Results for 'Diagonalization'

49 found
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  1. Diagonalization & Forcing FLEX: From Cantor to Cohen and Beyond. Learning from Leibniz, Cantor, Turing, Gödel, and Cohen; crawling towards AGI.Elan Moritz - manuscript
    The paper continues my earlier Chat with OpenAI’s ChatGPT with a Focused LLM Experiment (FLEX). The idea is to conduct Large Language Model (LLM) based explorations of certain areas or concepts. The approach is based on crafting initial guiding prompts and then follow up with user prompts based on the LLMs’ responses. The goals include improving understanding of LLM capabilities and their limitations culminating in optimized prompts. The specific subjects explored as research subject matter include a) diagonalization techniques as (...)
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  2. The Diagonal Lemma: An Informal Exposition.Richard Kimberly Heck - manuscript
    This is a completely informal presentation of the ideas behind the diagonal lemma. One really can't see this important result from too many different angles. This one aims at getting the main idea across. (For the cognoscenti, it is in the spirit of Quine's treatment in terms of "appended to its own quotation".).
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  3. Diagonal arguments and fixed points.Saeed Salehi - 2017 - Bulletin of the Iranian Mathematical Society 43 (5):1073-1088.
    ‎A universal schema for diagonalization was popularized by N. S‎. ‎Yanofsky (2003)‎, ‎based on a pioneering work of F.W‎. ‎Lawvere (1969)‎, ‎in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function‎. ‎It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema‎. ‎Here‎, ‎we fit more theorems in the universal‎ ‎schema of diagonalization‎, ‎such as Euclid's proof for the infinitude of the primes and (...)
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  4. Minimal Sartre: Diagonalization and Pure Reflection.John Bova - 2012 - Open Philosophy 1:360-379.
    These remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre’s theory of pure reflection, the linchpin of the works of Sartre’s early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Gödel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception (...)
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  5. Do Not Diagonalize.Cameron Kirk-Giannini - 2024 - In Ernie Lepore & Una Stojnic (eds.), The Oxford Handbook of Contemporary Philosophy of Language. Oxford University Press.
    Speakers assert in order to communicate information. It is natural, therefore, to hold that the content of an assertion is whatever information it communicates to its audience. In cases involving uncertainty about the semantic values of context-sensitive lexical items, moreover, it is natural to hold that the information an assertion communicates to its audience is whatever information audience members are in a position to recover from it by assuming that the proposition it semantically determines is true. This sort of picture (...)
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  6. (1 other version)Natural Deduction for Diagonal Operators.Fabio Lampert - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. New York: Birkhäuser. pp. 39-51.
    We present a sound and complete Fitch-style natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over some point (...)
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  7. Wittgenstein on Pseudo-Irrationals, Diagonal Numbers and Decidability.Timm Lampert - 2008 - In Lampert Timm (ed.), The Logica Yearbook 2008. pp. 95-111.
    In his early philosophy as well as in his middle period, Wittgenstein holds a purely syntactic view of logic and mathematics. However, his syntactic foundation of logic and mathematics is opposed to the axiomatic approach of modern mathematical logic. The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; (...)
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  8. Operational axioms for diagonalizing states.Giulio Chiribella & Carlo Maria Scandolo - 2015 - EPTCS 195:96-115.
    In quantum theory every state can be diagonalized, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This elementary structure plays an ubiquitous role in quantum mechanics, quantum information theory, and quantum statistical mechanics, where it provides the foundation for the notions of majorization and entropy. A natural question then arises: can we reconstruct these notions from purely operational axioms? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that (...)
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    A two-dimensional logic for diagonalization and the a priori.Melissa Fusco - 2020 - Synthese 198 (9):8307-8322.
    Two-dimensional semantics, which can represent the distinction between a priority and necessity, has wielded considerable influence in the philosophy of language. In this paper, I axiomatize the dagger operator of Stalnaker’s “Assertion” in the formal context of two-dimensional modal logic. The language contains modalities of actuality, necessity, and a priority, but is also able to represent diagonalization, a conceptually important operation in a variety of contexts, including models of the relative a priori and a posteriori often appealed to Bayesian (...)
