The use of evidence in medicine is something we should continuously seek to improve. This book seeks to develop our understanding of evidence of mechanism in evaluating evidence in medicine, public health, and social care; and also offers tools to help implement improved assessment of evidence of mechanism in practice. In this way, the book offers a bridge between more theoretical and conceptual insights and worries about evidence of mechanism and practical means to fit the results into evidence assessment procedures.

We argue that David Lewis’s principal principle implies a version of the principle of indifference. The same is true for similar principles that need to appeal to the concept of admissibility. Such principles are thus in accord with objective Bayesianism, but in tension with subjective Bayesianism. 1 The Argument2 Some Objections Met.

We argue that David Lewis’s principal principle implies a version of the principle of indifference. The same is true for similar principles that need to appeal to the concept of admissibility. Such principles are thus in accord with objective Bayesianism, but in tension with subjective Bayesianism. 1 The Argument2 Some Objections Met.

This paper highlights the role of Lewis’ Principal Principle and certain auxiliary conditions on admissibility as serving to explicate normal informal standards of what is reasonable. These considerations motivate the presuppositions of the argument that the Principal Principle implies the Principle of Indifference, put forward by Hawthorne et al.. They also suggest a line of response to recent criticisms of that argument, due to Pettigrew and Titelbaum and Hart, 621–632, 2020). The paper also shows that related concerns of Hart and (...) Titelbaum, 252–262, 2015) do not undermine the argument of Hawthorne et al.. (shrink)

This paper poses a problem for Lewis’ Principal Principle in a subjective Bayesian framework: we show that, where chances inform degrees of belief, subjective Bayesianism fails to validate normal informal standards of what is reasonable. This problem points to a tension between the Principal Principle and the claim that conditional degrees of belief are conditional probabilities. However, one version of objective Bayesianism has a straightforward resolution to this problem, because it avoids this latter claim. The problem, then, offers some support (...) to this version of objective Bayesianism. (shrink)

Direct inferences identify certain probabilistic credences or confirmation-function-likelihoods with values of objective chances or relative frequencies. The best known version of a direct inference principle is David Lewis’s Principal Principle. Certain kinds of statements undermine direct inferences. Lewis calls such statements inadmissible. We show that on any Bayesian account of direct inference several kinds of intuitively innocent statements turn out to be inadmissible. This may pose a significant challenge to Bayesian accounts of direct inference. We suggest some ways in which (...) these challenges may be addressed. (shrink)

This paper poses a problem for Lewis’ Principal Principle in a subjective Bayesian framework: we show that, where chances inform degrees of belief, subjective Bayesianism fails to validate normal informal standards of what is reasonable. This problem points to a tension between the Principal Principle and the claim that conditional degrees of belief are conditional probabilities. However, one version of objective Bayesianism has a straightforward resolution to this problem, because it avoids this latter claim. The problem, then, offers some support (...) to this version of objective Bayesianism. (shrink)

Probabilistic inference forms lead from point probabilities of the premises to interval probabilities of the conclusion. The probabilistic version of Modus Ponens, for example, licenses the inference from \({P(A) = \alpha}\) and \({P(B|A) = \beta}\) to \({P(B)\in [\alpha\beta, \alpha\beta + 1 - \alpha]}\) . We study generalized inference forms with three or more premises. The generalized Modus Ponens, for example, leads from \({P(A_{1}) = \alpha_{1}, \ldots, P(A_{n})= \alpha_{n}}\) and \({P(B|A_{1} \wedge \cdots \wedge A_{n}) = \beta}\) to an according interval for (...) P(B). We present the probability intervals for the conclusions of the generalized versions of Cut, Cautious Monotonicity, Modus Tollens, Bayes’ Theorem, and some SYSTEM O rules. Recently, Gilio has shown that generalized inference forms “degrade”—more premises lead to less precise conclusions, i.e., to wider probability intervals of the conclusion. We also study Adam’s probability preservation properties in generalized inference forms. Special attention is devoted to zero probabilities of the conditioning events. These zero probabilities often lead to different intervals in the coherence and the Kolmogorov approach. (shrink)

The conflict of narrowness and precision in direct inference occurs if a body of evidence contains estimates for frequencies in a certain reference class and less precise estimates for frequencies in a narrower reference class. To develop a solution to this conflict, I draw on ideas developed by Paul Thorn and John Pollock. First, I argue that Kyburg and Teng’s solution to the conflict of narrowness and precision leads to unreasonable direct inference probabilities. I then show that Thorn’s recent solution (...) to the conflict leads to unreasonable direct inference probabilities. Based on my analysis of Thorn’s approach, I propose a natural distribution for a Bayesian analysis of the data directly obtained from studying members of the narrowest reference class. (shrink)

Probability logic studies the properties resulting from the probabilistic interpretation of logical argument forms. Typical examples are probabilistic Modus Ponens and Modus Tollens. Argument forms with two premises usually lead from precise probabilities of the premises to imprecise or interval probabilities of the conclusion. In the contribution, we study generalized inference forms having three or more premises. Recently, Gilio has shown that these generalized forms ``degrade'' -- more premises lead to more imprecise conclusions, i. e., to wider intervals. We distinguish (...) different forms of degradation. We analyse Predictive Inference, Modus Ponens, Bayes' Theorem, and Modus Tollens. Special attention is devoted to the case where the conditioning events have zero probabilities. Finally, we discuss the relation of degradation to monotonicity. (shrink)

The paper investigates exchangeability in the context of probability logic. We study generalizations of basic inference rules and inferences involving cardinalities. We compare the results with those obtained in the case in which only identical probabilities are assumed.

