Search results for 'Axioms' (try it on Scholar)

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  1. Flavia Padovani (2011). Relativizing the Relativized a Priori: Reichenbach's Axioms of Coordination Divided. Synthese 181 (1):41 - 62.
    In recent years, Reichenbach's 1920 conception of the principles of coordination has attracted increased attention after Michael Friedman's attempt to revive Reichenbach's idea of a "relativized a priori". This paper follows the origin and development of this idea in the framework of Reichenbach's distinction between the axioms of coordination and the axioms of connection. It suggests a further differentiation among the coordinating axioms and accordingly proposes a different account of Reichenbach's "relativized a priori".
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  2.  36
    Rüdiger Schack (2003). Quantum Theory From Four of Hardy's Axioms. Foundations of Physics 33 (10):1461-1468.
    In a recent paper [e-print quant-ph/0101012], Hardy has given a derivation of “quantum theory from five reasonable axioms.” Here we show that Hardy's first axiom, which identifies probability with limiting frequency in an ensemble, is not necessary for his derivation. By reformulating Hardy's assumptions, and modifying a part of his proof, in terms of Bayesian probabilities, we show that his work can be easily reconciled with a Bayesian interpretation of quantum probability.
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  3.  10
    Matthew Harrison-Trainor, Wesley H. Holliday & Thomas F. Icard (2016). A Note on Cancellation Axioms for Comparative Probability. 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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  4.  22
    Joan Rand Moschovakis & Garyfallia Vafeiadou (2012). Some Axioms for Constructive Analysis. 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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  5.  18
    Daniel I. Fivel (2012). Derivation of the Rules of Quantum Mechanics From Information-Theoretic Axioms. Foundations of Physics 42 (2):291-318.
    Conventional quantum mechanics with a complex Hilbert space and the Born Rule is derived from five axioms describing experimentally observable properties of probability distributions for the outcome of measurements. Axioms I, II, III are common to quantum mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first noted by von Neumann (in Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955) and independently by Turing (Teuscher and Hofstadter, Alan Turing: Life and Legacy of a Great Thinker, (...)
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  6.  5
    Robert Sochacki (2009). Rejected Axioms for the “Nonsense-Logic” W and the K-Valued Logic of Sobociński. 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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  7.  23
    Itzhak Gilboa, Andrew Postlewaite & David Schmeidler (2012). Rationality of Belief Or: Why Savage's Axioms Are Neither Necessary nor Sufficient for Rationality. [REVIEW] Synthese 187 (1):11-31.
    Economic theory reduces the concept of rationality to internal consistency. As far as beliefs are concerned, rationality is equated with having a prior belief over a “Grand State Space”, describing all possible sources of uncertainties. We argue that this notion is too weak in some senses and too strong in others. It is too weak because it does not distinguish between rational and irrational beliefs. Relatedly, the Bayesian approach, when applied to the Grand State Space, is inherently incapable of describing (...)
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  8. Øystein Linnebo (2012). Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory. [REVIEW] Philosophy 87 (01):133-137.
  9.  1
    Peter C. Fishburn (1993). The Axioms and Algebra of Ambiguity. Theory and Decision 34 (2):119-137.
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  10.  4
    Riccardo Camerlo (2007). Finiteness Axioms on Fragments of Intuitionistic Set Theory. Notre Dame Journal of Formal Logic 48 (4):473-488.
    It is proved that in a suitable intuitionistic, locally classical, version of the theory ZFC deprived of the axiom of infinity, the requirement that every set be finite is equivalent to the assertion that every ordinal is a natural number. Moreover, the theory obtained with the addition of these finiteness assumptions is equivalent to a theory of hereditarily finite sets, developed by Previale in "Induction and foundation in the theory of hereditarily finite sets." This solves some problems stated there. The (...)
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  11.  4
    Mohammad Ardeshir & S. Mojtaba Mojtahedi (2014). Completeness of Intermediate Logics with Doubly Negated Axioms. Mathematical Logic Quarterly 60 (1-2):6-11.
