Jon Williamson

I put forward several desiderata that a philosophical theory of causality should satisfy: it should account for the objectivity of causality, it should underpin formalisms for causal reasoning, it should admit a viable epistemology, it should be able to cope with the great variety of causal claims that are made, and it should be ontologically parsimonious. I argue that Nancy Cartwright’s dispositional account of causality goes part way towards meeting these criteria but is lacking in important respects. I go on to argue that my epistemic account, which ties causal relationships to an agent’s knowledge and ignorance, performs well in the light of the desiderata. Such an account, I claim, is all we require from a theory of causality.

Objective Bayesianism has been criticised on the grounds that objective Bayesian updating, which on a finite outcome space appeals to the maximum entropy principle, differs from Bayesian conditionalisation. The main task of this paper is to show that this objection backfires: the difference between the two forms of updating reflects negatively on Bayesian conditionalisation rather than on objective Bayesian updating. The paper also reviews some existing criticisms and justifications of conditionalisation, arguing in particular that the diachronic Dutch book justification fails because diachronic Dutch book arguments are subject to a reductio: in certain circumstances one can Dutch book an agent however she changes her degrees of belief . One may also criticise objective Bayesianism on the grounds that its norms are not compulsory but voluntary, the result of a stance. It is argued that this second objection also misses the mark, since objective Bayesian norms are tied up in the very notion of degrees of belief.

Logic is a field studied mainly by researchers and students of philosophy, mathematics and computing. Inductive logic seeks to determine the extent to which the premises of an argument entail its conclusion, aiming to provide a theory of how one should reason in the face of uncertainty. It has applications to decision making and artificial intelligence, as well as how scientists should reason when not in possession of the full facts. In this work, Jon Williamson embarks on a quest to find a general, reasonable, applicable inductive logic (GRAIL), all the while examining why pioneers such as Ludwig Wittgenstein and Rudolf Carnap did not entirely succeed in this task.

In this chapter I discuss connections between machine learning and the philosophy of science. First I consider the relationship between the two disciplines. There is a clear analogy between hypothesis choice in science and model selection in machine learning. While this analogy has been invoked to argue that the two disciplines are essentially doing the same thing and should merge, I maintain that the disciplines are distinct but related and that there is a dynamic interaction operating between the two: a series of mutually beneficial interactions that changes over time. I will introduce some particularly fruitful interactions, in particular the consequences of automated scientific discovery for the debate on inductivism versus falsificationism in the philosophy of science, and the importance of philosophical work on Bayesian epistemology and causality for contemporary machine learning. I will close by suggesting the locus of a possible future interaction: evidence integration.

According to Russo and Williamson (Int Stud Philos Sci 21(2):157–170, 2007, Hist Philos Life Sci 33:389–396, 2011a, Philos Sci 1(1):47–69, 2011b ), in order to establish a causal claim of the form, ‘_C_ is a cause of _E_’, one typically needs evidence that there is an underlying mechanism between _C_ and _E_ as well as evidence that _C_ makes a difference to _E_. This thesis has been used to argue that hierarchies of evidence, as championed by evidence-based movements, tend to give primacy to evidence of difference making over evidence of mechanisms and are flawed because the two sorts of evidence are required and they should be treated on a par. An alternative approach gives primacy to evidence of mechanism over evidence of difference making. In this paper, we argue that this alternative approach is equally flawed, again because both sorts of evidence need to be treated on a par. As an illustration of this parity, we explain how scientists working in the ‘EnviroGenomarkers’ project constantly make use of the two evidential components in a dynamic and intertwined way. We argue that such an interplay is needed not only for causal assessment but also for policy purposes.

Objective Bayesian probability is often defined over rather simple domains, e.g., finite event spaces or propositional languages. This paper investigates the extension of objective Bayesianism to first-order logical languages. It is argued that the objective Bayesian should choose a probability function, from all those that satisfy constraints imposed by background knowledge, that is closest to a particular frequency-induced probability function which generalises the λ = 0 function of Carnap’s continuum of inductive methods.

Evidence-based medicine (EBM) makes use of explicit procedures for grading evidence for causal claims. Normally, these procedures categorise evidence of correlation produced by statistical trials as better evidence for a causal claim than evidence of mechanisms produced by other methods. We argue, in contrast, that evidence of mechanisms needs to be viewed as complementary to, rather than inferior to, evidence of correlation. In this paper we first set out the case for treating evidence of mechanisms alongside evidence of correlation in explicit protocols for evaluating evidence. Next we provide case studies which exemplify the ways in which evidence of mechanisms complements evidence of correlation in practice. Finally, we put forward some general considerations as to how the two sorts of evidence can be more closely integrated by EBM.

The mechanistic and causal accounts of explanation are often conflated to yield a ‘causal-mechanical’ account. This paper prizes them apart and asks: if the mechanistic account is correct, how can causal explanations be explanatory? The answer to this question varies according to how causality itself is understood. It is argued that difference-making, mechanistic, dualist and inferentialist accounts of causality all struggle to yield explanatory causal explanations, but that an epistemic account of causality is more promising in this regard.

How is probability related to logic? Should probability and logic be combined? If so, how? Bayesianism tells us we ought to reason probabilistically. In that sense, probability theory is logic. How then does probability theory relate to classical logic and the various non-classical logics that also stake a claim on normative reasoning? Is probability theory to be preferred over other logics or vice versa? Is probability theory to be used in some situations, and the other logics in other situations? Or should probability be combined with other logics?

This introduction to the volume begins with a manifesto that puts forward two theses: first, that the sciences are the best place to turn in order to understand causality; second, that scientifically-informed philosophical investigation can bring something to the sciences too. Next, the chapter goes through the various parts of the volume, drawing out relevant background and themes of the chapters in those parts. Finally, the chapter discusses the progeny of the papers and identifies some next steps for research into causality in the sciences.

I present a formalism that combines two methodologies: objective Bayesianism and Bayesian nets. According to objective Bayesianism, an agent’s degrees of belief (i) ought to satisfy the axioms of probability, (ii) ought to satisfy constraints imposed by background knowledge, and (iii) should otherwise be as non-committal as possible (i.e. have maximum entropy). Bayesian nets offer an efficient way of representing and updating probability functions. An objective Bayesian net is a Bayesian net representation of the maximum entropy probability function.