When making decisions, people naturally face uncertainty about the potential consequences of their actions due in part to limits in their capacity to represent, evaluate or deliberate. Nonetheless, they aim to make the best decisions possible. In Decision Theory with a Human Face, Richard Bradley develops new theories of agency and rational decision-making, offering guidance on how 'real' agents who are aware of their bounds should represent the uncertainty they face, how they should revise their opinions as a result of (...) experience and how they should make decisions when lacking full awareness of, or precise opinions on relevant contingencies. He engages with the strengths and flaws of Bayesian reasoning, and presents clear and comprehensive explorations of key issues in decision theory, from belief and desire to semantics and learning. His book draws on philosophy, economics, decision science and psychology, and will appeal to readers in all of these disciplines. (shrink)

According to the orthodox treatment of risk preferences in decision theory, they are to be explained in terms of the agent's desires about concrete outcomes. The orthodoxy has been criticised both for conflating two types of attitudes and for committing agents to attitudes that do not seem rationally required. To avoid these problems, it has been suggested that an agent's attitudes to risk should be captured by a risk function that is independent of her utility and probability functions. The main (...) problem with that approach is that it suggests that attitudes to risk are wholly distinct from people's (non-instrumental) desires. To overcome this problem, we develop a framework where an agent's utility function is defined over chance propositions (i.e., propositions describing objective probability distributions) as well as ordinary (non-chance) ones, and argue that one should explain different risk attitudes in terms of different forms of the utility function over such propositions. (shrink)

Adams’s Thesis, the claim that the probabilities of indicative conditionals equal the conditional probabilities of their consequents given their antecedents, has proven impossible to accommodate within orthodox possible-world semantics. This essay proposes a modification to the orthodoxy that removes this impossibility. The starting point is a proposal by Jeffrey and Stalnaker that conditionals take semantic values in the unit interval, interpreting these (à la McGee) as their expected truth-values at a world. Their theories imply a false principle, namely, that the (...) probability of a conditional is independent of any proposition inconsistent with its antecedent. But they also point to something important, namely, that our uncertainty about conditionals is not confined to uncertainty about the facts (what the actual world is like) but also expresses uncertainty about the counterfacts (what the world would be like if one or another supposition were true). To capture this observation, this essay proposes that the semantic contents of conditionals be treated as sets of vectors of possible worlds, not singleton worlds, with the coordinates of each specifying the world that is or would be true under the supposition that it represents. The probabilities of truth for conditionals will then depend on the joint probabilities of the facts and counterfacts, the latter in turn depending on the mode of supposition. The implication of this treatment is that the probabilities of conditionals are conditional probabilities whenever the mode of supposition is evidential. (shrink)

The desirability of what actually occurs is often influenced by what could have been. Preferences based on such value dependencies between actual and counterfactual outcomes generate a class of problems for orthodox decision theory, the best-known perhaps being the so-called Allais Paradox. In this paper we solve these problems by extending Richard Jeffrey's decision theory to counterfactual prospects, using a multidimensional possible-world semantics for conditionals, and showing that preferences that are sensitive to counterfactual considerations can still be desirability maximising. We (...) end the paper by investigating the conditions necessary and sufficient for a desirability function to be an expected utility. It turns out that the additional conditions imply highly implausible epistemic principles. (shrink)

Chance Neutrality is the thesis that, conditional on some proposition being true, its chance of being true should be a matter of practical indifference. The aim of this article is to examine whether Chance Neutrality is a requirement of rationality. We prove that given Chance Neutrality, the Principal Principle entails a thesis called Linearity; the centerpiece of von Neumann and Morgenstern’s expected utility theory. With this in mind, we argue that the Principal Principle is a requirement of practical rationality but (...) that Linearity is not and, hence, that Chance Neutrality is not rationally required. (shrink)

We present a general framework for representing belief-revision rules and use it to characterize Bayes's rule as a classical example and Jeffrey's rule as a non-classical one. In Jeffrey's rule, the input to a belief revision is not simply the information that some event has occurred, as in Bayes's rule, but a new assignment of probabilities to some events. Despite their differences, Bayes's and Jeffrey's rules can be characterized in terms of the same axioms: "responsiveness", which requires that revised beliefs (...) incorporate what has been learnt, and "conservativeness", which requires that beliefs on which the learnt input is "silent" do not change. To illustrate the use of non-Bayesian belief revision in economic theory, we sketch a simple decision-theoretic application. (shrink)

