
37Frank RamseyStanford Encyclopedia of Philosophy. 2019.Frank Plumpton Ramsey (1903–30) made seminal contributions to philosophy, mathematics and economics. Whilst he was acknowledged as a genius by his contemporaries, some of his most important ideas were not appreciated until decades later; now better appreciated, they continue to bear an influence upon contemporary philosophy. His historic significance was to usher in a new phase of analytic philosophy, which initially built upon the logical atomist doctrines of Bertrand Russell and Ludwig Wittgen…Read more

21J. 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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14Translation Invariance and Miller’s Weather ExampleJournal of Logic, Language and Information 28 (4): 489514. 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

33Franco Montagna, Giulia Simi, and Andrea Sorbi. Logic and probabilistic systems. Archive for mathematical logic, vol. 35 , pp. 225–261 (review)Bulletin of Symbolic Logic 6 (2): 223225. 2000.

12Pure inductive logic with functionsJournal of Symbolic Logic 84 (4): 13821402. 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.

30Six Problems in Pure Inductive LogicJournal of Philosophical Logic 48 (4): 731747. 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.

24From 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. 2001.

51Symmetry’s End?Erkenntnis 74 (1): 5367. 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

21Proof systems for probabilistic uncertain reasoningJournal of Symbolic Logic 63 (3): 10071039. 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

4Proof Systems for Probabilistic Uncertain ReasoningJournal of Symbolic Logic 63 (3): 10071039. 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.

9On the emergence of reasons in inductive logicLogic Journal of the IGPL 9 (2): 207216. 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'

32On parameter free induction schemasJournal of Symbolic Logic 53 (4): 10821097. 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

6Truth definitions without exponentiation and the Σ1 collection schemeJournal of Symbolic Logic 77 (2): 649. 2012.

41O is not enoughReview of Symbolic Logic 2 (2): 298309. 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

Pure 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.

34A 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 NixParis Continuum. We argue that t…Read more

235The Counterpart Principle of Analogical Support by Structural SimilarityErkenntnis 79 (S6): 116. 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.

72Regularity in models of arithmeticJournal of Symbolic Logic 49 (1): 272280. 1984.This paper investigates the quantifier "there exist unboundedly many" in the context of firstorder 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 secondorder theories associated with cuts in nonstandard models of arithmetic

58Rational Pavelka predicate logic is a conservative extension of łukasiewicz predicate logicJournal of Symbolic Logic 65 (2): 669682. 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

12Maximum Entropy Inference with Quantified KnowledgeLogic Journal of the IGPL 16 (1): 8598. 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

29Review: Franco Montagna, Giulia Simi, Andrea Sorbi, Logic and Probabilistic Systems (review)Bulletin of Symbolic Logic 6 (2): 223225. 2000.

2Deriving Information from Inconsistent Knowledge Bases: A Completeness Theorem for η▹ηLogic Journal of the IGPL 12 (5): 345353. 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

177Second Order Inductive Logic and Wilmers' PrincipleJournal of Applied Logic 12 (4): 462476. 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.

9Rational Pavelka Predicate Logic is a Conservative Extension of Lukasiewicz Predicate LogicJournal of Symbolic Logic 65 (2): 669682. 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.

University of ManchesterRegular Faculty
Areas of Interest
Logic and Philosophy of Logic 
Philosophy of Probability 