•  89
    Six Problems in Pure Inductive Logic
    with A. Vencovská
    Journal 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.
  •  131
    Symmetry’s End?
    with A. Vencovská
    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
  •  121
    Proof systems for probabilistic uncertain reasoning
    with A. Vencovska
    Journal 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
  •  166
    On parameter free induction schemas
    with R. Kaye and C. Dimitracopoulos
    Journal 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
  •  92
    Maximum Entropy Inference with Quantified Knowledge
    with Owen Barnett
    Logic 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.
  •  906
    Second Order Inductive Logic and Wilmers' Principle
    with M. S. Kliess
    Journal 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.
  •  130
    A note on the undefinability of cuts
    with C. Dimitracopoulos
    Journal of Symbolic Logic 48 (3): 564-569. 1983.
  •  106
    The theory of spectrum exchangeability
    with E. Howarth
    Review 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
  •  207
    European summer meeting of the Association for Symbolic Logic, Manchester, England, 1984
    with P. Aczel, A. J. Wilkie, G. M. Wilmers, and C. E. M. Yates
    Journal of Symbolic Logic 51 (2): 480-502. 1986.
  •  14
    Principles of Remembering and Forgetting
    with E. Howarth
    Logique 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.
  •  533
    ZF ⊦ Σ4 0 determinateness
    Journal of Symbolic Logic 37 (4): 661-667. 1972.
  •  45
    Measure and minimal degrees
    Annals of Mathematical Logic 11 (2): 203-216. 1977.
  •  202
    A Note on Binary Inductive Logic
    with C. J. Nix
    Journal 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.
  •  183
    Some observations on induction in predicate probabilistic reasoning
    with M. J. Hill and G. M. Wilmers
    Journal 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
  •  1175
    Ancient Indian Logic and Analogy
    with A. Vencovska
    In 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.
  •  169
    Symmetry in Polyadic Inductive Logic
    with A. Vencovská
    Journal 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
  •  123
    A Note on Irrelevance in Inductive Logic
    with Alena Vencovská
    Journal 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
  •  78
    Initial Segments of Models of Peano's Axioms
    with L. A. S. Kirby, A. Lachlan, M. Srebrny, and A. Zarach
    Journal of Symbolic Logic 48 (2): 482-483. 1983.
  •  151
    The Type Theoretic Interpretation of Constructive Set Theory
    with Peter Aczel, Angus Macintyre, and Leszek Pacholski
    Journal of Symbolic Logic 49 (1): 313-314. 1984.
  • The Finite Values Property
    with E. Howarth
    In C. Beierle, C. Brewka & M. Thimm (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.
  •  250
    O is not enough
    with R. Simmonds
    Review of Symbolic Logic 2 (2): 298-309. 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
  •  146
    Atom Exchangeability and Instantial Relevance
    with P. Waterhouse
    Journal of Philosophical Logic 38 (3): 313-332. 2009.
    We give an account of some relationships between the principles of Constant and Atom Exchangeability and various generalizations of the Principle of Instantial Relevance within the framework of Inductive Logic. In particular we demonstrate some surprising and somewhat counterintuitive dependencies of these relationships on ostensibly unimportant parameters, such as the number of predicates in the overlying language.
  •  123
    Rationality As Conformity
    Synthese 144 (2): 249-285. 2005.
    We argue in favour of identifying one aspect of rational choice with the tendency to conform to the choice you expect another like-minded, but non-communicating, agent to make and study this idea in the very basic case where the choice is from a non-empty subset K of 2 A and no further structure or knowledge of A is assumed.
  •  942
    An observation on Carnapʼs Continuum and stochastic independencies
    Journal of Applied Logic 11 (4): 421-429. 2013.
    We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the lan…Read more
  •  214
    Some independence results for peano arithmetic
    Journal of Symbolic Logic 43 (4): 725-731. 1978.
  •  189
    Common sense and maximum entropy
    Synthese 117 (1): 75-93. 1998.
    This paper concerns the question of how to draw inferences common sensically from uncertain knowledge. Since the early work of Shore and Johnson (1980), Paris and Vencovská (1990), and Csiszár (1989), it has been known that the Maximum Entropy Inference Process is the only inference process which obeys certain common sense principles of uncertain reasoning. In this paper we consider the present status of this result and argue that within the rather narrow context in which we work this complete a…Read more
  •  76
    Subsets of models of arithmetic
    Archive 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
  •  115
    Truth definitions without exponentiation and the Σ₁ collection scheme
    with Zofia Adamowicz and Leszek Aleksander Kołodziejczyk
    Journal 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