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44J. I. Friedman. Proper classes as members of extended sets. Mathematische Annalen, vol. 83 , pp. 232–240Journal of Symbolic Logic 40 (3): 462. 1975.
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42A characterization of the language invariant families satisfying spectrum exchangeability in polyadic inductive logicAnnals of Pure and Applied Logic 161 (6): 800-811. 2010.A necessary and sufficient condition in terms of a de Finetti style representation is given for a probability function in Polyadic Inductive Logic to satisfy being part of a Language Invariant family satisfying Spectrum Exchangeability. This theorem is then considered in relation to the unary Carnap and Nix–Paris Continua
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41The 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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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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39On 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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39A 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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38Six 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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38Truth 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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37Review: Franco Montagna, Giulia Simi, Andrea Sorbi, Logic and Probabilistic Systems (review)Bulletin of Symbolic Logic 6 (2): 223-225. 2000.
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35The Pigeonhole Principle and Fragments of ArithmeticMathematical Logic Quarterly 32 (1-5): 73-80. 1986.
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35The Type Theoretic Interpretation of Constructive Set TheoryJournal of Symbolic Logic 49 (1): 313-314. 1984.
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33Provability of the pigeonhole principle and the existence of infinitely many primesJournal of Symbolic Logic 53 (4): 1235-1244. 1988.
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31The 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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31Subsets 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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30Proof 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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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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26From the Johns Hopkins Baby to Baby Miller: What Have We Learned from Four Decades of Reflection on Neonatal Cases?Journal of Clinical Ethics 12 (3): 207-214. 2001.
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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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25Rethinking the End of Modernity: Empire, Hyper-Capitalism, and Cyberpunk DystopiasSocial Philosophy Today 21 173-189. 2005.This essay is comprised of two unusual pairings—Immanuel Wallerstein with Michael Hardt and Antonio Negri; and Don DeLillo with William Gibson—and a thesis: We live, today, in a period of transition between modernity and postmodernity that is best characterized as what I call hyper-capitalism. The end of modernity, as described both by Wallerstein’s world-systems theory and by the “postmodern” political philosophy of the authors of Empire, does not lead us into postmodernity proper, but into a p…Read more
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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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23Pure inductive logic with functionsJournal of Symbolic Logic 84 (4): 1382-1402. 2019.We consider the version of Pure Inductive Logic which obtains for the language with equality and a single unary function symbol giving a complete characterization of the probability functions on this language which satisfy Constant Exchangeability.
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20A Property of 2‐Sorted Peano Models and Program VerificationMathematical Logic Quarterly 30 (19-24): 325-334. 1984.
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19In defense of the maximum entropy inference processInternational Journal of Approximate Reasoning 17 (1): 77-103. 1997.This paper is a sequel to an earlier result of the authors that in making inferences from certain probabilistic knowledge bases the maximum entropy inference process, ME, is the only inference process respecting “common sense.” This result was criticized on the grounds that the probabilistic knowledge bases considered are unnatural and that ignorance of dependence should not be identified with statistical independence. We argue against these criticisms and also against the more general criticism…Read more
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University of ManchesterRegular Faculty
Areas of Interest
Logic and Philosophy of Logic |
Philosophy of Probability |