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Logical axiomatizations of spacetime. Samples from the literature
 In: NonEuclidean Geometries (J'anos Bolyai Memorial Volume
, 2005
"... Abstract We study relativity theory as a theory in the sense of mathematical logic. We use firstorder logic (FOL) as a framework to do so. We aim at an “analysis of the logical structure of relativity theories”. First we build up (the kinematics of) special relativity in FOL, then analyze it, and t ..."
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Cited by 10 (6 self)
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Abstract We study relativity theory as a theory in the sense of mathematical logic. We use firstorder logic (FOL) as a framework to do so. We aim at an “analysis of the logical structure of relativity theories”. First we build up (the kinematics of) special relativity in FOL, then analyze it, and then we experiment with generalizations in the direction of general relativity. The present paper gives samples from an ongoing broader research project which in turn is part of a research direction going back to Reichenbach and others in the 1920’s. We also try to give some perspective on the literature related in a broader sense. In the perspective of the present work, axiomatization is not a final goal. Axiomatization is only a first step, a tool. The goal is something like a conceptual analysis of relativity in the framework of logic. In section 1 we recall a complete FOLaxiomatization Specrel of special relativity from [5],[31]. In section 2 we answer questions from papers by Ax and Mundy concerning the logical status of faster than light motion (FTL) in relativity. We claim that already very small/weak fragments of Specrel prove “No FTL”. In section 3 we give a sketchy outlook for the possibility of generalizing Specrel to theories permitting accelerated observers (gravity). In section 4 we continue generalizing Specrel in the direction of general relativity by localizing it, i.e. by replacing it with a version still in firstorder logic but now local (in the sense of general relativity theory). In section 5 we give samples from the broader literature.
Firstorder logic foundation of relativity theories
 In New Logics for the XXIst Century II, Mathematical Problems from Applied Logics, volume 5 of International Mathematical Series
, 2006
"... Abstract. Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity— including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to ..."
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Cited by 8 (8 self)
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Abstract. Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity— including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction.
TWIN PARADOX AND THE LOGICAL FOUNDATION OF RELATIVITY THEORY
, 2005
"... Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a ..."
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Cited by 7 (6 self)
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Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a simple and streamlined FOLaxiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle ” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that the Twin Paradox becomes provable in AccRel, but it is not provable without IND. Key words: twin paradox, relativity theory, accelerated observers, firstorder logic, axiomatization, foundation of relativity theory 1.
Twin Paradox and the logical foundation of spacetime. Foundation of Physics
"... Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a ..."
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Cited by 7 (6 self)
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Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a simple and streamlined FOLaxiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle ” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that the Twin Paradox becomes provable in AccRel, but it is not provable without IND. 1.
Formal Theory Building Using Automated Reasoning Tools
 In
, 1998
"... The merits of representing scientific theories in formal logic are wellknown. Expressing a scientific theory in formal logic explicates the theory as a whole, and the logic provides formal criteria for evaluating the theory, such as soundness and consistency. On the one hand, these criteria corresp ..."
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Cited by 6 (6 self)
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The merits of representing scientific theories in formal logic are wellknown. Expressing a scientific theory in formal logic explicates the theory as a whole, and the logic provides formal criteria for evaluating the theory, such as soundness and consistency. On the one hand, these criteria correspond to natural questions to be asked about the theory: is the theory contradictionfree? (is the theory logically consistent?) is the theoretical argumentation valid? (can a theorem be soundly derived from the premises?) and other such questions. On the other hand, testing for these criteria amounts to making many specific proof attempts or model searches: respectively, does the theory have a model? can we find a proof of a particular theorem? As a result, testing for these criteria quickly defies manual processing. Fortunately, automated reasoning provides some valuable tools for this endeavor. This paper discusses the use of firstorder logic and existing automated rea...
On criteria for formal theory building: Applying logic and automated reasoning tools to the social sciences
 In Proc. AAAI’99
, 1999
"... This paper provides practical operationalizations of criteria for evaluating scientific theories, such as the consistency and falsifiability of theories and the soundness of inferences, that take into account definitions. The precise formulation of these criteria is tailored to the use of automated ..."
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Cited by 4 (3 self)
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This paper provides practical operationalizations of criteria for evaluating scientific theories, such as the consistency and falsifiability of theories and the soundness of inferences, that take into account definitions. The precise formulation of these criteria is tailored to the use of automated theorem provers and automated model generators—generic tools from the field of automated reasoning. The use of these criteria is illustrated by applying them to a first order logic representation of a classic organization theory, Thompson’s Organizations in Action.
AXIOMATIZING RELATIVISTIC DYNAMICS WITHOUT CONSERVATION POSTULATES
"... Abstract. A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within firstorder logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc 2 ..."
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Cited by 3 (3 self)
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Abstract. A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within firstorder logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc 2. The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and fourmomentum is also investigated. 1.
The logic of organizational markets: Thinking through resource partitioning theory
 Computational and Mathematical Organization Theory
, 2001
"... Resource partitioning theory claims that “Increasing concentration enhances the life chances of specialist organizations. ” We systematically think through this theory, specify implicit background assumptions, sharpen concepts, and rigorously check the theory’s logic. As a result, we increase the ..."
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Cited by 1 (1 self)
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Resource partitioning theory claims that “Increasing concentration enhances the life chances of specialist organizations. ” We systematically think through this theory, specify implicit background assumptions, sharpen concepts, and rigorously check the theory’s logic. As a result, we increase the theory’s explanatory power, and claim— contrary to received opinion—that under certain general conditions, “resource partitioning ” and the proliferation of specialists can take place independently of organizational mass and relative size effects, size localized competition, diversifying consumer tastes, increasing number of dimensions of the resource space, and changing niche widths. Our analysis makes furthermore clear that specialist and generalist strategies are asymmetric, and shows that not concentration enhances the life chances of specialists but economies of scale instead. Under the conditions explicated, we argue that if scale economies come to dominate, the number of organizations in the population increases, regardless of the incumbents ’ sizes. Key words: theory reconstruction; resource partitioning; competition; market concentration; economies of scale; niche; specialization; organizational ecology; logical formalization; applied logic. ∗We are grateful to Jaap Kamps, Michael Masuch, Patricia Thornton, and Jelka Hopster for their helpful comments on an earlier version, and to Glenn Carroll, Gábor Péli, Arjen van Witteloostuijn and Christophe Boone for discussions about resource partitioning. 1 1
Working Paper No. 731 The Common Error of Common Sense: An Essential Rectification of the Accounting Approach by
, 2012
"... Levy Institute scholars and conference participants. The purpose of the series is to disseminate ideas to and elicit comments from academics and professionals. Levy Economics Institute of Bard College, founded in 1986, is a nonprofit, nonpartisan, independently funded research organization devoted t ..."
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Levy Institute scholars and conference participants. The purpose of the series is to disseminate ideas to and elicit comments from academics and professionals. Levy Economics Institute of Bard College, founded in 1986, is a nonprofit, nonpartisan, independently funded research organization devoted to public service. Through scholarship and economic research it generates viable, effective public policy responses to important economic problems that profoundly affect the quality of life in the United States and abroad.