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61A 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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576The 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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92Regularity 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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81Rational 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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38Review: Franco Montagna, Giulia Simi, Andrea Sorbi, Logic and Probabilistic Systems (review)Bulletin of Symbolic Logic 6 (2): 223-225. 2000.
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26Maximum 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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10Deriving 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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354Second 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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45The 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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33On 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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41European summer meeting of the Association for Symbolic Logic, Manchester, England, 1984Journal of Symbolic Logic 51 (2): 480-502. 1986.
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27A natural prior probability distribution derived from the propositional calculusAnnals of Pure and Applied Logic 70 (3): 243-285. 1994.
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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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35A 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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490Ancient 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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26A 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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42The 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 “xis true” and satisfying the “dequotation schema”for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.
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36Provability of the pigeonhole principle and the existence of infinitely many primesJournal of Symbolic Logic 53 (4): 1235-1244. 1988.
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36The Type Theoretic Interpretation of Constructive Set TheoryJournal of Symbolic Logic 49 (1): 313-314. 1984.
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University of ManchesterRegular Faculty
Areas of Interest
Logic and Philosophy of Logic |
Philosophy of Probability |