Results for 'proof'

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  1.  90
    Basic Proof Theory.A. S. Troelstra - 2000 - Cambridge University Press.
    This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in (...)
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  2. Proofs and Refutations: The Logic of Mathematical Discovery.Imre Lakatos (ed.) - 1976 - Cambridge and London: Cambridge University Press.
    Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...)
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  3. On the Concept of Proof in Elementary Geometry Pirmin Stekeler-Weithofer.Proof In Elementary - 1992 - In Michael Detlefsen (ed.), Proof and Knowledge in Mathematics. Routledge.
     
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  4. Natural Deduction: A Proof-Theoretical Study.Dag Prawitz - 1965 - Stockholm, Sweden: Dover Publications.
    This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.
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  5. Evidence, Proofs, and Derivations.Andrew Aberdein - 2019 - ZDM 51 (5):825-834.
    The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation (...)
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  6.  57
    Proof Paradoxes, Agency, and Stereotyping.Aness Kim Webster - 2021 - Wiley: Philosophical Issues 31 (1):355-373.
    Philosophical Issues, Volume 31, Issue 1, Page 355-373, October 2021.
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  7.  75
    Structural Proof Theory.Sara Negri, Jan von Plato & Aarne Ranta - 2001 - Cambridge University Press.
    Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such (...)
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  8. Proofs, Pictures, and Euclid.John Mumma - 2010 - Synthese 175 (2):255 - 287.
    Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid's reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the received (...)
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  9.  2
    Proof Theory.Gaisi Takeuti - 1975 - Elsevier.
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  10.  47
    Proof Analysis: A Contribution to Hilbert's Last Problem.Sara Negri & Jan von Plato - 2011 - Cambridge and New York: Cambridge University Press.
    Machine generated contents note: Prologue: Hilbert's Last Problem; 1. Introduction; Part I. Proof Systems Based on Natural Deduction: 2. Rules of proof: natural deduction; 3. Axiomatic systems; 4. Order and lattice theory; 5. Theories with existence axioms; Part II. Proof Systems Based on Sequent Calculus: 6. Rules of proof: sequent calculus; 7. Linear order; Part III. Proof Systems for Geometric Theories: 8. Geometric theories; 9. Classical and intuitionistic axiomatics; 10. Proof analysis in elementary geometry; (...)
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  11.  46
    Proof and Falsity: A Logical Investigation.Nils Kürbis - 2019 - Cambridge, UK: Cambridge University Press.
    This book argues that the meaning of negation, perhaps the most important logical constant, cannot be defined within the framework of the most comprehensive theory of proof-theoretic semantics, as formulated in the influential work of Michael Dummett and Dag Prawitz. Nils Kürbis examines three approaches that have attempted to solve the problem - defining negation in terms of metaphysical incompatibility; treating negation as an undefinable primitive; and defining negation in terms of a speech act of denial - and concludes (...)
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  12. Proof Paradoxes and Normic Support: Socializing or Relativizing?Marcello Di Bello - 2020 - Mind 129 (516):1269-1285.
    Smith argues that, unlike other forms of evidence, naked statistical evidence fails to satisfy normic support. This is his solution to the puzzles of statistical evidence in legal proof. This paper focuses on Smith’s claim that DNA evidence in cold-hit cases does not satisfy normic support. I argue that if this claim is correct, virtually no other form of evidence used at trial can satisfy normic support. This is troublesome. I discuss a few ways in which Smith can respond.
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  13. Strategy-Proof Judgment Aggregation.Franz Dietrich & Christian List - 2005 - Economics and Philosophy 23 (3):269-300.
    Which rules for aggregating judgments on logically connected propositions are manipulable and which not? In this paper, we introduce a preference-free concept of non-manipulability and contrast it with a preference-theoretic concept of strategy-proofness. We characterize all non-manipulable and all strategy-proof judgment aggregation rules and prove an impossibility theorem similar to the Gibbard--Satterthwaite theorem. We also discuss weaker forms of non-manipulability and strategy-proofness. Comparing two frequently discussed aggregation rules, we show that “conclusion-based voting” is less vulnerable to manipulation than “premise-based (...)
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  14.  1
    Proof Theory.K. Schütte - 1977 - Springer Verlag.
