (and Searle) are not worth defending or attacking, whereas weaker versions remain problematic. Penrose (like Searle) regards the notion (...)
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  11.  21
    Peter Slezak (1984). Minds, Machines and Self-Reference. Dialectica 38 (1):17-34.
    SummaryJ.R. Lucas has argued that it follows from Godel's Theorem that the mind cannot be a machine or represented by any formal system. Although this notorious argument against the mechanism thesis has received considerable attention in the literature, it has not been decisively rebutted, even though mechanism is generally thought to be the only plausible view of the mind. In this paper I offer an analysis of Lucas's argument which shows that it derives its persuasiveness from a subtle confusion. (...)
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  12.  24
    David Coder (1969). Godel's Theorem and Mechanism. Philosophy 44 (September):234-7.
    In “Minds, Machines, and Gödel”, J. R. Lucas claims that Goedel's incompleteness theorem constitutes a proof “that Mechanism is false, that is, that minds cannot be explained as machines”. He claims further that “if the proof of the falsity of mechanism is valid, it is of the greatest consequence for the whole of philosophy”. It seems to me that both of these claims are exaggerated. It is true that no minds can be explained as machines. But it is (...)
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  13. Albert E. Lyngzeidetson (1990). Massively Parallel Distributed Processing and a Computationalist Foundation for Cognitive Science. British Journal for the Philosophy of Science 41 (March):121-127.
    My purpose in this brief paper is to consider the implications of a radically different computer architecure to some fundamental problems in the foundations of Cognitive Science. More exactly, I wish to consider the ramifications of the 'Gödel-Minds-Machines' controversy of the late 1960s on a dynamically changing computer architecture which, I venture to suggest, is going to revolutionize which 'functions' of the human mind can and cannot be modelled by (non-human) computational automata. I will proceed on the presupposition that the (...)
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  14. I. J. Good (1969). Godel's Theorem is a Red Herring. British Journal for the Philosophy of Science 19 (February):357-8.
  15.  40
    Peter Slezak (1982). Godel's Theorem and the Mind. British Journal for the Philosophy of Science 33 (March):41-52.
  16.  82
    William H. Hanson (1971). Mechanism and Godel's Theorem. British Journal for the Philosophy of Science 22 (February):9-16.
  17.  5
    Graham Priest (1994). Godel's Theorem and the Mind... Again. In M. Michael & John O'Leary-Hawthorne (eds.), Philosophy in Mind: The Place of Philosophy in the Study of Mind. Kluwer 41-52.
  18. John R. Lucas (1961). Minds, Machines and Godel. Philosophy 36 (April-July):112-127.
    Goedel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulae which cannot be proved-in-the-system, but which we can see to be true. Essentially, we consider the formula which says, in effect, "This formula is unprovable-in-the-system". If this formula were provable-in-the-system, we should have a contradiction: for if it were provablein-the-system, then it would not be unprovable-in-the-system, so that "This formula is unprovable-in-the-system" would be false: equally, if it were provable-in-the-system, (...)
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  19. Johanna N. Y. Franklin & Frank Stephan (2010). Van Lambalgen's Theorem and High Degrees. Notre Dame Journal of Formal Logic 52 (2):173-185.
    We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.
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  20. Franz Dietrich & Christian List (2007). Arrow's Theorem in Judgment Aggregation. Social Choice and Welfare 29 (1):19-33.
    In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue for the converse claim. After proving two impossibility theorems on judgment aggregation (using "systematicity" and "independence" conditions, respectively), we construct an embedding of preference aggregation into judgment aggregation and prove Arrow’s theorem (stated for strict preferences) as a corollary of our second result. Although we (...)
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  21. Einar Duenger Bohn (forthcoming). Composition as Identity and Plural Cantor's Theorem. Logic and Logical Philosophy.
    I argue that Composition as Identity blocks the plural version of Cantor's Theorem, and that therefore the plural version of Cantor's Theorem can no longer be uncritically appealed to. As an example, I show how this result blocks a recent argument by Hawthorne and Uzquiano.
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  22. Christian List & Robert E. Goodin (2001). Epistemic Democracy: Generalizing the Condorcet Jury Theorem. Journal of Political Philosophy 9 (3):277–306.