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  12.  4
    M. Carmen Sánchez (1998). Rational Choice on Non-Finite Sets by Means of Expansion-Contraction Axioms. Theory and Decision 45 (1):1-17.
    The rationalization of a choice function, in terms of assumptions that involve expansion or contraction properties of the feasible set, over non-finite sets is analyzed. Schwartz's results, stated in the finite case, are extended to this more general framework. Moreover, a characterization result when continuity conditions are imposed on the choice function, as well as on the binary relation that rationalizes it, is presented.
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  13. A. S. Troelstra (1975). Axioms for Intuitionistic Mathematics Incompatible with Classical Logic. Mathematisch Instituut.
     
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  14.  78
    Kai Hauser & W. Hugh Woodin (2014). Strong Axioms of Infinity and the Debate About Realism. 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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  15. Penelope Maddy (2011). Defending the Axioms: On the Philosophical Foundations of Set Theory. 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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  16. Kooi, Barteld (2007). Expressivity and Completeness for Public Update Logics Via Reduction Axioms. Journal of Applied Non-Classical Logics 17 (2):231-253.
    In this paper, we present several extensions of epistemic logic with update operators modelling public information change. Next to the well-known public announcement operators, we also study public substitution operators. We prove many of the results regarding expressivity and completeness using so-called reduction axioms. We develop a general method for using reduction axioms and apply it to the logics at hand.
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  17.  17
    Georg Schiemer (2012). Carnap on Extremal Axioms, "Completeness of the Models," and Categoricity. Review of Symbolic Logic 5 (4):613-641.
    This paper provides a historically sensitive discussion of Carnaps theory will be assessed with respect to two interpretive issues. The first concerns his mathematical sources, that is, the mathematical axioms on which his extremal axioms were based. The second concerns Carnapcompleteness of the modelss different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time.
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  18.  31
    Alexander Wilce (2012). Four and a Half Axioms for Finite-Dimensional Quantum Probability. In Yemima Ben-Menahem & Meir Hemmo (eds.), Probability in Physics. Springer 281--298.
    It is an old idea, lately out of fashion but now experiencing a revival, that quantum mechanics may best be understood, not as a physical theory with a problematic probabilistic interpretation, but as something closer to a probability calculus per se. However, from this angle, the rather special C *-algebraic apparatus of quantum probability theory stands in need of further motivation. One would like to find additional principles, having clear physical and/or probabilistic content, on the basis of which this apparatus (...)
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  19. Kenny Easwaran (2008). The Role of Axioms in Mathematics. Erkenntnis 68 (3):381 - 391.
    To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept (...)
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  20.  3
    Keith E. Stanovich (2012). Why Humans Are (Sometimes) Less Rational Than Other Animals: Cognitive Complexity and the Axioms of Rational Choice. Thinking and Reasoning 19 (1):1 - 26.
    (2013). Why humans are (sometimes) less rational than other animals: Cognitive complexity and the axioms of rational choice. Thinking & Reasoning: Vol. 19, No. 1, pp. 1-26. doi: 10.1080/13546783.2012.713178.
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  21.  17
    Sara Negri & Jan Von Plato (1998). Cut Elimination in the Presence of Axioms. Bulletin of Symbolic Logic 4 (4):418-435.
    A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of (...)
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  22.  45
    Igor L. Aleksander & B. Dunmall (2003). Axioms and Tests for the Presence of Minimal Consciousness in Agents I: Preamble. Journal of Consciousness Studies 10 (4):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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  23.  26
    Solomon Feferman, Godel's Program for New Axioms: Why, Where, How and What?
    From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there (...)
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  24.  12
    Robert Shaver (2014). Sidgwick's Axioms and Consequentialism. Philosophical Review 123 (2):173-204.
    Sidgwick gives various tests for highest certainty. When he applies these tests to commonsense morality, he finds nothing of highest certainty. In contrast, when he applies these tests to his own axioms, he finds these axioms to have highest certainty. The axioms culminate in Benevolence: “Each one is morally bound to regard the good of any other individual as much as his own, except in so far as he judges it to be less, when impartially viewed, or (...)
