Results for 'axioms'

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  1. Strong Axioms of Infinity and the Debate About Realism.Kai Hauser & W. Hugh Woodin - 2014 - Journal of Philosophy 111 (8):397-419.
    One of the most distinctive and intriguing developments of modern set theory has been the realization that, despite widely divergent incentives for strengthening the standard axioms, there is essentially only one way of ascending the higher reaches of infinity. To the mathematical realist the unexpected convergence suggests that all these axiomatic extensions describe different aspects of the same underlying reality.
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  2.  34
    Strong Axioms of Infinity and Elementary Embeddings.Robert M. Solovay - 1978 - Annals of Mathematical Logic 13 (1):73.
  3.  4
    From Axiom to Dialogue: A Philosophical Study of Logics and Argumentation.Else Margarete Barth & Erik C. W. Krabbe - 1982 - Berlin and New York: De Gruyter.
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  4.  1
    From Axiom to Dialogue: A Philosophical Study of Logics and Argumentation.E. M. Barth - 1982 - Berlin and New York: W. De Gruyter.
  5.  57
    Local Axioms in Disguise: Hilbert on Minkowski Diagrams.Ivahn Smadja - 2012 - Synthese 186 (1):315-370.
    While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski’s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating “rigorously” with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as “drawn formulas”, and formulas as (...)
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  6. Axioms in Mathematical Practice.Dirk Schlimm - 2013 - Philosophia Mathematica 21 (1):37-92.
    On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. (...)
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  7.  81
    Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles.Dan Willard - 2001 - Journal of Symbolic Logic 66 (2):536-596.
    We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive (...)
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  8. From Axiom to Dialogue.E. M. Barth & E. C. W. Krabbe - 1985 - Studia Logica 44 (2):228-230.
  9. Natural Axioms for Classical Mereology.Aaron Cotnoir & Achille C. Varzi - 2019 - Review of Symbolic Logic 12 (1):201-208.
    We present a new axiomatization of classical mereology in which the three components of the theory—ordering, composition, and decomposition prin-ciples—are neatly separated. The equivalence of our axiom system with other, more familiar systems is established by purely deductive methods, along with additional results on the relative strengths of the composition and decomposition axioms of each theory.
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  10.  24
    The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal.W. Hugh Woodin - 2002 - Bulletin of Symbolic Logic 8 (1):91-93.
  11. New Axioms for Probability and Likelihood Ratio Measures.Vincenzo Crupi, Nick Chater & Katya Tentori - 2013 - British Journal for the Philosophy of Science 64 (1):189-204.
    Probability ratio and likelihood ratio measures of inductive support and related notions have appeared as theoretical tools for probabilistic approaches in the philosophy of science, the psychology of reasoning, and artificial intelligence. In an effort of conceptual clarification, several authors have pursued axiomatic foundations for these two families of measures. Such results have been criticized, however, as relying on unduly demanding or poorly motivated mathematical assumptions. We provide two novel theorems showing that probability ratio and likelihood ratio measures can be (...)
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  12.  16
    Resurrection Axioms and Uplifting Cardinals.Joel David Hamkins & Thomas A. Johnstone - 2014 - Archive for Mathematical Logic 53 (3-4):463-485.
    We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.
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  13.  84
    Axioms of Symmetry: Throwing Darts at the Real Number Line.Chris Freiling - 1986 - Journal of Symbolic Logic 51 (1):190-200.
    We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will (...)
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  14.  71
    Rejected Axioms for the “Nonsense-Logic” W and the K-Valued Logic of Sobociński.Robert Sochacki - 2008 - Logic and Logical Philosophy 17 (4):321-327.
    In this paper rejection systems for the “nonsense-logic” W and the k-valued implicational-negational sentential calculi of Sobociński are given. Considered systems consist of computable sets of rejected axioms and only one rejection rule: the rejection version of detachment rule.
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  15.  81
    Beyond the Axioms: The Question of Objectivity in Mathematics.W. TaitW - 2001 - Philosophia Mathematica 9 (1):21-36.
    This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. a (...)
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  16.  37
    Reduction Axioms for Epistemic Actions.Johan van Benthem & Barteld Kooi - unknown
    Current dynamic epistemic logics often become cumbersome and opaque when common knowledge is added. In this paper we propose new versions that extend the underlying static epistemic language in such a way that dynamic completeness proofs can be obtained by perspicuous reduction axioms.
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  17.  61
    Axioms for the Part Relation.Nicholas Rescher - 1955 - Philosophical Studies 6 (1):8-11.
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  18. Axioms for Deliberative Stit.Ming Xu - 1998 - Journal of Philosophical Logic 27 (5):505-552.
    Based on a notion of "companions to stit formulas" applied in other papers dealing with astit logics, we introduce "choice formulas" and "nested choice formulas" to prove the completeness theorems for dstit logics in a language with the dstit operator as the only non-truth-functional operator. The main logic discussed in this paper is the basic logic of dstit with multiple agents, other logics discussed include the basic logic of dstit with a single agent and some logics of dstit with multiple (...)