    This paper generalises the classical Condorcet jury theorem from majority voting over two options to plurality voting over multiple options. The paper further discusses the debate between epistemic and procedural democracy and situates its formal results in that debate. The paper finally compares a number of different social choice procedures for many-option choices in terms of their epistemic merits. An appendix explores the implications of some of the present mathematical results for the question of how probable majority cycles (as (...)
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  23.  30
    Nikolaos Galatos & Hiroakira Ono (2006). Algebraization, Parametrized Local Deduction Theorem and Interpolation for Substructural Logics Over FL. Studia Logica 83 (1-3):279 - 308.
    Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.
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  24.  21
    Heinz-Dieter Ebbinghaus (1996). Mathematical Logic. Springer.
    This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...)
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  25.  23
    Sigmund Wagner-Tsukamoto (2007). Moral Agency, Profits and the Firm: Economic Revisions to the Friedman Theorem. [REVIEW] Journal of Business Ethics 70 (2):209 - 220.
    The paper reconstructs in economic terms Friedman's theorem that the only social responsibility of firms is to increase their profits while staying within legal and ethical rules. A model of three levels of moral conduct is attributed to the firm: (1) self-interested engagement in the market process itself, which reflects according to classical and neoclassical economics an ethical ideal; (2) the obeying of the "rules of the game," largely legal ones; and (3) the creation of ethical capital, which allows (...)
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  26. Jacob Stegenga (2013). An Impossibility Theorem for Amalgamating Evidence. Synthese 190 (12):2391-2411.
    Amalgamating evidence of different kinds for the same hypothesis into an overall confirmation is analogous, I argue, to amalgamating individuals’ preferences into a group preference. The latter faces well-known impossibility theorems, most famously “Arrow’s Theorem”. Once the analogy between amalgamating evidence and amalgamating preferences is tight, it is obvious that amalgamating evidence might face a theorem similar to Arrow’s. I prove that this is so, and end by discussing the plausibility of the axioms required for the theorem.
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  27.  99
    Joe Henson (2013). Non-Separability Does Not Relieve the Problem of Bell's Theorem. Foundations of Physics 43 (8):1008-1038.
    This paper addresses arguments that “separability” is an assumption of Bell’s theorem, and that abandoning this assumption in our interpretation of quantum mechanics (a position sometimes referred to as “holism”) will allow us to restore a satisfying locality principle. Separability here means that all events associated to the union of some set of disjoint regions are combinations of events associated to each region taken separately.In this article, it is shown that: (a) localised events can be consistently defined without implying (...)
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  28. Peter A. Cholak, Carl G. Jockusch & Theodore A. Slaman (2001). On the Strength of Ramsey's Theorem for Pairs. Journal of Symbolic Logic 66 (1):1-55.
    We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n (...)
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  29.  42
    Raul Hakli & Sara Negri (2012). Does the Deduction Theorem Fail for Modal Logic? Synthese 187 (3):849-867.
    Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a (...)
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  30. Sjoerd D. Zwart & Maarten Franssen (2007). An Impossibility Theorem for Verisimilitude. Synthese 158 (1):75 - 92.
    In this paper, we show that Arrow’s well-known impossibility theorem is instrumental in bringing the ongoing discussion about verisimilitude to a more general level of abstraction. After some preparatory technical steps, we show that Arrow’s requirements for voting procedures in social choice are also natural desiderata for a general verisimilitude definition that places content and likeness considerations on the same footing. Our main result states that no qualitative unifying procedure of a functional form can simultaneously satisfy the requirements of (...)
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  31. Saul A. Kripke (2013). The Church-Turing ‘Thesis’ as a Special Corollary of Gödel’s Completeness Theorem. In B. J. Copeland, C. Posy & O. Shagrir (eds.), Computability: Turing, Gödel, Church, and Beyond. MIT Press
    Traditionally, many writers, following Kleene (1952), thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing’s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis, what Turing (1936) calls “argument I,” has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing (...)
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  32.  46
    Serguei Kaniovski (2010). Aggregation of Correlated Votes and Condorcet's Jury Theorem. Theory and Decision 69 (3):453-468.
    This paper proves two theorems for homogeneous juries that arise from different solutions to the problem of aggregation of dichotomous choice. In the first theorem, negative correlation increases the competence of the jury, while positive correlation has the opposite effect. An enlargement of the jury with positive correlation can be detrimental up to a certain size, beyond which it becomes beneficial. The second theorem finds a family of distributions for which correlation has no effect on a jury’s competence. (...)
