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17Translation Invariance and Miller’s Weather ExampleJournal of Logic, Language and Information 28 (4): 489-514. 2019.In his 1974 paper “Popper’s qualitative theory of verisimilitude” published in the British Journal for the Philosophy of Science David Miller gave his so called ‘Weather Example’ to argue that the Hamming distance between constituents is flawed as a measure of proximity to truth since the former is not, unlike the latter, translation invariant. In this present paper we generalise David Miller’s Weather Example in both the unary and polyadic cases, characterising precisely which permutations of c…Read more
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15On some formalized conservation results in arithmeticArchive for Mathematical Logic 30 (4): 201-218. 1990.IΣ n andBΣ n are well known fragments of first-order arithmetic with induction and collection forΣ n formulas respectively;IΣ n 0 andBΣ n 0 are their second-order counterparts. RCA0 is the well known fragment of second-order arithmetic with recursive comprehension;WKL 0 isRCA 0 plus weak König's lemma. We first strengthen Harrington's conservation result by showing thatWKL 0 +BΣ n 0 is Π 1 1 -conservative overRCA 0 +BΣ n 0 . Then we develop some model theory inWKL 0 and illustrate the use of for…Read more
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14A mathematical incompleteness in Peano arithmeticIn Jon Barwise (ed.), Handbook of mathematical logic, North-holland. pp. 90--1133. 1977.
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14Truth definitions without exponentiation and the Σ1 collection schemeJournal of Symbolic Logic 77 (2): 649. 2012.
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11Principles of uncertain reasoningIn and J. Larrazabal J. Ezquerro A. Clark (ed.), Philosophy and Cognitive Science: Categories, Consciousness, and Reasoning, Kluwer Academic Publishers. pp. 221--259. 1996.
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10On the emergence of reasons in inductive logicLogic Journal of the IGPL 9 (2): 207-216. 2001.We apply methods of abduction derived from propositional probabilistic reasoning to predicate probabilistic reasoning, in particular inductive logic, by treating finite predicate knowledge bases as potentially infinite propositional knowledge bases. It is shown that for a range of predicate knowledge bases and several key propositional inference processes this procedure is well defined, and furthermore yields an explanation for the validity of the induction in terms of 'reasons'
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10Common sense and stochastic independenceIn David Corfield & Jon Williamson (eds.), Foundations of Bayesianism, Kluwer Academic Publishers. pp. 203--240. 2001.
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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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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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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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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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6What You See Is What You GetEntropy 16 (11): 6186-6194. 2014.This paper corrects three widely held misunderstandings about Maxent when used in common sense reasoning: That it is language dependent; That it produces objective facts; That it subsumes, and so is at least as untenable as, the paradox-ridden Principle of Insufficient Reason.
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6Inexact and Inductive ReasoningIn Jens Erik Fenstad, Ivan Timofeevich Frolov & Risto Hilpinen (eds.), Logic, Methodology, and Philosophy of Science Viii: Proceedings of the Eighth International Congress of Logic, Methodology, and Philosophy of Science, Moscow, 1987, Sole Distributors For the U.s.a. and Canada, Elsevier Science. 1989.
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5Proof 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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3Logic Colloquium '84: Proceedings of the Colloquium Held in Manchester, U.K., July 1984 (edited book)North Holland. 1986.This proceedings volume contains most of the invited talks presented at the colloquium. The main topics treated are the model theory of arithmetic and algebra, the semantics of natural languages, and applications of mathematical logic to complexity theory. The volume contains both surveys by acknowledged experts and original research papers presenting advances in these disciplines.
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2Review: K. McAloon, Modeles de l'arithmetique, Siminaire Paris VII (review)Journal of Symbolic Logic 48 (2): 483-484. 1983.
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The Finite Values PropertyIn Beierle C., Brewka C. & Thimm M. (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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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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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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University of ManchesterRegular Faculty
Areas of Interest
Logic and Philosophy of Logic |
Philosophy of Probability |