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.score: 24.0
    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. Daniel I. Fivel (2012). Derivation of the Rules of Quantum Mechanics From Information-Theoretic Axioms. Foundations of Physics 42 (2):291-318.score: 24.0
    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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  3. Rüdiger Schack (2003). Quantum Theory From Four of Hardy's Axioms. Foundations of Physics 33 (10):1461-1468.score: 24.0
    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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  4. Joan Rand Moschovakis & Garyfallia Vafeiadou (2012). Some Axioms for Constructive Analysis. Archive for Mathematical Logic 51 (5-6):443-459.score: 24.0
    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. 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.score: 24.0
    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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  6. Riccardo Camerlo (2007). Finiteness Axioms on Fragments of Intuitionistic Set Theory. Notre Dame Journal of Formal Logic 48 (4):473-488.score: 22.0
    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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  7. Øystein Linnebo (2012). Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory. [REVIEW] Philosophy 87 (01):133-137.score: 21.0
  8. 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.score: 21.0
    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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  9. Mohammad Ardeshir & S. Mojtaba Mojtahedi (2014). Completeness of Intermediate Logics with Doubly Negated Axioms. Mathematical Logic Quarterly 60 (1-2):6-11.score: 21.0
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  10. M. Carmen Sánchez (1998). Rational Choice on Non-Finite Sets by Means of Expansion-Contraction Axioms. Theory and Decision 45 (1):1-17.score: 21.0
    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 (1976), 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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  11. Peter C. Fishburn (1993). The Axioms and Algebra of Ambiguity. Theory and Decision 34 (2):119-137.score: 21.0
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  12. A. S. Troelstra (1975). Axioms for Intuitionistic Mathematics Incompatible with Classical Logic. Mathematisch Instituut.score: 21.0
     
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  13. Athanassios Tzouvaras (2010). Localizing the Axioms. Archive for Mathematical Logic 49 (5):571-601.score: 20.0
    We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}$ consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ0-Collection and minus ${\in}$ -induction scheme. ZFC+ “there is (...)
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  14. Kenny Easwaran (2008). The Role of Axioms in Mathematics. Erkenntnis 68 (3):381 - 391.score: 18.0
    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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  15. 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.score: 18.0
    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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  16. Karl-Georg Niebergall (2000). On the Logic of Reducibility: Axioms and Examples. [REVIEW] Erkenntnis 53 (1-2):27-61.score: 18.0
    This paper is an investigation into what could be a goodexplication of ``theory S is reducible to theory T''''. Ipresent an axiomatic approach to reducibility, which is developedmetamathematically and used to evaluate most of the definitionsof ``reducible'''' found in the relevant literature. Among these,relative interpretability turns out to be most convincing as ageneral reducibility concept, proof-theoreticalreducibility being its only serious competitor left. Thisrelation is analyzed in some detail, both from the point of viewof the reducibility axioms and of modal (...)
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  17. J. H. Harris (1982). What's So Logical About the “Logical” Axioms? Studia Logica 41 (2-3):159 - 171.score: 18.0
    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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  18. Wilfried Sieg & Dirk Schlimm (2005). Dedekind's Analysis of Number: Systems and Axioms. Synthese 147 (1):121 - 170.score: 18.0
    Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.
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  19. Jamie Tappenden (2000). Frege on Axioms, Indirect Proof, and Independence Arguments in Geometry: Did Frege Reject Independence Arguments? Notre Dame Journal of Formal Logic 41 (3):271-315.score: 18.0
    It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct (...)