Inconsistencies between scientific theories have been studied, by and large, from the perspective of paraconsistent logic. This approach considered the formal properties of theories and the structure of inferences one can legitimately draw from theories. However, inconsistencies can be also analysed from the perspective of modelling practices, in particular how modelling practices may lead scientists to form opinions and attitudes that are different, but not necessarily inconsistent. In such cases, it is preferable to talk about disagreement, rather than inconsistency. Disagreement (...) may originate in, or concern, a number of epistemic, socio-political or psychological factors. In this paper, we offer an account of the ‘loci and reasons’ for disagreement at different stages of the scientific process. We then present a controversial episode in the health sciences: the studies on hypercholesterolemia. The causes and effects of high levels of cholesterol in blood have been long and hotly debated, to the point of deserving the name of ‘cholesterol wars’; the debate, to be sure, isn’t settled yet. In this contribution, we focus on some selected loci and reasons for disagreement that occurred between 1920 and 1994 in the studies on hypercholesterolemia. We hope that our analysis of ‘loci and reasons’ for disagreement may shed light on the cholesterol wars, and possibly on other episodes of scientific disagreement. (shrink)

In this paper we develop an abstract theory of adequacy. In the same way as the theory of consequence operations is a general theory of logic, this theory of adequacy is a general theory of the interactions and connections between consequence operations and its sound and complete semantics. Addition of axioms for the connectives of propositional logic to the basic axioms of consequence operations yields a unifying framework for different systems of classical propositional logic. We present an abstract model-theoretical semantics (...) based on model mappings and theory mappings. Between the classes of models and theories, i.e., the set of sentences verified by a model, it obtains a connection that is well-known within algebra as Galois correspondence. Many basic semantical properties can be derived from this observation. A sentence A is a semantical consequence of T if every model of T is also a model of A. A model mapping is adequate for a consequence operation if its semantical inference operation is identical with the consequence operation. We study how properties of an adequate model mapping reflect the properties of the consequence operation and vice versa. In particular, we show how every concept of the theory of consequence operations can be formulated semantically. (shrink)

This article deals with algebraic logic. In particular, it discusses the theory of consequence operations and the general concept of logical independency. The advantage of this general view is its gr- eat applicability: The stated properties of consequence operations hold for almost every logical system. The notion of independency is well known and important in logic, philosophy of science and mathematics. Roughly speaking, a set is independent with respect to a consequence operation, if none of its elements is a consequence (...) of the other elements. The property of being an independent set guarantees therefore that none of its elements is superfluous. In particular, I'm going to show fundamental results for every consequence operation, and hence for every logic: no infinite independent set is finite axiomatizable, and every finite axiomatizable set has relative to a finitary consequence operation an independent axiom system. The main result is that in sentential logic every set of formulas has an independend axiom system. (shrink)

We give an elementary introduction into the theory of consequence operations. We proof some elementary results concerning basic notions of logic like tautology, consistency, independence and completeness. We show in particular that every finite axiomatizable set is independent axiomatizable and that every consistent set has relative to a finitary consequence operation a maximal consistent extension. Finally we provide an abstract semantics for consequence operations.

This article deals with algebraic logic. In particular, it discusses the theory of consequence operations and the general concept of logical independency. The advantage of this general view is its gr- eat applicability: The stated properties of consequence operations hold for almost every logical system. The notion of independency is well known and important in logic, philosophy of science and mathematics. Roughly speaking, a set is independent with respect to a consequence operation, if none of its elements is a consequence (...) of the other elements. The property of being an independent set guarantees therefore that none of its elements is superfluous. In particular, I'm going to show fundamental results for every consequence operation, and hence for every logic: no infinite independent set is finite axiomatizable, and every finite axiomatizable set has relative to a finitary consequence operation an independent axiom system. The main result is that in sentential logic every set of formulas has an independend axiom system. (shrink)

We give an elementary introduction into the theory of consequence operations. We proof some elementary results concerning basic notions of logic like tautology, consistency, independence and completeness. We show in particular that every finite axiomatizable set is independent axiomatizable and that every consistent set has relative to a finitary consequence operation a maximal consistent extension. Finally we provide an abstract semantics for consequence operations.