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  19.  78
    Axioms and Tests for the Presence of Minimal Consciousness in Agents I: Preamble.Igor L. Aleksander & B. Dunmall - 2003 - Journal of Consciousness Studies 10 (4-5):7-18.
    This paper relates to a formal statement of the mechanisms that are thought minimally necessary to underpin consciousness. This is expressed in the form of axioms. We deem this to be useful if there is ever to be clarity in answering questions about whether this or the other organism is or is not conscious. As usual, axioms are ways of making formal statements of intuitive beliefs and looking, again formally, at the consequences of such beliefs. The use of (...)
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  20.  19
    Defending the Axioms: On the Philosophical Foundations of Set Theory.Penelope Maddy - 2011 - Oxford, England: Oxford University Press.
    Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a (...)
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  21. Axioms for Determinateness and Truth.Solomon Feferman - 2008 - Review of Symbolic Logic 1 (2):204-217.
    elaboration of the last part of my Tarski Lecture, “Truth unbound”, UC Berkeley, 3 April 2006, and of the lecture, “A nicer formal theory of non-hierarchical truth”, Workshop on Mathematical Methods in Philosophy, Banff , 18-23 Feb. 2007.
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  22.  15
    Weak Axioms of Determinacy and Subsystems of Analysis II.Kazuyuki Tanaka - 1991 - Annals of Pure and Applied Logic 52 (1-2):181-193.
    In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, but the (...)
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  23.  49
    The Axiom of Choice is False Intuitionistically.Charles Mccarty, Stewart Shapiro & Ansten Klev - forthcoming - Bulletin of Symbolic Logic:1-26.
  24.  50
    Fundamental Axioms for Preference Relations.Bengt Hansson - 1968 - Synthese 18 (4):423 - 442.
    The basic theory of preference relations contains a trivial part reflected by axioms A1 and A2, which say that preference relations are preorders. The next step is to find other axims which carry the theory beyond the level of the trivial. This paper is to a great part a critical survey of such suggested axioms. The results are much in the negative — many proposed axioms imply too strange theorems to be acceptable as axioms in a (...)
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  25. The Axiom of Choice.John L. Bell - 2008 - Stanford Encyclopedia of Philosophy.
    The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...)
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  26. Axioms for Actuality.Harold T. Hodes - 1984 - Journal of Philosophical Logic 13 (1):27 - 34.
  27. The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories.John L. Bell - 2008 - Mathematical Logic Quarterly 54 (2):194-201.
    A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.
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  28.  19
    Axioms for Abstract Model Theory.K. Jon Barwise - 1974 - Annals of Mathematical Logic 7 (2-3):221-265.
  29.  19
    The Axioms of Constructive Geometry.Jan von Plato - 1995 - Annals of Pure and Applied Logic 76 (2):169-200.
    Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions (...)
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  30.  8
    The Axioms of Subjective Probability.Peter C. Fishburn - 1986 - Statistical Science 1 (3):335-358.
  31.  39
    Which Set Existence Axioms Are Needed to Prove the Cauchy/Peano Theorem for Ordinary Differential Equations?Stephen G. Simpson - 1984 - Journal of Symbolic Logic 49 (3):783-802.
    We investigate the provability or nonprovability of certain ordinary mathematical theorems within certain weak subsystems of second order arithmetic. Specifically, we consider the Cauchy/Peano existence theorem for solutions of ordinary differential equations, in the context of the formal system RCA 0 whose principal axioms are ▵ 0 1 comprehension and Σ 0 1 induction. Our main result is that, over RCA 0 , the Cauchy/Peano Theorem is provably equivalent to weak Konig's lemma, i.e. the statement that every infinite {0, (...)
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  32. Reduction Axioms for Epistemic Actions. Kooi, Barteld & van Benthem, Johan - unknown
    Current dynamic epistemic logics often become cumbersome and opaque when common knowledge is added. In this paper we propose new versions that extend the underlying static epistemic language in such a way that dynamic completeness proofs can be obtained by perspicuous reduction axioms.
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  33.  32
    An Axiom System for the Modular Logic.Jerzy Kotas - 1967 - Studia Logica 21 (1):17 - 38.
  34.  26
    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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  35. The Axiom of Choice in Quine's New Foundations for Mathematical Logic.Ernst P. Specker - 1954 - Journal of Symbolic Logic 19 (2):127-128.
  36.  47
    Axioms for Grounded Truth.Thomas Schindler - 2014 - Review of Symbolic Logic 7 (1):73-83.
    We axiomatize Leitgeb's (2005) theory of truth and show that this theory proves all arithmetical sentences of the system of ramified analysis up to $\epsilon_0$. We also give alternative axiomatizations of Kripke's (1975) theory of truth (Strong Kleene and supervaluational version) and show that they are at least as strong as the Kripke-Feferman system KF and Cantini's VF, respectively.