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  20. Victoria Gitman & Joel David Hamkins (2010). A Natural Model of the Multiverse Axioms. Notre Dame Journal of Formal Logic 51 (4):475-484.score: 18.0
    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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  21. Ivahn Smadja (2012). Local Axioms in Disguise: Hilbert on Minkowski Diagrams. Synthese 186 (1):315-370.score: 18.0
    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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  22. W. W. Tait (2001). Beyond the Axioms: The Question of Objectivity in Mathematics. Philosophia Mathematica 9 (1):21-36.score: 18.0
    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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  23. Solomon Feferman, Presentation to the Panel, “Does Mathematics Need New Axioms?” Asl 2000 Meeting, Urbana Il, June 5, 2000.score: 18.0
    The point of departure for this panel is a somewhat controversial paper that I published in the American Mathematical Monthly under the title “Does mathematics need new axioms?” [4]. The paper itself was based on a lecture that I gave in 1997 to a joint session of the American Mathematical Society and the Mathematical Association of America, and it was thus written for a general mathematical audience. Basically, it was intended as an assessment of Gödel’s program for new (...) that he had advanced most prominently in his 1947 paper for the Monthly, entitled “What is Cantor’s continuum problem?” [7]. My paper aimed to be an assessment of that program in the light of research in mathematical logic in the intervening years, beginning in the 1960s, but especially in more recent years. (shrink)
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  24. N. Hadjisavvas & F. Thieffine (1984). Piron's Axioms for Quantum Mechanics: A Reply to Foulis and Randall. [REVIEW] Foundations of Physics 14 (1):83-88.score: 18.0
    In their paper “A note on Misunderstandings of Piron's Axioms for Quantum Mechanics,” Foulis and Randall undertake a reply to our critique of Piron's “question-proposition system” (qp-s) which appeared in previous issues of this journal. In the present paper, we want briefly to refute the points of criticism raised by Foulis and Randall (FR). We argue that the “misunderstandings” are not ours, and we prove it.
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  25. H. Helmholtz (1876). The Origin and Meaning of Geometrical Axioms. Mind 1 (3):301-321.score: 18.0
    The object in this article is to discuss the philosophical bearing of recent inquiries concerning geometrical axioms and the possibility of working out analytically other systems of geometry with other axioms than Euclid's. Digital edition compiled by Gabriele Dörflinger, Heidelberg University Library.
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  26. Chris Freiling (1986). Axioms of Symmetry: Throwing Darts at the Real Number Line. Journal of Symbolic Logic 51 (1):190-200.score: 18.0
    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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  27. Wei Xiong (2011). Implications of the Dutch Book: Following Ramsey's Axioms. Frontiers of Philosophy in China 6 (2):334-344.score: 18.0
    The Dutch Book Argument shows that an agent will lose surely in a gamble (a Dutch Book is made) if his degrees of belief do not satisfy the laws of the probability. Yet a question arises here: What does the Dutch Book imply? This paper firstly argues that there exists a utility function following Ramsey’s axioms. And then, it explicates the properties of the utility function and degree of belief respectively. The properties show that coherence in partial beliefs for (...)
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  28. Rudolf Carnap, Friedrich Bachmann & H. G. Bohnert (1981). On Extremal Axioms. History and Philosophy of Logic 2 (1-2):67-85.score: 18.0
    In the paper translated here, Carnap and Bachmann shows that the apparently metalinguistic ?extremal' axioms that are added to some axiom systems to the effect that the foregoing axioms are to apply as broadly, or as narrowly, as possible may be formulated directly as proper axioms. They analyze such axioms into four fundamental types, with the help of a concept of ?complete? isomorphism.
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  29. D. Schlimm (2013). Axioms in Mathematical Practice. Philosophia Mathematica 21 (1):37-92.score: 18.0
    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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  30. Itzhak Gilboa, Andrew Postlewaite & David Schmeidler (2009). Is It Always Rational to Satisfy Savage's Axioms? Economics and Philosophy 25 (3):285-296.score: 18.0
    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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  31. Jairo José Da Silva (2002). The Axioms of Set Theory. Axiomathes 13 (2):107-126.score: 18.0
    In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that (...)
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  32. Solomon Feferman, Godel's Program for New Axioms: Why, Where, How and What?score: 18.0
    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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  33. Louis Narens (1974). Measurement Without Archimedean Axioms. Philosophy of Science 41 (4):374-393.score: 18.0
    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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  34. Wilfried Sieg, Church Without Dogma: Axioms for Computability.score: 18.0
    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 work (...)
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  35. Francesca Biagioli (2014). What Does It Mean That “Space Can Be Transcendental Without the Axioms Being So”? Journal for General Philosophy of Science 45 (1):1-21.score: 18.0
    In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that (...)
