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89Six Problems in Pure Inductive LogicJournal of Philosophical Logic 48 (4): 731-747. 2019.We present six significant open problems in Pure Inductive Logic, together with their background and current status, with the intention of raising awareness and leading ultimately to their resolution.
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131Symmetry’s End?Erkenntnis 74 (1): 53-67. 2011.We examine the idea that similar problems should have similar solutions (to paraphrase van Fraassen’s slogan ‘Problems which are essentially the same must receive essentially the same solution’, see van Fraassen in Laws and symmetry, Oxford Univesity Press, Oxford, 1989, p. 236) in the context of symmetries of sentence algebras within Inductive Logic and conclude that by itself this is too generous a notion upon which to found the rational assignment of probabilities. We also argue that within o…Read more
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121Proof systems for probabilistic uncertain reasoningJournal of Symbolic Logic 63 (3): 1007-1039. 1998.The paper describes and proves completeness theorems for a series of proof systems formalizing common sense reasoning about uncertain knowledge in the case where this consists of sets of linear constraints on a probability function
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166On parameter free induction schemasJournal of Symbolic Logic 53 (4): 1082-1097. 1988.We present a comprehensive study of the axiom schemas IΣ - n , BΣ - n (induction and collection schemas for parameter free Σ n formulas) and some closely related schemas
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202A Note on Binary Inductive LogicJournal of Philosophical Logic 36 (6): 735-771. 2007.We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.
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183Some observations on induction in predicate probabilistic reasoningJournal of Philosophical Logic 31 (1): 43-75. 2002.We consider the desirability, or otherwise, of various forms of induction in the light of certain principles and inductive methods within predicate uncertain reasoning. Our general conclusion is that there remain conflicts within the area whose resolution will require a deeper understanding of the fundamental relationship between individuals and properties
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1175Ancient Indian Logic and AnalogyIn S. Ghosh & S. Prasad (eds.), Logic and its Applications, Lecture Notes in Computer Science 10119, Springer. pp. 198-210. 2017.B.K.Matilal, and earlier J.F.Staal, have suggested a reading of the `Nyaya five limb schema' (also sometimes referred to as the Indian Schema or Hindu Syllogism) from Gotama's Nyaya-Sutra in terms of a binary occurrence relation. In this paper we provide a rational justification of a version of this reading as Analogical Reasoning within the framework of Polyadic Pure Inductive Logic.
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169Symmetry in Polyadic Inductive LogicJournal of Logic, Language and Information 21 (2): 189-216. 2012.A family of symmetries of polyadic inductive logic are described which in turn give rise to the purportedly rational Permutation Invariance Principle stating that a rational assignment of probabilities should respect these symmetries. An equivalent, and more practical, version of this principle is then derived
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123A Note on Irrelevance in Inductive LogicJournal of Philosophical Logic 40 (3). 2011.We consider two formalizations of the notion of irrelevance as a rationality principle within the framework of (Carnapian) Inductive Logic: Johnson's Sufficientness Principle, JSP, which is classically important because it leads to Carnap's influential Continuum of Inductive Methods and the recently proposed Weak Irrelevance Principle, WIP. We give a complete characterization of the language invariant probability functions satisfying WIP which generalizes the Nix-Paris Continuum. We argue that t…Read more
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151The Type Theoretic Interpretation of Constructive Set TheoryJournal of Symbolic Logic 49 (1): 313-314. 1984.
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The Finite Values PropertyIn C. Beierle, C. Brewka & M. Thimm (eds.), Computational Models of Rationality, Essays Dedicated to Gabriele Kern-Isberner on the Occasion of her 60th Birthday, College Publications. pp. 316-331. 2016.We argue that the simplicity condition on a probability function on sentences of a predicate language L that it takes only finitely many values on the sentences of any finite sublanguage of L can be viewed as rational. We then go on to investigate consequences of this condition, linking it to the model theoretic notion of quantifier elimination.
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67The Twin Continua of Inductive MethodsIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. pp. 355-366. 2015.
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250O is not enoughReview of Symbolic Logic 2 (2): 298-309. 2009.We examine the closure conditions of the probabilistic consequence relation of Hawthorne and Makinson, specifically the outstanding question of completeness in terms of Horn rules, of their proposed (finite) set of rules O. We show that on the contrary no such finite set of Horn rules exists, though we are able to specify an infinite set which is complete
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146Atom Exchangeability and Instantial RelevanceJournal of Philosophical Logic 38 (3): 313-332. 2009.We give an account of some relationships between the principles of Constant and Atom Exchangeability and various generalizations of the Principle of Instantial Relevance within the framework of Inductive Logic. In particular we demonstrate some surprising and somewhat counterintuitive dependencies of these relationships on ostensibly unimportant parameters, such as the number of predicates in the overlying language.