  15.  82
    A Proof-Theoretic Defence of Meaning-Invariant Logical Pluralism.Bogdan Dicher - 2016 - Mind 125 (499):727-757.
    In this paper I offer a proof-theoretic defence of meaning-invariant logical pluralism. I argue that there is a relation of co-determination between the operational and structural aspects of a logic. As a result, some features of the consequence relation are induced by the connectives. I propose that a connective is defined by those rules which are conservative and unique, while at the same time expressing only connective-induced structural information. This is the key to stabilizing the meaning of the connectives (...)
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  16. Proof-Theoretic Semantics for a Natural Language Fragment.Nissim Francez & Roy Dyckhoff - 2010 - Linguistics and Philosophy 33 (6):447-477.
    The paper presents a proof-theoretic semantics (PTS) for a fragment of natural language, providing an alternative to the traditional model-theoretic (Montagovian) semantics (MTS), whereby meanings are truth-condition (in arbitrary models). Instead, meanings are taken as derivability-conditions in a dedicated natural-deduction (ND) proof-system. This semantics is effective (algorithmically decidable), adhering to the meaning as use paradigm, not suffering from several of the criticisms formulated by philosophers of language against MTS as a theory of meaning. In particular, Dummett’s manifestation argument (...)
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  17. Proof of an External World.G. E. Moore - 1939 - H. Milford.
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  18. Proof Analysis in Modal Logic.Sara Negri - 2005 - Journal of Philosophical Logic 34 (5-6):507-544.
    A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.
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  19. A Proof of Gamma.Saul A. Kripke - 2022 - In Katalin Bimbó (ed.), Relevance Logics and other Tools for Reasoning. Essays in Honor of J. Michael Dunn. London: College Publications. pp. 261-265.
    This paper is dedicated to the memory of Mike Dunn. His untimely death is a loss not only to logic, computer science, and philosophy, but to all of us who knew and loved him. The paper gives an argument for closure under γ in standard systems of relevance logic (first proved by Meyer and Dunn 1969). For definiteness, I chose the example of R. The proof also applies to E and to the quantified systems RQ and EQ. The argument (...)
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  20.  53
    Proofs and Countermodels in Non-Classical Logics.Sara Negri - 2014 - Logica Universalis 8 (1):25-60.
    Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame (...)
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  21. Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality.Nils Kürbis - 2015 - Journal of Philosophical Logic 44 (6):713-727.
    This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings (...)
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  22. Informal Proofs and Mathematical Rigour.Marianna Antonutti Marfori - 2010 - Studia Logica 96 (2):261-272.
    The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.
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  23.  39
    Proof-Theoretic Semantics, Paradoxes and the Distinction Between Sense and Denotation.Luca Tranchini - forthcoming - Journal of Logic and Computation 2014.
    In this paper we show how Dummett-Prawitz-style proof-theoretic semantics has to be modified in order to cope with paradoxical phenomena. It will turn out that one of its basic tenets has to be given up, namely the definition of the correctness of an inference as validity preservation. As a result, the notions of an argument being valid and of an argument being constituted by correct inference rules will no more coincide. The gap between the two notions is accounted for (...)
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  24. Proof and Truth.Christopher Peacocke - 1993 - In John Haldane & Crispin Wright (eds.), Reality, Representation, and Projection. Oxford University Press. pp. 165--190.
     
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  25. Philosophical Proofs Against Common Sense.Bryan Frances - 2021 - Analysis 81 (1):18-26.
    Many philosophers are sceptical about the power of philosophy to refute commonsensical claims. They look at the famous attempts and judge them inconclusive. I prove that, even if those famous attempts are failures, there are alternative successful philosophical proofs against commonsensical claims. After presenting the proofs I briefly comment on their significance.
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  26.  77
    Proof-Theoretic Semantics, Self-Contradiction, and the Format of Deductive Reasoning.Peter Schroeder-Heister - 2012 - Topoi 31 (1):77-85.
    From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a (...)
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  27.  40
    Proof-Theoretic Validity.Stephen Read - 2015 - In Colin R. Caret & Ole T. Hjortland (eds.), Foundations of Logical Consequence. Oxford, UK: pp. 136-158.