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  37. The Axiom of Infinity and Transformations J: V → V.Paul Corazza - 2010 - Bulletin of Symbolic Logic 16 (1):37-84.
    We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? (...)
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  38.  28
    Equivocation Axiom on First Order Languages.Soroush Rafiee Rad - 2017 - Studia Logica 105 (1):121-152.
    In this paper we investigate some mathematical consequences of the Equivocation Principle, and the Maximum Entropy models arising from that, for first order languages. We study the existence of Maximum Entropy models for these theories in terms of the quantifier complexity of the theory and will investigate some invariance and structural properties of such models.
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  39.  47
    Axioms of Set Theory.Joseph R. Shoenfield - 1977 - In Jon Barwise & H. Jerome Keisler (eds.), Handbook of Mathematical Logic. North-Holland Pub. Co.. pp. 90.
  40. Some Axioms for Constructive Analysis.Joan Rand Moschovakis & Garyfallia Vafeiadou - 2012 - Archive for Mathematical Logic 51 (5-6):443-459.
    This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CFd asserting that every decidable (...)
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  41.  18
    Intricate Axioms as Interaction Axioms.Guillaume Aucher - 2015 - Studia Logica 103 (5):1035-1062.
    In epistemic logic, some axioms dealing with the notion of knowledge are rather convoluted and difficult to interpret intuitively, even though some of them, such as the axioms.2 and.3, are considered to be key axioms by some epistemic logicians. We show that they can be characterized in terms of understandable interaction axioms relating knowledge and belief or knowledge and conditional belief. In order to show it, we first sketch a theory dealing with the characterization of (...) in terms of interaction axioms in modal logic. We then apply the main results and methods of this theory to obtain specific results related to epistemic and doxastic logics. (shrink)
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  42.  68
    Axioms for Classical, Intuitionistic, and Paraconsistent Hybrid Logic.Torben Braüner - 2006 - Journal of Logic, Language and Information 15 (3):179-194.
    In this paper we give axiom systems for classical and intuitionistic hybrid logic. Our axiom systems can be extended with additional rules corresponding to conditions on the accessibility relation expressed by so-called geometric theories. In the classical case other axiomatisations than ours can be found in the literature but in the intuitionistic case no axiomatisations have been published. We consider plain intuitionistic hybrid logic as well as a hybridized version of the constructive and paraconsistent logic N4.
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  43. A Note on Cancellation Axioms for Comparative Probability.Matthew Harrison-Trainor, Wesley H. Holliday & Thomas F. Icard - 2016 - Theory and Decision 80 (1):159-166.
    We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Rios Insua and Alon and Lehrer is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott, relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature.
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  44. Quantum Nonlocality as an Axiom.Sandu Popescu & Daniel Rohrlich - 1994 - Foundations of Physics 24 (3):379-385.
    In the conventional approach to quantum mechanics, indeterminism is an axiom and nonlocality is a theorem. We consider inverting the logical order, making nonlocality an axiom and indeterminism a theorem. Nonlocal “superquantum” correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.
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  45.  14
    Choosing Axioms of Correlativity.Andrew Halpin - 2019 - American Journal of Jurisprudence 64 (2):225-258.
    This article explores an axiomatic approach to distinguishing different usages of correlativity and investigates Hurd and Moore’s disagreement with Hohfeldian correlativity, in terms of a choice of axioms. Detailed critical consideration is provided of three negative steps, ascribing theoretical positions to Hohfeld that Hurd and Moore wish to amend or depart from; and three positive steps taken towards vindicating their stated objectives of avoiding moral combat and providing recognition to active rights. The conclusion is reached that the actual state (...)
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  46. Die Axiome der Geometry Eine Philosophische Untersuchung der Riemann-Helmholtz'schen Raumtheorie.Benno Erdmann - 1877 - L. Voss.
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  47. Axioms for Collections of Indistinguishable Objects.Décio Krause - 1996 - Logique Et Analyse 153 (154):69-93.
  48.  4
    Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory.Kurt Gödel - 1940 - Princeton, NJ, USA: Princeton University Press.
  49.  44
    Qualitative Axioms of Uncertainty as a Foundation for Probability and Decision-Making.Patrick Suppes - 2016 - Minds and Machines 26 (1-2):185-202.
    Although the concept of uncertainty is as old as Epicurus’s writings, and an excellent quantitative theory, with entropy as the measure of uncertainty having been developed in recent times, there has been little exploration of the qualitative theory. The purpose of the present paper is to give a qualitative axiomatization of uncertainty, in the spirit of the many studies of qualitative comparative probability. The qualitative axioms are fundamentally about the uncertainty of a partition of the probability space of events. (...)
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  50.  31
    An Axiom System for Deontic Logic.Nicholas Rescher - 1958 - Philosophical Studies 9 (1-2):24 - 30.
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