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  36. Torben Braüner (2006). Axioms for Classical, Intuitionistic, and Paraconsistent Hybrid Logic. Journal of Logic, Language and Information 15 (3):179-194.score: 18.0
    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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  37. Harvey Friedman, Does Normal Mathematics Need New Axioms?score: 18.0
    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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  38. Alberto Naibo, Mattia Petrolo & Thomas Seiller, On the Computational Meaning of Axioms.score: 18.0
    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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  39. Georg Schiemer (2012). Carnap on Extremal Axioms, "Completeness of the Models," and Categoricity. Review of Symbolic Logic 5 (04):613-641.score: 18.0
    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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  40. James Ax (1976). Group-Theoretic Treatment of the Axioms of Quantum Mechanics. Foundations of Physics 6 (4):371-399.score: 18.0
    This axiomatization is based on the observation that ifG is the group of automorphisms of the states (induced, e.g., by suitable evolutions), then we can define a spherical function by mapping each element ofG to the matrix of its transition probabilities. Starting from five physically conservative axioms, we utilize the correspondence between spherical functions and representations to apply the structure theory for compact Lie groups and their orbits in representation spaces to arrive at the standard complex Hilbert space structure (...)
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  41. Jairo José Da Silva (2002). The Axioms of Set Theory. Axiomathes 13 (2):107-126.score: 18.0
    In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that (...)
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  42. Dmitry Sustretov (2009). Hybrid Logics of Separation Axioms. Journal of Logic, Language and Information 18 (4):541-558.score: 18.0
    We study hybrid logics in topological semantics. We prove that hybrid logics of separation axioms are complete with respect to certain classes of finite topological models. This characterisation allows us to obtain several further results. We prove that aforementioned logics are decidable and PSPACE-complete, the logics of T 1 and T 2 coincide, the logic of T 1 is complete with respect to two concrete structures: the Cantor space and the rational numbers.
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  43. H. Andréka, Á Kurucz & I. Németi (1994). Connections Between Axioms of Set Theory and Basic Theorems of Universal Algebra. Journal of Symbolic Logic 59 (3):912-923.score: 18.0
    One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class K of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of K. G. Gratzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in ZF. Surprisingly, the Axiom of Foundation plays a crucial role (...)
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  44. Bengt Hansson (1968). Fundamental Axioms for Preference Relations. Synthese 18 (4):423 - 442.score: 18.0
    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. R. D. Kopperman (1981). First-Order Topological Axioms. Journal of Symbolic Logic 46 (3):475-489.score: 18.0
    We exhibit a finite list of first-order axioms which may be used to define topological spaces. For most separation axioms we discover a first-order equivalent statement.
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  46. Daniel R. Brooks & Richard T. O'Grady (1986). Nonequilibrium Thermodynamics and Different Axioms of Evolution. Acta Biotheoretica 35 (1-2).score: 18.0
    Proponents of two axioms of biological evolutionary theory have attempted to find justification by reference to nonequilibrium thermodynamics. One states that biological systems and their evolutionary diversification are physically improbable states and transitions, resulting from a selective process; the other asserts that there is an historically constrained inherent directionality in evolutionary dynamics, independent of natural selection, which exerts a self-organizing influence. The first, the Axiom of Improbability, is shown to be nonhistorical and thus, for a theory of change through (...)
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  47. Kooi, Barteld (2007). Expressivity and Completeness for Public Update Logics Via Reduction Axioms. Journal of Applied Non-Classical Logics 17 (2):231-253.score: 18.0
    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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  48. Sara Negri & Jan Von Plato (1998). Cut Elimination in the Presence of Axioms. Bulletin of Symbolic Logic 4 (4):418-435.score: 18.0
    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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  49. T. Thacher Robinson (1968). Independence of Two Nice Sets of Axioms for the Propositional Calculus. Journal of Symbolic Logic 33 (2):265-270.score: 18.0
    Kanger [4] gives a set of twelve axioms for the classical propositional Calculus which, together with modus ponens and substitution, have the following nice properties: (0.1) Each axiom contains $\supset$ , and no axiom contains more than two different connectives. (0.2) Deletions of certain of the axioms yield the intuitionistic, minimal, and classical refutability1 subsystems of propositional calculus. (0.3) Each of these four systems of axioms has the separation property: that if a theorem is provable in such (...)
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  50. Robert Shaver (2014). Sidgwick's Axioms and Consequentialism. Philosophical Review 123 (2):173-204.score: 18.0
    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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