According to the Ramsey Test hypothesis the conditional claim that if A then B is credible just in case it is credible that B, on the supposition that A. If true the hypothesis helps explain the way in which we evaluate and use ordinary language conditionals. But impossibility results for the Ramsey Test hypothesis in its various forms suggest that it is untenable. In this paper, I argue that these results do not in fact have this implication, on the grounds (...) that similar results can be proved without recourse to the Ramsey test hypothesis. Instead they show that a number of well entrenched principles of rational belief and belief revision do not apply to conditionals. (shrink)

We distinguish three qualitatively different types of uncertainty—ethical, option and state space uncertainty—that are distinct from state uncertainty, the empirical uncertainty that is typically measured by a probability function on states of the world. Ethical uncertainty arises if the agent cannot assign precise utilities to consequences. Option uncertainty arises when the agent does not know what precise consequence an act has at every state. Finally, state space uncertainty exists when the agent is unsure how to construct an exhaustive state space. (...) These types of uncertainty are characterised along three dimensions—nature, object and severity—and the relationship between them is examined. We conclude that these different forms of uncertainty cannot be reduced to empirical uncertainty about the state of the world without inducing an increase in its severity. (shrink)

Richard Jeffrey espoused an antifoundationalist variant of Bayesian thinking that he termed ‘Radical Probabilism’. Radical Probabilism denies both the existence of an ideal, unbiased starting point for our attempts to learn about the world and the dogma of classical Bayesianism that the only justified change of belief is one based on the learning of certainties. Probabilistic judgment is basic and irreducible. Bayesian conditioning is appropriate when interaction with the environment yields new certainty of belief in some proposition but leaves one’s (...) conditional beliefs untouched (the ‘Rigidity’ condition). Although Richard Jeffrey denied the general applicability of this condition, one of his main contributions to probabilistic thinking is a form of belief updating—now typically called ‘Jeffrey conditioning’ or ‘probability kinematics’—that is appropriate in circumstances in which Rigidity is satisfied, but where the interaction causes one to reevaluate one’s probability judgments over some partition of the possibility space without conferring certainty on any particular element. The most familiar occasion for Jeffrey conditioning is receipt of uncertain evidence: things partially perceived or remembered. But it also serves to illuminate belief updating occasioned by a change in one’s degrees of conditional belief, a kind of belief change largely ignored by classical Bayesianism. I argue that such changes in conditional belief can also be basic (in the sense of not being analyzable as a consequence of conditioning on factual information) and offer a kinematical model for a particular kind change in conditional belief. Both are applied to changes in preference. Finally, I argue that Rigidity can fail when changes of belief give inferential grounds for changes in conditional belief (and vice versa). These failures show that conditioning methods are properly regarded, not as valid rules of inference, but as tools in the ‘art of judgment’. (shrink)

On Hume’s account of motivation, beliefs and desires are very different kinds of propositional attitudes. Beliefs are cognitive attitudes, desires emotive ones. An agent’s belief in a proposition captures the weight he or she assigns to this proposition in his or her cognitive representation of the world. An agent’s desire for a proposition captures the degree to which he or she prefers its truth, motivating him or her to act accordingly. Although beliefs and desires are sometimes entangled, they play very (...) different roles in rational agency. In two classic papers (Lewis 1988, 1996), David Lewis discusses several challenges to this Humean picture, but ultimately rejects them. We think that his discussion of a central anti-Humean alternative – the desire-as-belief thesis – is in need of refinnement. On this thesis, the desire for proposition p is given by the belief that p is desirable. Lewis claims that ‘[e]xcept in trivial cases, [this thesis] collapses into contradiction’(Lewis 1996, p. 308). The problem, he argues, is that the thesis is inconsistent with the purportedly plausible requirement that one’s desire for a proposition should not change upon learning that the proposition is true; call this the invariance requirement. In this paper, we revisit Lewis’s argument. We show that, if one carefully distinguishes between non-evaluative and evaluative propositions, the desire-asbelief thesis can be rendered consistent with the invariance requirement. Lewis’s conclusion holds only under certain conditions: the desire-as-belief thesis conflicts with the invariance requirement if and only if there are certain correlations between non-evaluative and evaluative propositions. But when there are such correlations, we suggest, the invariance requirement loses its plausibility. Thus Lewis’s argument against the desire-as-belief thesis appears to be valid only in cases in which it is unsound. (shrink)