    The idea of proof-theoretic validity originated in the work of Gentzen, when he suggested that the meaning of each logical expression was encapsulated in its introduction-rules. The idea was developed by Prawitz and Dummett, but came under attack by Prior under the soubriquet 'analytic validity'. Logical truths and logical consequences are deemed analytically valid by virtue of following, in a way which the present chapter clarifies, from the meaning of the logical constants. But different logics are based on different (...)
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  28.  95
    New Proofs for the Existence of God: Contributions of Contemporary Physics and Philosophy.Robert J. Spitzer (ed.) - 2010 - William B. Eerdmans.
    New Proofs for the Existence of God responds to these glaring omissions. / From universal space-time asymmetry to cosmic coincidences to the intelligibility of ...
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  29. Proof of an External World.George Edward Moore - 1939 - Proceedings of the British Academy 25 (5):273--300.
  30.  53
    Ideas and Results in Proof Theory.Dag Prawitz & J. E. Fenstad - 1971 - Journal of Symbolic Logic 40 (2):232-234.
  31. Proof Theory of Finite-Valued Logics.Richard Zach - 1993 - Dissertation, Technische Universität Wien
    The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of (...)
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  32.  82
    Proofs and Refutations (IV).I. Lakatos - 1963 - British Journal for the Philosophy of Science 14 (56):296-342.
  33. Legal Burdens of Proof and Statistical Evidence.Georgi Gardiner - forthcoming - In James Chase & David Coady (eds.), The Routledge Handbook of Applied Epistemology. Routledge.
    In order to perform certain actions – such as incarcerating a person or revoking parental rights – the state must establish certain facts to a particular standard of proof. These standards – such as preponderance of evidence and beyond reasonable doubt – are often interpreted as likelihoods or epistemic confidences. Many theorists construe them numerically; beyond reasonable doubt, for example, is often construed as 90 to 95% confidence in the guilt of the defendant. -/- A family of influential cases (...)
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  34.  58
    Truth Through Proof: A Formalist Foundation for Mathematics.Alan Weir - 2010 - Oxford, England: Oxford University Press.
    Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance.
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  35.  14
    Proof Theory of Paraconsistent Weak Kleene Logic.Francesco Paoli & Michele Pra Baldi - 2020 - Studia Logica 108 (4):779-802.
    Paraconsistent Weak Kleene Logic is the 3-valued propositional logic defined on the weak Kleene tables and with two designated values. Most of the existing proof systems for PWK are characterised by the presence of linguistic restrictions on some of their rules. This feature can be seen as a shortcoming. We provide a cut-free calculus for PWK that is devoid of such provisos. Moreover, we introduce a Priest-style tableaux calculus for PWK.
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  36. The Proof-Structure of Kant's Transcendental Deduction.Dieter Henrich - 1969 - Review of Metaphysics 22 (4):640-659.
    Hence, there is still controversy over which of the two versions of the deduction deserves priority and whether indeed any distinction between them can be maintained that would go beyond questions of presentation and involve the structure of the proof itself. Schopenhauer and Heidegger held that the first edition alone fully expresses Kant's unique philosophy, while Kant himself, as well as many other Kantians, have only seen a difference in the method of presentation.
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  37.  15
    Proofs and Refutations.Imre Lakatos - 1980 - Noûs 14 (3):474-478.
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  38.  38
    Motivated Proofs: What They Are, Why They Matter and How to Write Them.Rebecca Lea Morris - 2020 - Review of Symbolic Logic 13 (1):23-46.
    Mathematicians judge proofs to possess, or lack, a variety of different qualities, including, for example, explanatory power, depth, purity, beauty and fit. Philosophers of mathematical practice have begun to investigate the nature of such qualities. However, mathematicians frequently draw attention to another desirable proof quality: being motivated. Intuitively, motivated proofs contain no "puzzling" steps, but they have received little further analysis. In this paper, I begin a philosophical investigation into motivated proofs. I suggest that a proof is motivated (...)
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  39.  32
    Proof Theory and Logical Complexity. [REVIEW]Helmut Pfeifer - 1991 - Annals of Pure and Applied Logic 53 (4):197.
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  40.  42
    The Proof by Cases Property and its Variants in Structural Consequence Relations.Petr Cintula & Carles Noguera - 2013 - Studia Logica 101 (4):713-747.
    This paper is a contribution to the study of the rôle of disjunction inAlgebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this (...)
