Inductive logic admits a variety of semantics (Haenni et al., 2011, Part 1). This paper develops semantics based on the norms of Bayesian epistemology (Williamson, 2010, Chapter 7). §1 introduces the semantics and then, in §2, the paper explores methods for drawing inferences in the resulting logic and compares the methods of this paper with the methods of Barnett and Paris (2008). §3 then evaluates this Bayesian inductive logic in the light of four traditional critiques of inductive logic, arguing (i) that it is language independent in a key sense, (ii) that it admits connections with the Principle of Indifference but these connections do not lead to paradox, (iii) that it can capture the phenomenon of learning from experience, and (iv) that while the logic advocates scepticism with regard to some universal hypotheses, such scepticism is not problematic from the point of view of scientific theorising.

Jon Williamson: Review of ‘Reliable Reasoning’ by Gilbert Harman and Sanjeev Kulkarni, Mind 121:1073-1076, 2013. doi: 10.1093/mind/fzt006.

Jon Williamson: Inductive logic, The Reasoner 6(11):176-7, 2012.

Gregory Wheeler & Jon Williamson: Evidential probability and objective Bayesian epistemology, in Prasanta S. Bandyopadhyay & Malcolm R.Forster (eds): Philosophy of statistics, Handbook of the Philosophy of Science volume 7, Elsevier, pp. 307-331, 2011.

In this chapter we draw connections between two seemingly opposing approaches to probability and statistics: evidential probability on the one hand and objective Bayesian epistemology on the other.

Jan-Willem Romeijn, Rolf Haenni, Gregory Wheeler and Jon Williamson: Logical Relations in a Statistical Problem, in B. Lowe, E. Pacuit & J.W. Romeijn (eds): Foundations of the Formal Sciences VI, Reasoning about Probabilities and Probabilistic Reasoning, London: College Publications, pp. 49-79, 2009.

This paper presents the progicnet programme. It proposes a general framework for probabilistic logic that can guide inference based on both logical and probabilistic input. After an introduction to the framework as such, it is illustrated by means of a toy example from psychometrics. It is shown that the framework can accommodate a number of approaches to probabilistic reasoning: Bayesian statistical inference, evidential probability, probabilistic argumentation, and objective Bayesianism. The framework thus provides insight into the relations between these approaches, it illustrates how the results of different approaches can be combined, and it provides a basis for doing efficient inference in each of the approaches.

Jon Williamson: Aggregating judgements by merging evidence, Journal of Logic and Computation 19(3), pp. 461-473, 2009.

The theory of belief revision and merging has recently been applied to judgement aggregation. In this paper I argue that judgements are best aggregated by merging the evidence on which they are based, rather than by directly merging the judgements themselves. This leads to a three-step strategy for judgement aggregation. First, merge the evidence bases of the various agents using some method of belief merging. Second, determine which degrees of belief one should adopt on the basis of this merged evidence base, by applying objective Bayesian theory. Third, determine which judgements are appropriate given these degrees of belief by applying a decision-theoretic account of rational judgement formation.

Rolf Haenni, Jan-Willem Romeijn, Gregory Wheeler and Jon Williamson: Possible Semantics for a Common Framework of Probabilistic Logics, in V. N. Huynh (ed.): Interval / Probabilistic Uncertainty and Non-Classical Logics, Advances in Soft Computing Series, Springer 2008, pp. 268-279.

This paper proposes a common framework for various probabilistic logics. It consists of a set of uncertain premises with probabilities attached to them. This raises the question of the strength of a conclusion, but without imposing a particular semantics, no general solution is possible. The paper discusses several possible semantics by looking at it from the perspective of probabilistic argumentation.

Jon Williamson: A note on probabilistic logics and probabilistic networks, The Reasoner 2(5), pp. 4-5, 2008.

Jon Williamson: Objective Bayesian probabilistic logic, Journal of Algorithms in Cognition, Informatics and Logic 63: 167-183, 2008.

This paper develops connections between objective Bayesian epistemology – which holds that the strengths of an agent’s beliefs should be representable by probabilities, should be calibrated with evidence of empirical probability, and should otherwise be equivocal – and probabilistic logic. After introducing objective Bayesian epistemology over propositional languages, the formalism is extended to handle predicate languages. A rather general probabilistic logic is formulated and then given a natural semantics in terms of objective Bayesian epistemology. The machinery of objective Bayesian nets and objective credal nets is introduced and this machinery is applied to provide a calculus for probabilistic logic that meshes with the objective Bayesian semantics.

Jon Williamson: Combining probability and logic: introduction, Journal of Logic, Language and Information 15(1-2), special issue on Combining Probability and Logic, pp. 1-3, 2006.

Jon Williamson & Dov Gabbay: Combining probability and logic – editorial, Journal of Applied Logic 1(3-4), Special Issue on Combining probability and logic, 2003, pp. 135-138.

Jon Williamson: Abduction and its distinctions , Review of Lorenzo Magnani [2001]: Abduction, reason and science: processes of discovery and explanation, British Journal for the Philosophy of Science 54(2), 2003, pp.353-358.

Jon Williamson: Bayesian networks for logical reasoning, in Carla Gomes & Toby Walsh (eds) [2001]: Proceedings of the AAAI Fall Symposium on using Uncertainty within Computation, AAAI Press Technical Report FS-01-04, pp. 136-143.

By identifying and pursuing analogies between causal and logical influence I show how the Bayesian network formalism can be applied to reasoning about logical deductions.

Jon Williamson: Probability logic, in Dov Gabbay, Ralph Johnson, Hans Jurgen Ohlbach & John Woods (eds)[2002]: Handbook of the Logic of Inference and Argument: The Turn Toward the Practical, Studies in Logic and Practical Reasoning Volume 1, Elsevier, pp. 397-424.

I examine the idea of incorporating probability into logic for a logic of practical reasoning. I introduce probability and its interpretations, give an account of the development of the logical approach to probability, its immediate problems, and improved formulations. Then I discuss inference in probabilistic logic, and propose the use of Bayesian networks for inference in both causal logics and proof planning.