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  33. Michael E. Cuffaro (forthcoming). On the Significance of the Gottesman-Knill Theorem. British Journal for the Philosophy of Science:axv016.
    According to the Gottesman-Knill theorem, quantum algorithms which utilise only the operations belonging to a certain restricted set are efficiently simulable classically. Since some of the operations in this set generate entangled states, it is commonly concluded that entanglement is insufficient to enable quantum computers to outperform classical computers. I argue in this paper that this conclusion is misleading. First, the statement of the theorem (that the particular set of quantum operations in question can be simulated using a (...)
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  34.  6
    Eric G. Cavalcanti & Raymond Lal (2014). On Modifications of Reichenbach's Principle of Common Cause in Light of Bell's Theorem. Journal of Physics A: Mathematical and Theoretical 47 (42):424018.
    Bellʼs 1964 theorem causes a severe problem for the notion that correlations require explanation, encapsulated in Reichenbachʼs principle of common cause. Despite being a hallmark of scientific thought, dropping the principle has been widely regarded as much less bitter medicine than the perceived alternative—dropping relativistic causality. Recently, however, some authors have proposed that modified forms of Reichenbachʼs principle could be maintained even with relativistic causality. Here we break down Reichenbachʼs principle into two independent assumptions—the principle of common cause proper (...)
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  35.  8
    Warren J. Ewens (2014). Grafen, the Price Equations, Fitness Maximization, Optimisation and the Fundamental Theorem of Natural Selection. Biology and Philosophy 29 (2):197-205.
    This paper is a commentary on the focal article by Grafen and on earlier papers of his on which many of the results of this focal paper depend. Thus it is in effect a commentary on the “formal Darwinian project”, the focus of this sequence of papers. Several problems with this sequence are raised and discussed. The first of these concerns fitness maximization. It is often claimed in these papers that natural selection leads to a maximization of fitness and that (...)
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  36.  21
    Christian List (2003). A Possibility Theorem on Aggregation Over Multiple Interconnected Propositions. Mathematical Social Sciences 45 (1):1-13.
    Drawing on the so-called “doctrinal paradox”, List and Pettit (2002) have shown that, given an unrestricted domain condition, there exists no procedure for aggregating individual sets of judgments over multiple interconnected propositions into corresponding collective ones, where the procedure satisfies some minimal conditions similar to the conditions of Arrow’s theorem. I prove that we can avoid the paradox and the associated impossibility result by introducing an appropriate domain restriction: a structure condition, called unidimensional alignment, is shown to open up (...)
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  37.  65
    Nicholaos Jones (2012). An Arrovian Impossibility Theorem for the Epistemology of Disagreement. Logos and Episteme 3 (1):97-115.
    According to conciliatory views about the epistemology of disagreement, when epistemic peers have conflicting doxastic attitudes toward a proposition and fully disclose to one another the reasons for their attitudes toward that proposition (and neither has independent reason to believe the other to be mistaken), each peer should always change his attitude toward that proposition to one that is closer to the attitudes of those peers with which there is disagreement. According to pure higher-order evidence views, higher-order evidence for a (...)
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  38.  52
    Robert E. Goodin & Christian List (2006). A Conditional Defense of Plurality Rule: Generalizing May's Theorem in a Restricted Informational Environment. American Journal of Political Science 50 (4):940-949.
    May's theorem famously shows that, in social decisions between two options, simple majority rule uniquely satisfies four appealing conditions. Although this result is often cited in support of majority rule, it has never been extended beyond decisions based on pairwise comparisons of options. We generalize May's theorem to many-option decisions where voters each cast one vote. Surprisingly, plurality rule uniquely satisfies May's conditions. This suggests a conditional defense of plurality rule: If a society's balloting procedure collects only a (...)
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  39.  72
    Louis Vervoort (2013). Bell's Theorem: Two Neglected Solutions. Foundations of Physics 43 (6):769-791.