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  10. Wittgenstein’s analysis on Cantor’s diagonal argument.Chaohui Zhuang - manuscript
    In Zettel, Wittgenstein considered a modified version of Cantor’s diagonal argument. According to Wittgenstein, Cantor’s number, different with other numbers, is defined based on a countable set. If Cantor’s number belongs to the countable set, the definition of Cantor’s number become incomplete. Therefore, Cantor’s number is not a number at all in this context. We can see some examples in the form of recursive functions. The definition "f(a)=f(a)" can not decide anything about the value of f(a). The definiton is incomplete. (...)
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  11. The Exotic – An example of a diagonal category.Mihai Nadin - 1976 - Revue Roumaine des Sciences Sociales 20 (1).
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  12. A Two-Dimensional Logic for Two Paradoxes of Deontic Modality.Fusco Melissa & Kocurek Alexander - 2022 - Review of Symbolic Logic 15 (4):991-1022.
    In this paper, we axiomatize the deontic logic in Fusco 2015, which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross’s Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results help make explicit the restrictions (...)
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  13. Gödel Incompleteness and Turing Completeness.Ramón Casares - manuscript
    Following Post program, we will propose a linguistic and empirical interpretation of Gödel’s incompleteness theorem and related ones on unsolvability by Church and Turing. All these theorems use the diagonal argument by Cantor in order to find limitations in finitary systems, as human language, which can make “infinite use of finite means”. The linguistic version of the incompleteness theorem says that every Turing complete language is Gödel incomplete. We conclude that the incompleteness and unsolvability theorems find limitations in our finitary (...)
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  14. Cantor's Illusion.Hudson Richard L. - manuscript
    This analysis shows Cantor's diagonal definition in his 1891 paper was not compatible with his horizontal enumeration of the infinite set M. The diagonal sequence was a counterfeit which he used to produce an apparent exclusion of a single sequence to prove the cardinality of M is greater than the cardinality of the set of integers N.
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  15. Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory.Valérie Lynn Therrien - 2012 - Ithaque 10:43-65.
    In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...)
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  16. On the Reality of the Continuum Discussion Note: A Reply to Ormell, ‘Russell's Moment of Candour’, Philosophy.Anne Newstead - 2008 - Philosophy 83 (1):117-127.
    In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means excluding as (...)
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  17. On a Perceived Expressive Inadequacy of Principia Mathematica.Burkay T. Öztürk - 2011 - Florida Philosophical Review 12 (1):83-92.
    This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and Principia Mathematica.
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  18. The Algebraic Creativity in The Neutrosophic Square Matrices‏.Mohammad Abobala, Ahmed Hatip, A. A. Salama, Necati Olgun, Broumi Said & Huda E. Khaled - 2021 - Neutrosophic Sets and Systems 40:1-11.
    The objective of this paper is to study algebraic properties of neutrosophic matrices, where a necessary and sufficient condition for the invertibility of a square neutrosophic matrix is presented by defining the neutrosophic determinant. On the other hand, this work introduces the concept of neutrosophic Eigen values and vectors with an easy algorithm to compute them. Also, this article finds a necessary and sufficient condition for the diagonalization of a neutrosophic matrix.
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  19. David Wolpert on impossibility, incompleteness, the liar paradox, the limits of computation, a non-quantum mechanical uncertainty principle and the universe as computer—the ultimate theorem in Turing Machine Theory.Michael Starks - manuscript
    I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, and even (...)
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  20. Deontic Modality and the Semantics of Choice.Melissa Fusco - 2015 - Philosophers' Imprint 15.
    I propose a unified solution to two puzzles: Ross's puzzle and free choice permission. I begin with a pair of cases from the decision theory literature illustrating the phenomenon of act dependence, where what an agent ought to do depends on what she does. The notion of permissibility distilled from these cases forms the basis for my analysis of 'may' and 'ought'. This framework is then combined with a generalization of the classical semantics for disjunction — equivalent to Boolean disjunction (...)
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  21. Actuality and the a priori.Fabio Lampert - 2018 - Philosophical Studies 175 (3):809-830.
    We consider a natural-language sentence that cannot be formally represented in a first-order language for epistemic two-dimensional semantics. We also prove this claim in the “Appendix” section. It turns out, however, that the most natural ways to repair the expressive inadequacy of the first-order language render moot the original philosophical motivation of formalizing a priori knowability as necessity along the diagonal.
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  22. De Morgan on Euclid’s fourth postulate.John Corcoran & Sriram Nambiar - 2014 - Bulletin of Symbolic Logic 20 (2):250-1.