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  41. Proof-Theoretic Semantics for Subsentential Phrases.Nissim Francez, Roy Dyckhoff & Gilad Ben-Avi - 2010 - Studia Logica 94 (3):381-401.
    The paper briefly surveys the sentential proof-theoretic semantics for fragment of English. Then, appealing to a version of Frege’s context-principle (specified to fit type-logical grammar), a method is presented for deriving proof-theoretic meanings for sub-sentential phrases, down to lexical units (words). The sentential meaning is decomposed according to the function-argument structure as determined by the type-logical grammar. In doing so, the paper presents a novel proof-theoretic interpretation of simple type, replacing Montague’s model-theoretic type interpretation (in arbitrary Henkin (...)
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  42.  61
    Proof Theory for Reasoning with Euler Diagrams: A Logic Translation and Normalization.Ryo Takemura - 2013 - Studia Logica 101 (1):157-191.
    Proof-theoretical notions and techniques, developed on the basis of sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables us (...)
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  43. The Proof Structure of Kant's A-Edition Objective Deduction.Corey W. Dyck - forthcoming - In Giuseppe Motta & Dennis Schulting (eds.), Kant’s Deduction From Apperception: An Essay on the Transcendental Deduction of the Categories. Berlin: DeGruyter.
    Kant's A-Edition objective deduction is naturally (and has traditionally been) divided into two arguments: an " argument from above" and one that proceeds " von unten auf." This would suggest a picture of Kant's procedure in the objective deduction as first descending and ascending the same ladder, the better, perhaps, to test its durability or to thoroughly convince the reader of its soundness. There are obvious obstacles to such a reading, however; and in this chapter I will argue that the (...)
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  44.  42
    Normal Proofs, Cut Free Derivations and Structural Rules.Greg Restall - 2014 - Studia Logica 102 (6):1143-1166.
    Different natural deduction proof systems for intuitionistic and classical logic —and related logical systems—differ in fundamental properties while sharing significant family resemblances. These differences become quite stark when it comes to the structural rules of contraction and weakening. In this paper, I show how Gentzen and Jaśkowski’s natural deduction systems differ in fine structure. I also motivate directed proof nets as another natural deduction system which shares some of the design features of Genzen and Jaśkowski’s systems, but which (...)
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  45.  86
    Proof Analysis in Intermediate Logics.Roy Dyckhoff & Sara Negri - 2012 - Archive for Mathematical Logic 51 (1-2):71-92.
    Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than (...)
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  46. Proof Theory for Modal Logic.Sara Negri - 2011 - Philosophy Compass 6 (8):523-538.
    The axiomatic presentation of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generalizations of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective.
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  47.  22
    Pragmatic Encroachment and Legal Proof.Sarah Moss - 2021 - Wiley: Philosophical Issues 31 (1):258-279.
    This paper uses some modest claims about knowledge to identify a significant problem for contemporary American trial procedure. First, suppose that legal proof requires knowledge. In particular, suppose that the defendant in a jury trial is proven guilty only if the jury knows that the defendant is guilty. Second, suppose that knowledge is subject to pragmatic encroachment. In particular, whether the jury knows the defendant is guilty depends on what’s at stake in their decision to convict, including the consequences (...)
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  48.  3
    Proofs and Types.Jean-Yves Girard - 1989 - Cambridge University Press.
    This text is an outgrowth of notes prepared by J. Y. Girard for a course at the University of Paris VII. It deals with the mathematical background of the application to computer science of aspects of logic (namely the correspondence between proposition & types). Combined with the conceptual perspectives of Girard's ideas, this sheds light on both the traditional logic material & its prospective applications to computer science. The book covers a very active & exciting research area, & it will (...)
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  49. Proof Theory.Gaisi Takeuti - 1990 - Studia Logica 49 (1):160-161.
     
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  50. Legal proof and statistical conjunctions.Lewis D. Ross - 2020 - Philosophical Studies 178 (6):2021-2041.
    A question, long discussed by legal scholars, has recently provoked a considerable amount of philosophical attention: ‘Is it ever appropriate to base a legal verdict on statistical evidence alone?’ Many philosophers who have considered this question reject legal reliance on bare statistics, even when the odds of error are extremely low. This paper develops a puzzle for the dominant theories concerning why we should eschew bare statistics. Namely, there seem to be compelling scenarios in which there are multiple sources of (...)
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