    Bell’s theorem admits several interpretations or ‘solutions’, the standard interpretation being ‘indeterminism’, a next one ‘nonlocality’. In this article two further solutions are investigated, termed here ‘superdeterminism’ and ‘supercorrelation’. The former is especially interesting for philosophical reasons, if only because it is always rejected on the basis of extra-physical arguments. The latter, supercorrelation, will be studied here by investigating model systems that can mimic it, namely spin lattices. It is shown that in these systems the Bell inequality can be (...)
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  40.  27
    Peter Øhrstrøm, Jörg Zeller & Ulrik Sandborg-Petersen (2012). Prior's Defence of Hintikka's Theorem. A Discussion of Prior's 'The Logic of Obligation and the Obligations of the Logician'. Synthese 188 (3):449-454.
    In his paper, The logic of obligation and the obligations of the logician, A.N. Prior considers Hintikka's theorem, according to which a statement cannot be both impossible and permissible. This theorem has been seen as problematic for the very idea of a logic of obligation. However, Prior rejects the view that the logic of obligation cannot be formalised. He sees this resistance against such a view as an important part of what could be called the obligation of the (...)
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  41.  4
    Guido Gherardi & Alberto Marcone (2009). How Incomputable Is the Separable Hahn-Banach Theorem? Notre Dame Journal of Formal Logic 50 (4):393-425.
    We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and the (...)
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  42.  30
    Anya Plutynski (2006). What Was Fisher's Fundamental Theorem of Natural Selection and What Was It For? Studies in History and Philosophy of Science Part C 37 (1):59-82.
    Fisher’s ‘fundamental theorem of natural selection’ is notoriously abstract, and, no less notoriously, many take it to be false. In this paper, I explicate the theorem, examine the role that it played in Fisher’s general project for biology, and analyze why it was so very fundamental for Fisher. I defend Ewens (1989) and Lessard (1997) in the view that the theorem is in fact a true theorem if, as Fisher claimed, ‘the terms employed’ are ‘used strictly (...)
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  43.  96
    Federico Laudisa (2008). Non-Local Realistic Theories and the Scope of the Bell Theorem. Foundations of Physics 38 (12):1110-1132.
    According to a widespread view, the Bell theorem establishes the untenability of so-called ‘local realism’. On the basis of this view, recent proposals by Leggett, Zeilinger and others have been developed according to which it can be proved that even some non-local realistic theories have to be ruled out. As a consequence, within this view the Bell theorem allows one to establish that no reasonable form of realism, be it local or non-local, can be made compatible with the (...)
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  44.  18
    Albert Visser (2012). The Second Incompleteness Theorem and Bounded Interpretations. Studia Logica 100 (1-2):399-418.
    In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n , all translations of V -sentences are U -provably equivalent to sentences of complexity less than n . We call a sequential sentence with consistency power over T a pro-consistency statement for (...)
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  45. Manuel Bremer, Frege's Basic Law V and Cantor's Theorem.
    The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These (...)
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  46. Chin-Liang Chang (1973). Symbolic Logic and Mechanical Theorem Proving. Academic Press.
    This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4–9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.
     
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  47. Adán Cabello (2005). Bell's Theorem Without Inequalities and Without Unspeakable Information. Foundations of Physics 35 (11):1927-1934.
    A proof of Bell’s theorem without inequalities is presented in which distant local setups do not need to be aligned, since the required perfect correlations are achieved for any local rotation of the local setups.
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  48.  11
    Karin U. Katz, Mikhail G. Katz & Taras Kudryk (2014). Toward a Clarity of the Extreme Value Theorem. Logica Universalis 8 (2):193-214.
    We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementarity rather than a rivalry of the approaches.
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  49.  14
    Josef Berger & Hajime Ishihara (2005). Brouwer's Fan Theorem and Unique Existence in Constructive Analysis. Mathematical Logic Quarterly 51 (4):360-364.
    Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. (...)
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  50.  68
    Michael Detlefsen & Mark Luker (1980). The Four-Color Theorem and Mathematical Proof. Journal of Philosophy 77 (12):803-820.
    I criticize a recent paper by Thomas Tymoczko in which he attributes fundamental philosophical significance and novelty to the lately-published computer-assisted proof of the four color theorem (4CT). Using reasoning precisely analogous to that employed by Tymoczko, I argue that much of traditional mathematical proof must be seen as resting on what Tymoczko must take as being "empirical" evidence. The new proof of the 4CT, with its use of what Tymoczko calls "empirical" evidence is therefore not so novel as (...)
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