    This paper will annoy modern logicians who follow Bertrand Russell in taking pleasure in denigrating Aristotle for [allegedly] being ignorant of relational propositions. To be sure this paper does not clear Aristotle of the charge. On the contrary, it shows that such ignorance, which seems unforgivable in the current century, still dominated the thinking of one of the greatest modern logicians as late as 1831. Today it is difficult to accept the proposition that Aristotle was blind to the fact that, (...)
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  23. Self-reference and the languages of arithmetic.Richard Heck - 2007 - Philosophia Mathematica 15 (1):1-29.
    I here investigate the sense in which diagonalization allows one to construct sentences that are self-referential. Truly self-referential sentences cannot be constructed in the standard language of arithmetic: There is a simple theory of truth that is intuitively inconsistent but is consistent with Peano arithmetic, as standardly formulated. True self-reference is possible only if we expand the language to include function-symbols for all primitive recursive functions. This language is therefore the natural setting for investigations of self-reference.
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  24. From Pictures to Employments: Later Wittgenstein on 'the Infinite'.Philip Bold - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that (...)
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  25. ‘Let No-One Ignorant of Geometry…’: Mathematical Parallels for Understanding the Objectivity of Ethics.James Franklin - 2023 - Journal of Value Inquiry 57 (2):365-384.
    It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so on). That (...)
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  26. The negative theology of absolute infinity: Cantor, mathematics, and humility.Rico Gutschmidt & Merlin Carl - 2024 - International Journal for Philosophy of Religion 95 (3):233-256.
    Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to the recent debate (...)
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  27. A Liar Paradox.Richard G. Heck - 2012 - Thought: A Journal of Philosophy 1 (1):36-40.
    The purpose of this note is to present a strong form of the liar paradox. It is strong because the logical resources needed to generate the paradox are weak, in each of two senses. First, few expressive resources required: conjunction, negation, and identity. In particular, this form of the liar does not need to make any use of the conditional. Second, few inferential resources are required. These are: (i) conjunction introduction; (ii) substitution of identicals; and (iii) the inference: From ¬(p (...)
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  28. Linguistic Matrices.W. B. Vasantha Kandasamy, K. Ilanthenral & Florentin Smarandache - 2022 - Miami, FL, USA: Global Knowledge.
    In this book, the authors introduce the linguistic set associated with a linguistic variable and the structure of matrices, which they define as linguistic matrices. The authors build linguistic matrices only for those linguistic variables which yield a linguistic continuum or an ordered linguistic set. This book is organised into three chapters. The first chapter is introductory, in which we introduce all the basic concepts of linguistic variables and the associated linguistic set to make this book self-contained. Chapter two introduces (...)
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  29. Douglas Hofstadter's Gödelian Philosophy of Mind.Theodor Nenu - 2022 - Journal of Artificial Intelligence and Consciousness 9 (2):241-266.
    Hofstadter [1979, 2007] offered a novel Gödelian proposal which purported to reconcile the apparently contradictory theses that (1) we can talk, in a non-trivial way, of mental causation being a real phenomenon and that (2) mental activity is ultimately grounded in low-level rule-governed neural processes. In this paper, we critically investigate Hofstadter’s analogical appeals to Gödel’s [1931] First Incompleteness Theorem, whose “diagonal” proof supposedly contains the key ideas required for understanding both consciousness and mental causation. We maintain that bringing sophisticated (...)
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  30. Graph of Socratic Elenchos.John Bova - manuscript
    From my ongoing "Metalogical Plato" project. The aim of the diagram is to make reasonably intuitive how the Socratic elenchos (the logic of refutation applied to candidate formulations of virtues or ruling knowledges) looks and works as a whole structure. This is my starting point in the project, in part because of its great familiarity and arguable claim to being the inauguration of western philosophy; getting this point less wrong would have broad and deep consequences, including for philosophy’s self-understanding. -/- (...)
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  31. (1 other version)Wolpert, Chaitin and Wittgenstein on impossibility, incompleteness, the liar paradox, theism, the limits of computation, a non-quantum mechanical uncertainty principle and the universe as computer—the ultimate theorem in Turing Machine Theory (revised 2019).Michael Starks - 2019 - In Suicidal Utopian Delusions in the 21st Century -- Philosophy, Human Nature and the Collapse of Civilization-- Articles and Reviews 2006-2019 4th Edition Michael Starks. Las Vegas, NV USA: Reality Press. pp. 294-299.
    I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv dot org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, (...)
