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91Six 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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134Symmetry’s End?Erkenntnis 74 (1): 53-67. 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
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129Proof 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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175On parameter free induction schemasJournal of Symbolic Logic 53 (4): 1082-1097. 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
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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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113On the scheme of induction for bounded arithmetic formulasAnnals of Pure and Applied Logic 35 (C): 261-302. 1987.
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148On 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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183A 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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79An examination of the SEP candidate analogical inference rule within pure inductive logicJournal of Applied Logic 14 (C): 22-45. 2016.
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40Pure 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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1205The 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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49Deriving 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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153A 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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96Maximum 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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913Second 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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153Provability of the pigeonhole principle and the existence of infinitely many primesJournal of Symbolic Logic 53 (4): 1235-1244. 1988.
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108The 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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211European summer meeting of the Association for Symbolic Logic, Manchester, England, 1984Journal of Symbolic Logic 51 (2): 480-502. 1986.
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16Principles 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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211A Note on Binary Inductive LogicJournal of Philosophical Logic 36 (6): 735-771. 2007.We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.
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189Some observations on induction in predicate probabilistic reasoningJournal of Philosophical Logic 31 (1): 43-75. 2002.We consider the desirability, or otherwise, of various forms of induction in the light of certain principles and inductive methods within predicate uncertain reasoning. Our general conclusion is that there remain conflicts within the area whose resolution will require a deeper understanding of the fundamental relationship between individuals and properties
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1189Ancient 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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170Symmetry 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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126A 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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157The 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 |