Many policy decisions take input from collections of scientific models. Such decisions face significant and often poorly understood uncertainty. We rework the so-called confidence approach to tackle decision-making under severe uncertainty with multiple models, and we illustrate the approach with a case study: insurance pricing using hurricane models. The confidence approach has important consequences for this case and offers a powerful framework for a wide class of problems. We end by discussing different ways in which model ensembles can feed information (...) into the approach, appropriate to different collections of models. (shrink)

How should we revise our beliefs in response to the expressed probabilistic opinions of experts on some proposition when these experts are in disagreement? In this paper I examine the suggestion that in such circumstances we should adopt a linear average of the experts’ opinions and consider whether such a belief revision policy is compatible with Bayesian conditionalisation. By looking at situations in which full or partial deference to the expressed opinions of others is required by Bayesianism I show that (...) only in trivial circumstances are the requirements imposed by linear averaging compatible with it. (shrink)

Many fine-grained decisions concerning climate change involve significant, even severe, uncertainty. Here, we focus on modelling the decisions of single agents, whether individual persons or groups perceived as corporate entities. We offer a taxonomy of the sources and kinds of uncertainty that arise in framing these decision problems, as well as strategies for making a choice in spite of uncertainty. The aim is to facilitate a more transparent and structured treatment of uncertainty in climate decision making.

Decision-making typically requires judgments about causal relations: we need to know the causal effects of our actions and the causal relevance of various environmental factors. We investigate how several individuals' causal judgments can be aggregated into collective causal judgments. First, we consider the aggregation of causal judgments via the aggregation of probabilistic judgments, and identify the limitations of this approach. We then explore the possibility of aggregating causal judgments independently of probabilistic ones. Formally, we introduce the problem of causal-network aggregation. (...) Finally, we revisit the aggregation of probabilistic judgments when this is constrained by prior aggregation of qualitative causal judgments. (shrink)

The Desire-as-Belief thesis (DAB) states that any rational person desires a proposition exactly to the degree that she believes or expects the proposition to be good. Many people take David Lewis to have shown the thesis to be inconsistent with Bayesian decision theory. However, as we show, Lewis's argument was based on an Invariance condition that itself is inconsistent with the (standard formulation of the) version of Bayesian decision theory that he assumed in his arguments against DAB. The aim of (...) this paper is to explore what impact the rejection of Invariance has on the DAB thesis. Without assuming Invariance, we first refute all versions of DAB that entail that there are only two levels of goodness. We next consider two theses according to which rational desires are intimately connected to expectations of (multi-levelled) goodness, and show that these are consistent with Bayesian decision theory as long as we assume that the contents of 'value propositions' are not fixed. We explain why this conclusion is independently plausible, and show how to construct such propositions. (shrink)

What value should we put on our chances of obtaining a good? This paper argues that, contrary to the widely accepted theory of von Neumann and Morgenstern, the value of a chance of some good G may be a nonlinear function of the value of G. In particular, chances may have diminishing marginal utility, a property that is termed chance uncertainty aversion. The hypothesis that agents are averse to uncertainy about chances explains a pattern of preferences often observed in the (...) Ellsberg paradox. While these preferences have typically been taken to refute Bayesian decision theory, it is shown that chance risk aversion is perfectly compatible with it. (shrink)

Richard Jeffrey regarded the version of Bayesian decision theory he floated in ‘The Logic of Decision’ and the idea of a probability kinematics—a generalisation of Bayesian conditioning to contexts in which the evidence is ‘uncertain’—as his two most important contributions to philosophy. This paper aims to connect them by developing kinematical models for the study of preference change and practical deliberation. Preference change is treated in a manner analogous to Jeffrey’s handling of belief change: not as mechanical outputs of combinations (...) of intrinsic desires plus information, but as a matter of judgement and of making up one’s mind. In the first section Jeffrey’s probability kinematics is motivated and extended to the treatment of changes in conditional belief. In the second, analogous kinematical models are developed for preference change and in particular belief-induced change that depends on an invariance condition for conditional preference. The two are the brought together in the last section in a tentative model of pratical deliberation. (shrink)