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  32. Formal Background for the Incompleteness and Undefinability Theorems.Richard Kimberly Heck - manuscript
    A teaching document I've used in my courses on truth and on incompleteness. Aimed at students who have a good grasp of basic logic, and decent math skills, it attempts to give them the background they need to understand a proper statement of the classic results due to Gödel and Tarski, and sketches their proofs. Topics covered include the notions of language and theory, the basics of formal syntax and arithmetization, formal arithmetic (Q and PA), representability, diagonalization, and the (...)
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  33. Operators in the paradox of the knower.Patrick Grim - 1993 - Synthese 94 (3):409 - 428.
    Predicates are term-to-sentence devices, and operators are sentence-to-sentence devices. What Kaplan and Montague's Paradox of the Knower demonstrates is that necessity and other modalities cannot be treated as predicates, consistent with arithmetic; they must be treated as operators instead. Such is the current wisdom.A number of previous pieces have challenged such a view by showing that a predicative treatment of modalities neednot raise the Paradox of the Knower. This paper attempts to challenge the current wisdom in another way as well: (...)
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  34. Logical Form and the Development of Russell’s Logicism.Kevin C. Klement - 2022 - In F. Boccuni & A. Sereni (eds.), Origins and Varieties of Logicism. Routledge. pp. 147–166.
    Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth, and the other not a truth at all. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not at all (...)
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  35. Theoremizing Yablo's Paradox.Ahmad Karimi & Saeed Salehi - manuscript
    To counter a general belief that all the paradoxes stem from a kind of circularity (or involve some self--reference, or use a diagonal argument) Stephen Yablo designed a paradox in 1993 that seemingly avoided self--reference. We turn Yablo's paradox, the most challenging paradox in the recent years, into a genuine mathematical theorem in Linear Temporal Logic (LTL). Indeed, Yablo's paradox comes in several varieties; and he showed in 2004 that there are other versions that are equally paradoxical. Formalizing these versions (...)
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  36. Grundgesetze and the Sense/Reference Distinction.Kevin C. Klement - 2019 - In Philip A. Ebert & Marcus Rossberg (eds.), Essays on Frege's Basic Laws of Arithmetic. Oxford: Oxford University Press. pp. 142-166.
    Frege developed the theory of sense and reference while composing his Grundgesetze and considering its philosophical implications. The Grundgesetze is thus the most important test case for the application of this theory of meaning. I argue that evidence internal and external to the Grundgesetze suggests that he thought of senses as having a structure isomorphic to the Grundgesetze expressions that would be used to express them, which entails a theory about the identity conditions of senses that is relatively fine-grained, though (...)
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  37. On Classical and Quantum Logical Entropy.David Ellerman - manuscript
    The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as "propositional" logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of (...)
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  38. Truth vs. Necessary Truth in Aristotle’s Sciences.Thomas V. Upton - 2004 - Review of Metaphysics 57 (4):741-753.
    AT POSTERIOR ANALYTICS 1.1.71B15 AND FOLLOWING, Aristotle identifies six characteristics of the first principles from which demonstrative science proceeds. These are traditionally grouped into two sets of three: group A: ex alêthôn, prôtôn, amêsôn; group B: gnôrimôterôn, proterôn, and aitiôn. The characteristic, which I believe has been underrated and somewhat misinterpreted by scholars and commentators from Philoponus to the present day, is the characteristic of truth. In this paper I propose to present a textually based interpretation of truth that shows (...)
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  39. Identification of antinomies by complementary analysis.Andrzej Burkiet - manuscript
    It has been noticed that self-referential, ambiguous definitional formulas are accompanied by complementary self-referential antinomy formulas, which gives rise to contradictions. This made it possible to re-examine ancient antinomies and Cantor’s Diagonal Argument (CDA), as well as the method of nested intervals, which is the basis for evaluating the existence of uncountable sets. Using Georg Cantor’s remark that every real number can be represented as an infinite digital expansion (usually decimal or binary), a simplified system for verifying the definitions of (...)
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  40. Epistemicism without metalinguistic safety.Justin Khoo - 2023 - In Abrol Fairweather & Carlos Montemayor (eds.), Linguistic Luck: Safeguards and Threats to Linguistic Communication. Oxford, GB: Oxford University Press.
