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57On the scheme of induction for bounded arithmetic formulasAnnals of Pure and Applied Logic 35 (C): 261-302. 1987.
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10A property of 2‐sorted peano models and program verificationMathematical Logic Quarterly 30 (19‐24): 325-334. 1984.
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74Symmetry 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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10Pure 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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55O 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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525The 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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57A 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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88Regularity in models of arithmeticJournal of Symbolic Logic 49 (1): 272-280. 1984.This paper investigates the quantifier "there exist unboundedly many" in the context of first-order arithmetic. An alternative axiomatization is found for Peano arithmetic based on an axiom schema of regularity: The union of boundedly many bounded sets is bounded. We also obtain combinatorial equivalents of certain second-order theories associated with cuts in nonstandard models of arithmetic
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78Rational Pavelka predicate logic is a conservative extension of łukasiewicz predicate logicJournal of Symbolic Logic 65 (2): 669-682. 2000.Rational Pavelka logic extends Lukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0, 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963
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23Maximum Entropy Inference with Quantified KnowledgeLogic Journal of the IGPL 16 (1): 85-98. 2008.We investigate uncertain reasoning with quantified sentences of the predicate calculus treated as the limiting case of maximum entropy inference applied to finite domains
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36Review: Franco Montagna, Giulia Simi, Andrea Sorbi, Logic and Probabilistic Systems (review)Bulletin of Symbolic Logic 6 (2): 223-225. 2000.
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329Second Order Inductive Logic and Wilmers' PrincipleJournal of Applied Logic 12 (4): 462-476. 2014.We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity.
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8Deriving 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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40The theory of spectrum exchangeabilityReview of Symbolic Logic 8 (1): 108-130. 2015.Spectrum Exchangeability, Sx, is an irrelevance principle of Pure Inductive Logic, and arguably the most natural extension of Atom Exchangeability to polyadic languages. It has been shown1that all probability functions which satisfy Sx are comprised of a mixture of two essential types of probability functions; heterogeneous and homogeneous functions. We determine the theory of Spectrum Exchangeability, which for a fixed languageLis the set of sentences ofLwhich must be assigned probability 1 by …Read more
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9Rational Pavelka Predicate Logic is a Conservative Extension of Lukasiewicz Predicate LogicJournal of Symbolic Logic 65 (2): 669-682. 2000.Rational Pavelka logic extends Lukasiewicz infinitely valued logic $by adding truth constants \bar{r} for rationals in [0, 1].$ We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.
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65European summer meeting of the Association for Symbolic Logic, Manchester, England, 1984Journal of Symbolic Logic 51 (2): 480-502. 1986.
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38On 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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Principles of Remembering and ForgettingLogique Et Analyse 57 (228): 489-511. 2014.We propose two principles of inductive reasoning related to how observed information is handled by conditioning, and justify why they may be said to represent aspects of rational reasoning. A partial classification is given of the probability functions which satisfy these principles.
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25A natural prior probability distribution derived from the propositional calculusAnnals of Pure and Applied Logic 70 (3): 243-285. 1994.
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54A Continuum of Inductive Methods Arising from a Generalized Principle of Instantial RelevanceJournal of Philosophical Logic 35 (1): 83-115. 2006.In this paper we consider a natural generalization of the Principle of Instantial Relevance and give a complete characterization of the probabilistic belief functions satisfying this principle as a family of discrete probability functions parameterized by a single real δ ∊ [0, 1)
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110The liar paradox and fuzzy logicJournal of Symbolic Logic 65 (1): 339-346. 2000.Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying "x is true" and satisfying the "dequotation schema" $\varphi \equiv \text{Tr}(\bar{\varphi})$ for all sentences φ? This problem is investigated in the frame of Lukasiewicz infinitely valued logic
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29Review: J. I. Friedman, Proper Classes as Members of Extended Sets (review)Journal of Symbolic Logic 40 (3): 462-462. 1975.
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University of ManchesterRegular Faculty
Areas of Interest
Logic and Philosophy of Logic |
Philosophy of Probability |