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  25.  22
    Itzhak Gilboa, Andrew Postlewaite & David Schmeidler (2009). Is It Always Rational to Satisfy Savage's Axioms? Economics and Philosophy 25 (3):285-296.
    This note argues that, under some circumstances, it is more rational not to behave in accordance with a Bayesian prior than to do so. The starting point is that in the absence of information, choosing a prior is arbitrary. If the prior is to have meaningful implications, it is more rational to admit that one does not have sufficient information to generate a prior than to pretend that one does. This suggests a view of rationality that requires a compromise between (...)
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  26.  11
    Yde Venema (1993). Derivation Rules as Anti-Axioms in Modal Logic. Journal of Symbolic Logic 58 (3):1003-1034.
    We discuss a `negative' way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the `non-ξ rules', styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas and Λ+ is the extension of Λ with a set of non-ξ rules, then Λ+ (...)
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  27.  11
    Stephen G. Simpson (1984). Which Set Existence Axioms Are Needed to Prove the Cauchy/Peano Theorem for Ordinary Differential Equations? 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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  28. Gerhard Jäger & Thomas Studer (2002). Extending the System T0 of Explicit Mathematics: The Limit and Mahlo Axioms. Annals of Pure and Applied Logic 114 (1-3):79-101.
    In this paper we discuss extensions of Feferman's theory T 0 for explicit mathematics by the so-called limit and Mahlo axioms and present a novel approach to constructing natural recursion-theoretic models for systems of explicit mathematics which is based on nonmonotone inductive definitions.
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  29.  5
    Patrick Suppes (forthcoming). Qualitative Axioms of Uncertainty as a Foundation for Probability and Decision-Making. Minds and Machines:1-18.
    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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  30.  62
    Wilfried Sieg & Dirk Schlimm (2005). Dedekind's Analysis of Number: Systems and Axioms. Synthese 147 (1):121 - 170.
    Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.
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  31.  31
    D. Schlimm (2013). Axioms in Mathematical Practice. 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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  32.  39
    Victoria Gitman & Joel David Hamkins (2010). A Natural Model of the Multiverse Axioms. Notre Dame Journal of Formal Logic 51 (4):475-484.
    If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins.
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  33.  9
    Johan van Benthem & Barteld Kooi, Reduction Axioms for Epistemic Actions.
    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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  34. Timothy J. Surendonk (2001). Canonicity for Intensional Logics with Even Axioms. Journal of Symbolic Logic 66 (3):1141-1156.
    This paper looks at the concept of neighborhood canonicity introduced by BRIAN CHELLAS [2]. We follow the lead of the author's paper [9] where it was shown that every non-iterative logic is neighborhood canonical and here we will show that all logics whose axioms have a simple syntactic form-no intensional operator is in boolean combination with a propositional letter-and which have the finite model property are neighborhood canonical. One consequence of this is that KMcK, the McKinsey logic, is neighborhood (...)
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  35.  52
    Alberto Naibo, Mattia Petrolo & Thomas Seiller, On the Computational Meaning of Axioms.
    An anti-realist theory of meaning suitable for both logical and proper axioms is investigated. As opposed to other anti-realist accounts, like Dummett-Prawitz verificationism, the standard framework of classical logic is not called into question. In particular, semantical features are not limited solely to inferential ones, but also computational aspects play an essential role in the process of determination of meaning. In order to deal with such computational aspects, a relaxation of syntax is shown to be necessary. This leads to (...)
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  36.  52
    Heiko Herrmann, W. Muschik, G. Rückner & H.-H. Von Borzeszkowski (2004). Spin Axioms in Different Geometries of Relativistic Continuum Physics. Foundations of Physics 34 (6):1005-1021.
    The 24 components of the relativistic spin tensor consist of 3 + 3 basic spin fields and 9 + 9 constitutive fields. Empirically only three basic spin fields and nine constitutive fields are known. This empirem can be expressed by two spin axioms, one of them denying purely relativistic spin fields, and the other one relating the three additional basic fields and the nine additional constitutive fields to the known (and measurable) ones. This identification by the spin axioms (...)