This paper provides new foundations for Bayesian Decision Theory based on a representation theorem for preferences defined on a set of prospects containing both factual and conditional possibilities. This use of a rich set of prospects not only provides a framework within which the main theoretical claims of Savage, Ramsey, Jeffrey and others can be stated and compared, but also allows for the postulation of an extended Bayesian model of rational belief and desire from which they can be derived as (...) special cases. The main theorem of the paper establishes the existence of a such a Bayesian representation of preferences over conditional prospects, i.e. the existence of a pair of real-valued functions respectively measuring the agent’s degrees of belief and desire and which satisfy the postulated rationality conditions on partial belief and desire. The representation of partial belief is shown to be unique and that of partial desire, unique up to a linear transformation. (shrink)

This paper explores some aspects of the relation between different ways of achieving a consensus on the judgemental values of a group of indviduals; in particular, aggregation and deliberation. We argue firstly that the framing of an aggregation problem itself generates information that individuals are rationally obliged to take into account. And secondly that outputs of the deliberative process that this initiates is in tension with constraints on consensual values typically imposed by aggregation theory, at least when deliberation is modelled (...) as process of learning from others compatible with Bayesian updating principles. (shrink)

The Intergovernmental Panel on Climate Change has developed a novel framework for assessing and communicating uncertainty in the findings published in their periodic assessment reports. But how should these uncertainty assessments inform decisions? We take a formal decision-making perspective to investigate how scientific input formulated in the IPCC’s novel framework might inform decisions in a principled way through a normative decision model.

Bayesian models typically assume that agents are rational, logically omniscient and opinionated. The last of these has little descriptive or normative appeal, however, and limits our ability to describe how agents make up their minds (as opposed to changing them) or how they can suspend or withdraw their opinions. To address these limitations this paper represents the attitudinal states of non-opinionated agents by sets of (permissible) probability and desirability functions. Several basic ways in which such states of mind can be (...) changed are then characterised and compared with those found in AGM style models of attitude revision. Finally these models are employed to describe how agents make up their mind when deliberating. (shrink)

How should an impartial social observer judge distributions of well-being across different individuals when there is uncertainty regarding the state of the world? I explore this question by imposing very weak conditions of rationality and benevolent sympathy on impartial betterness judgments under uncertainty. Although weak enough to be consistent with all the main theories of rationality, these conditions prove to be sufficient to rule out any heterogeneity in what is good for individuals, to require a neutral attitude to uncertainty on (...) the part of the social observer, and to require that both individual and social betterness be strongly separable. (shrink)

Conditional attitudes are not the attitudes an agent is disposed to acquire in event of learning that a condition holds. Rather they are the components of agent's current attitudes that derive from the consideration they give to the possibility that the condition is true. Jeffrey's decision theory can be extended to include quantitative representation of the strength of these components. A conditional desirability measure for degrees of conditional desire is proposed and shown to imply that an agent's degrees of conditional (...) belief are conditional probabilities. Rational conditional preference is axiomatised and by application of Bolker's representation theorem for rational preferences it is shown that conditional preference rankings determine the existence of probability and desirability measures that agree with them. It is then proven that every conditional desirability function agrees with an agent's conditional preferences and, under certain assumptions, every desirability function agreeing with an agent's conditional preferences is a conditional desirability function agreeing with her unconditional preferences. (shrink)

Adams Thesis has much evidence in its favour, but David Lewis famously showed that it cannot be true, in all but the most trivial of cases, if conditionals are proprositions and their probabilities are classical probabilities of truth. In this paper I show thatsimilar results can be constructed for a much wider class of conditionals. The fact that these results presuppose that the logic of conditionals is Boolean motivates a search for a non-Boolean alternative. It is argued that the exact (...) proposition expressed by a conditional depends on the context in which it is uttered. Consequentlyits probability of truth will depend not only on the probabilities of the various propositions it might express, but also on the probabilities of the contexts determining which proposition it does in fact express.The semantic theory developed from this is then shown to explain why agents degrees of belief satisfyAdams Thesis. Finally the theory is compared with proposals for a three-valued logic of conditionals. (shrink)

This paper investigates the role of conditionals in hypothetical reasoning and rational decision making. Its main result is a proof of a representation theorem for preferences defined on sets of sentences (and, in particular, conditional sentences), where an agent’s preference for one sentence over another is understood to be a preference for receiving the news conveyed by the former. The theorem shows that a rational preference ordering of conditional sentences determines probability and desirability representations of the agent’s degrees of belief (...) and desire that satisfy, in the case of non-conditional sentences, the axioms of Jeffrey’s decision theory and, in the case of conditional sentences, Adams’ expression for the probabilities of conditionals. Furthermore, the probability representation is shown to be unique and the desirability representation unique up to positive linear transformation. (shrink)