    Epistemicists claim that vague predicates have precise but unknow- able cutoffs. I argue against against the standard, Williamsonian, answer, that appeals to metalinguistic safety: we can know that p even if our true belief that p is metalinguistically lucky. I then propose that epistemicists should be diagonalized epistemicists and show how this alternative formulation of the view avoids the chal- lenge. However, in an M. Night Shyamalan-style twist, I then argue we should not be diagonalized epistemicists either.
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  41. Wolpert, Chaitin and Wittgenstein on impossibility, incompleteness, the limits of computation, theism and the universe as computer-the ultimate Turing Theorem.Michael Starks - 2017 - Philosophy, Human Nature and the Collapse of Civilization Michael Starks 3rd Ed. (2017).
    I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, and even (...)
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  42. A Cantorian argument against Frege's and early Russell's theories of descriptions.Kevin C. Klement - 2008 - In Nicholas Griffin & Dale Jacquette (eds.), Russell Vs. Meinong: The Legacy of "on Denoting". London and New York: Routledge. pp. 65-77.
    It would be an understatement to say that Russell was interested in Cantorian diagonal paradoxes. His discovery of the various versions of Russell’s paradox—the classes version, the predicates version, the propositional functions version—had a lasting effect on his views in philosophical logic. Similar Cantorian paradoxes regarding propositions—such as that discussed in §500 of The Principles of Mathematics—were surely among the reasons Russell eventually abandoned his ontology of propositions.1 However, Russell’s reasons for abandoning what he called “denoting concepts”, and his rejection (...)
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  43. The Two-Dimensional Content of Consciousness.Simon Prosser - 2007 - Philosophical Studies 136 (3):319 - 349.
    In this paper I put forward a representationalist theory of conscious experience based on Robert Stalnaker's version of two-dimensional modal semantics. According to this theory the phenomenal character of an experience correlates with a content equivalent to what Stalnaker calls the diagonal proposition. I show that the theory is closely related both to functionalist theories of consciousness and to higher-order representational theories. It is also more compatible with an anti-Cartesian view of the mind than standard representationalist theories.
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  44. Three Essays on Later Wittgenstein's Philosophy of Mathematics: Reality, Determination, and Infinity.Philip Bold - 2022 - Dissertation, University of North Carolina, Chapel Hill
    This dissertation provides a careful reading of the later Wittgenstein’s philosophy of mathematics centered around three major themes: reality, determination, and infinity. The reading offered gives pride of place to Wittgenstein’s therapeutic conception of philosophy. This conception views questions often taken as fundamental in the philosophy of mathematics with suspicion and attempts to diagnose the confusions which lead to them. In the first essay, I explain Wittgenstein’s approach to perennial issues regarding the alleged reality to which mathematical truths or propositions (...)
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  45. Noesis and Logos in Plato's Statesman, with a Focus on the Visitor's Jokes at 266a-d.Mitchell Miller - 2017 - In John Sallis (ed.), Plato's Statesman: Dialectic, Myth, and Politics. Albany, NY: Suny Series in Contemporary Company. pp. 107-136.
    In his “Noesis and Logos in the Eleatic Trilogy, with a Focus on the Visitor’s Jokes at Statesman 266a-d,” Mitchell Miller explores the interplay of intuition and discourse in the Statesman. He prepares by considering the orienting provocations provided by Socrates’ refutations of the proposed definition of knowledge — namely, “true judgment and a logos” — in the closing pages of the Theaetetus, by the Eleatic Visitor’s obscure schematization at Sophist 253d-e of the kinds of eidetic field discerned by dialectic, (...)
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  46. KK is Wrong Because We Say So.Simon Goldstein & John Hawthorne - forthcoming - Mind.
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  47. This sentence does not contain the symbol X.Samuel Alexander - 2013 - The Reasoner 7 (9):108.
    A suprise may occur if we use a similar strategy to the Liar's paradox to mathematically formalize "This sentence does not contain the symbol X".
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  48. Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?Bhupinder Singh Anand - 2004 - Neuroquantology 2:60-100.
    Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...)
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  49. Rethinking Cantor: Infinite Iterations and the Cardinality of the Reals.Manus Ross - manuscript
    In this paper, I introduce an iterative method aimed at exploring numbers within the interval [0, 1]. Beginning with a foundational set, S0, a series of algorithms are employed to expand and refine this set. Each algorithm has its designated role, from incorporating irrational numbers to navigating non-deterministic properties. With each successive iteration, our set grows, and after infinite iterations, its cardinality is proposed to align with that of the real numbers. This work is an initial exploration into this approach, (...)
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