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  37. Kooi, Barteld & van Benthem, Johan, Reduction Axioms for Epistemic Actions.
    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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  38.  4
    Matthew H. Kramer (2007). Why The Axioms and Theorems of Arithmetic Are Not Legal Norms. Oxford Journal of Legal Studies 27 (3):555-562.
    Ronald Dworkin has long criticized legal positivists for their efforts to distinguish between legal and non-legal standards of conduct that are incumbent on people. Recently, Dworkin has broached this criticism in his hostile account of the debates between Incorporationist Legal Positivists and Exclusive Legal Positivists. Specifically, he has maintained that Incorporationists cannot avoid the unpalatable conclusion that the axioms and theorems of arithmetic are legal norms. This article shows why such a conclusion is indeed avoidable and why Dworkin's criticism (...)
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  39.  1
    Jan von Plato (1995). The Axioms of Constructive Geometry. 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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  40.  27
    Ivahn Smadja (2012). Local Axioms in Disguise: Hilbert on Minkowski Diagrams. 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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  41.  48
    J. H. Harris (1982). What's So Logical About the “Logical” Axioms? Studia Logica 41 (2-3):159 - 171.
    Intuitionists and classical logicians use in common a large number of the logical axioms, even though they supposedly mean different things by the logical connectives and quantifiers — conquans for short. But Wittgenstein says The meaning of a word is its use in the language. We prove that in a definite sense the intuitionistic axioms do indeed characterize the logical conquans, both for the intuitionist and the classical logician.
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  42.  20
    Wilfried Sieg, Church Without Dogma: Axioms for Computability.
    Church's and Turing's theses dogmatically assert that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses. To investigate effective calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting (...)
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  43.  6
    Joel David Hamkins (2001). The Wholeness Axioms and V=HOD. Archive for Mathematical Logic 40 (1):1-8.
    If the Wholeness Axiom wa $_0$ is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa $_0$ is finitely axiomatizable.
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  44.  17
    Bengt Hansson (1968). Fundamental Axioms for Preference Relations. 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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  45. Harvey Friedman, Does Normal Mathematics Need New Axioms?
    We present a range of mathematical theorems whose proofs require unexpectedly strong logical methods, which in some cases go well beyond the usual axioms for mathematics.
     
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  46.  31
    Chris Freiling (1986). Axioms of Symmetry: Throwing Darts at the Real Number Line. 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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  47.  1
    Joan Bagaria (2000). Bounded Forcing Axioms as Principles of Generic Absoluteness. Archive for Mathematical Logic 39 (6):393-401.
    We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a $\Sigma_1$ sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters is (...)
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  48.  39
    W. W. Tait (2001). Beyond the Axioms: The Question of Objectivity in Mathematics. 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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  49.  23
    Louis Narens (1974). Measurement Without Archimedean Axioms. Philosophy of Science 41 (4):374-393.
    Axiomatizations of measurement systems usually require an axiom--called an Archimedean axiom--that allows quantities to be compared. This type of axiom has a different form from the other measurement axioms, and cannot--except in the most trivial cases--be empirically verified. In this paper, representation theorems for extensive measurement structures without Archimedean axioms are given. Such structures are represented in measurement spaces that are generalizations of the real number system. Furthermore, a precise description of "Archimedean axioms" is given and it (...)
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  50. John K. Dagsvik & Stine Røine Hoff (2011). Justification of Functional Form Assumptions in Structural Models: Applications and Testing of Qualitative Measurement Axioms. [REVIEW] Theory and Decision 70 (2):215-254.
    In both theoretical and applied modeling in behavioral sciences, it is common to choose a mathematical specification of functional form and distribution of unobservables on grounds of analytic convenience without support from explicit theoretical postulates. This article discusses the issue of deriving particular qualitative hypotheses about functional form restrictions in structural models from intuitive theoretical axioms. In particular, we focus on a family of postulates known as dimensional invariance. Subsequently, we discuss how specific qualitative postulates can be reformulated so (...)
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