Reports of the Intergovernmental Panel on Climate Change (IPCC) employ an evolving framework of calibrated language for assessing and communicating degrees of certainty in findings. A persistent challenge for this framework has been ambiguity in the relationship between multiple degree-of-certainty metrics. We aim to clarify the relationship between the likelihood and confidence metrics used in the Fifth Assessment Report (2013), with benefits for mathematical consistency among multiple findings and for usability in downstream modeling and decision analysis. We discuss how our (...) proposal meshes with current and proposed practice in IPCC uncertainty assessment. (shrink)

The UK has been ‘following the science’ in response to the COVID-19 pandemic in line with the national framework for the use of scientific advice in assessment of risk. We argue that the way in which it does so is unsatisfactory in two important respects. Firstly, pandemic policy making is not based on a comprehensive assessment of policy impacts. And secondly, the focus on reasonable worst-case scenarios as a way of managing uncertainty results in a loss of decision-relevant information and (...) does not provide a coherent basis for policy making. (shrink)

Diversity of opinion both presents problems and aff ords opportunities. Diff erences of opinion can stand in the way of reaching an agreement within a group on what decisions to take. But at the same time, the fact that the differences in question could derive from access to different information or from the exercise of diff erent judgemental skills means that they present individuals with the opportunity to improve their own opinions. This paper explores the implications for solutions to the (...) former (aggregation) problem of supposing that individuals exploit these opportunities. In particular, it argues that rational individual revision of opinion implies that aggregation problems are unstable in a certain sense and that solving them by exploiting the information embedded in individual opinion has profound implications for the conditions that we should impose on aggregation procedures. (shrink)

This paper uses the framework of Popper and Miller's work on axiom systems for conditional probabilities to explore Adams' thesis concerning the probabilities of conditionals. It is shown that even very weak axiom systems have only a very restricted set of models satisfying a natural generalisation of Adams' thesis, thereby casting severe doubt on the possibility of developing a non-Boolean semantics for conditionals consistent with it.

This article examines Becker's thesis that the hypothesis that choices maximize expected utility relative to fixed and universal tastes provides a general framework for the explanation of behaviour. Three different models of preference revision are presented and their scope evaluated. The first, the classical conditioning model, explains all changes in preferences in terms of changes in the information held by the agent, holding fundamental beliefs and desires fixed. The second, the Jeffrey conditioning model, explains them in terms of changes in (...) both the information held by the agent and changes in her prior beliefs, holding her fundamental desires fixed. The final model, that of generalized conditioning, allows for explanations in terms of changes in the values of all three variables. Key Words: preference change • decision theory • probability • desirability • attitude change. (shrink)

This paper reconstructs and evaluates the representation theorem presented by Ramsey in his essay 'Truth and Probability', showing how its proof depends on a novel application of Hölder's theory of measurement. I argue that it must be understood as a solution to the problem of measuring partial belief, a solution that in many ways remains unsurpassed. Finally I show that the method it employs may be interpreted in such a way as to avoid a well known objection to it due (...) to Richard Jeffrey. (shrink)

Models of collective deliberation often assume that the chief aim of a deliberative exchange is the sharing of information. In this paper, we argue that an equally important role of deliberation is to draw participants’ attention to pertinent questions, which can aid the assembly and processing of distributed information by drawing deliberators’ attention to new issues. The assumption of logical omniscience renders classical models of agents’ informational states unsuitable for modelling this role of deliberation. Building on recent insights from psychology, (...) linguistics and philosophy about the role of questions in speech and thought, we propose a different model in which beliefs are treated as answers directed at specific questions. Here, questions are formally represented as partitions of the space of possibilities and individuals’ information states as sets of questions and corresponding partial answers to them. The state of conversation is then characterised by individuals’ information together with the questions under discussion, which can be steered by various deliberative inputs. Using this model, deliberation is then shown to shape collective decisions in ways that classical models cannot capture, allowing for novel explanations of how group consensus is achieved. (shrink)