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123Rationality As ConformitySynthese 144 (2): 249-285. 2005.We argue in favour of identifying one aspect of rational choice with the tendency to conform to the choice you expect another like-minded, but non-communicating, agent to make and study this idea in the very basic case where the choice is from a non-empty subset K of 2 A and no further structure or knowledge of A is assumed.
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942An observation on Carnapʼs Continuum and stochastic independenciesJournal of Applied Logic 11 (4): 421-429. 2013.We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the lan…Read more
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189Common sense and maximum entropySynthese 117 (1): 75-93. 1998.This paper concerns the question of how to draw inferences common sensically from uncertain knowledge. Since the early work of Shore and Johnson (1980), Paris and Vencovská (1990), and Csiszár (1989), it has been known that the Maximum Entropy Inference Process is the only inference process which obeys certain common sense principles of uncertain reasoning. In this paper we consider the present status of this result and argue that within the rather narrow context in which we work this complete a…Read more
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76Subsets of models of arithmeticArchive for Mathematical Logic 32 (1): 65-73. 1992.We define certain properties of subsets of models of arithmetic related to their codability in end extensions and elementary end extensions. We characterize these properties using some more familiar notions concerning cuts in models of arithmetic
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115Truth definitions without exponentiation and the Σ₁ collection schemeJournal of Symbolic Logic 77 (2): 649-655. 2012.We prove that: • if there is a model of I∆₀ + ¬ exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then I∆₀ + ¬ exp +¬BΣ₁ is consistent, • there is a model of I∆₀ Ω₁ + ¬ exp with cofinal Σ₁-definable elements, both a Σ₂ and a ∏₂ truth definition for Σ₁ sentences, and for each n > 2, a Σ n truth definition for Σ n sentences. The latter result is obtained by constructing a model with a recursive truth-preserving translation of Σ₁ sentences into boolean combinations …Read more
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Predicate Exchangeability and Language Invariance in Pure Inductive LogicLogique Et Analyse 57 (228): 513-540. 2014.In Pure Inductive Logic, the rational principle of Predicate Exchangeability states that permuting the predicates in a given language L and replacing each occurrence of a predicate in an L-sentence phi according to this permutation should not change our belief in the truth of phi. In this paper we study when a prior probability function w on a purely unary language L satisfying Predicate Exchangeability also satisfies the principle of Unary Language Invariance.
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107On the scheme of induction for bounded arithmetic formulasAnnals of Pure and Applied Logic 35 (C): 261-302. 1987.
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133On LP -models of arithmeticJournal of Symbolic Logic 73 (1): 212-226. 2008.We answer some problems set by Priest in [11] and [12], in particular refuting Priest's Conjecture that all LP-models of Th(N) essentially arise via congruence relations on classical models of Th(N). We also show that the analogue of Priest's Conjecture for I δ₀ + Exp implies the existence of truth definitions for intervals [0,a] ⊂ₑ M ⊨ I δ₀ + Exp in any cut [0,a] ⊂e K ⊆ M closed under successor and multiplication
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171A Note on Priest's Finite Inconsistent ArithmeticsJournal of Philosophical Logic 35 (5): 529-537. 2006.We give a complete characterization of Priest's Finite Inconsistent Arithmetics observing that his original putative characterization included arithmetics which cannot in fact be realized
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74An examination of the SEP candidate analogical inference rule within pure inductive logicJournal of Applied Logic 14 (C): 22-45. 2016.
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37Pure Inductive LogicCambridge University Press. 2011.Pure Inductive Logic is the study of rational probability treated as a branch of mathematical logic. This monograph, the first devoted to this approach, brings together the key results from the past seventy years, plus the main contributions of the authors and their collaborators over the last decade, to present a comprehensive account of the discipline within a single unified context.
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1180The Counterpart Principle of Analogical Support by Structural SimilarityErkenntnis 79 (S6): 1-16. 2014.We propose and investigate an Analogy Principle in the context of Unary Inductive Logic based on a notion of support by structural similarity which is often employed to motivate scientific conjectures.
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47Deriving Information from Inconsistent Knowledge Bases: A Completeness Theorem for η▹ηLogic Journal of the IGPL 12 (5): 345-353. 2004.The logical consequence relations η▹η provide a very attractive way of inferring new facts from inconsistent knowledge bases without compromising standards of credibility. In this short note we provide proof theories and completeness theorems for these consequence relations which may have some applicability in small examples
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University of ManchesterRegular Faculty
Areas of Interest
| Logic and Philosophy of Logic |
| Philosophy of Probability |