Foundations of Bayesianism is an authoritative collection of papers addressing the key challenges that face the Bayesian interpretation of probability today. Some of these papers seek to clarify the relationships between Bayesian, causal and logical reasoning. Others consider the application of Bayesianism to artificial intelligence, decision theory, statistics and the philosophy of science and mathematics. The volume includes important criticisms of Bayesian reasoning and also gives an insight into some of the points of disagreement amongst advocates of the Bayesian approach. (...) The upshot is a plethora of new problems and directions for Bayesians to pursue.The book will be of interest to graduate students or researchers who wish to learn more about Bayesianism than can be provided by introductory textbooks to the subject. Those involved with the applications of Bayesian reasoning will find essential discussion on the validity of Bayesianism and its limits, while philosophers and others interested in pure reasoning will find new ideas on normativity and the logic of belief. (shrink)

This paper explores the problem of comparing the strengths of different individual's attitudes, and especially their evaluative attitudes, by looking at how measures of these quantities are obtained. I argue that comparisons of both strengths of belief and relative strengths of preference and desire are justified by the causal role they play in the production of action.

Case-based reasoning is a familiar method of evaluating sentences. But when applied to conditionals, it seems to lead to implausible conclusions. In this paper I argue that the problem arises from equating the probability of a conditional sentence on the evidential supposition of some condition with the conditional probability of the former, given the latter.

This paper explores the problem of comparing the strengths of different individual's attitudes, and especially their evaluative attitudes, by looking at how measures of these quantities are obtained. I argue that comparisons of both strengths of belief and relative strengths of preference and desire are justified by the causal role they play in the production of action.

This paper reconstructs and evaluates the representation theorem presented by Ramsey in his essay ‘Truth and Probability’, showing how its proof depends on a novel application of Hölder's theory of measurement. I argue that it must be understood as a solution to the problem of measuring partial belief, a solution that in many ways remains unsurpassed. Finally I show that the method it employs may be interpreted in such a way as to avoid a well known objection to it due (...) to Richard Jeffrey. (shrink)

In this paper Richard Jeffrey's 'Logic of Decision' is extended by examination of agents' attitudes to the sorts of possibilities identified by indicative conditional sentences. An expression for the desirability of conditionals is proposed and, along with Adams' thesis that the probability of a conditional equals the conditional probability of its antecedent given its consequent, is defended by informally deriving it from Jeffrey's notion of desirability and some weak constraints on rational preference for conditional possibilities. Finally a statement is given (...) of a representation theorem establishing the conditions under which a rational agent's preferences for conditionals determines the existence of unique measures (up to choice of scale) of her degrees of belief and desire. (shrink)

I argue that two of the standard axioms of the AGM theory of belief revision stand in the way of it serving as the basis for an adequate account of defeasible reasoning, because they respectively disallow the adoption of beliefs not logically entailed by those previously learned and the abandonment of those not contradicted by them.

Multiple-vote majority rule is a procedure for making group decisions in which individuals weight their votes on issues in accordance with how competent they are on them. When individuals are motivated by the truth and know their relative competence on different issues, multiple-vote majority rule performs nearly as well, epistemically speaking, as rule by an expert oligarchy, but is still acceptable from the point of view of equal participation in the political process.Send article to KindleTo send this article to your (...) Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.A CASE FOR MULTIPLE-VOTE MAJORITY RULEVolume 9, Issue 1Richard Bradley and Christopher ThompsonDOI: https://doi.org/10.1017/epi.2011.5Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. A CASE FOR MULTIPLE-VOTE MAJORITY RULEVolume 9, Issue 1Richard Bradley and Christopher ThompsonDOI: https://doi.org/10.1017/epi.2011.5Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. A CASE FOR MULTIPLE-VOTE MAJORITY RULEVolume 9, Issue 1Richard Bradley and Christopher ThompsonDOI: https://doi.org/10.1017/epi.2011.5Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

Adams' famous thesis that the probabilities of conditionals are conditional probabilities is incompatible with standard probability theory. Indeed it is incompatible with any system of monotonic conditional probability satisfying the usual multiplication rule for conditional probabilities. This paper explores the possibility of accommodating Adams' thesis in systems of non-monotonic probability of varying strength. It shows that such systems impose many familiar lattice theoretic properties on their models as well as yielding interesting logics of conditionals, but that a standard complementation operation (...) cannot be defined within them, on pain of collapsing probability into bivalence. (shrink)