(This is for the series Elements of Decision Theory published by Cambridge University Press and edited by Martin Peterson) -/- Our beliefs come in degrees. I believe some things more strongly than I believe others. I believe very strongly that global temperatures will continue to rise during the coming century; I believe slightly less strongly that the European Union will still exist in 2029; and I believe much less strongly that Cardiff is east of Edinburgh. My credence in something is (...) a measure of the strength of my belief in it; it represents my level of confidence in it. These are the states of mind we report when we say things like ‘I’m 20% confident I switch off the gas before I left' or ‘I’m 99.9% confident that it is raining outside'. -/- There are laws that govern these credences. For instance, I shouldn't be more confident that sea levels will rise by over 2 metres in the next 100 years than I am that they'll rise by over 1 metre, since the latter is true if the former is. This book is about a particular way we might try to establish these laws of credence: the Dutch Book arguments (For briefer overviews of these arguments, see Alan Hájek’s entry in the Oxford Handbook of Rational and Social Choice and Susan Vineberg’s entry in the Stanford Encyclopaedia.) -/- We begin, in Chapter 2, with the standard formulation of the various Dutch Book arguments that we'll consider: arguments for Probabilism, Countable Additivity, Regularity, and the Principal Principle. In Chapter 3, we subject this standard formulation to rigorous stress-testing, and make some small adjustments so that it can withstand various objections. What we are left with is still recognisably the orthodox Dutch Book argument. In Chapter 4, we set out the Dutch Strategy argument for Conditionalization. In Chapters 5 and 6, we consider two objections to Dutch Book arguments that cannot be addressed by making small adjustments. Instead, we must completely redesign those arguments, replacing them with ones that share a general approach but few specific details. In Chapter 7, we consider a further objection to which I do not have a response. In Chapter 8, we'll ask what happens to the Dutch Book arguments if we change certain features of the basic framework in which we've been working: first, we ask how Dutch Book arguments fare when we consider credences in self-locating propositions, such as It is Monday; second, we lift the assumption that the background logic is classical and explore Dutch Book arguments for non-classical logics; third, we lift the assumption that an agent's credal state can be represented by a single assignment of numerical values to the propositions she considers. In Chapter 9, we present the mathematical results that underpin these arguments. (shrink)

The most immediately appealing model for formal constraints on degrees of belief is provided by probability theory, which tells us, for instance, that the probability of P can never be greater than that of (P v Q). But while this model has much intuitive appeal, many have been concerned to provide arguments showing that ideally rational degrees of belief would conform to the calculus of probabilities. The arguments most frequently used to make this claim plausible are the so-called (...) "Dutch Book" arguments. (shrink)

The Reflection Principle can be defended with a Diachronic Dutch Book Argument (DBA), but it is also defeated by numerous compelling counter-examples. It seems then that Diachronic DBAs can lead us astray. Should we reject them en masse—including Lewis’s Diachronic DBA for Conditionalization? Rachael Briggs’s “suppositional test” is supposed to differentiate between Diachronic DBAs that we can safely ignore (including the DBA for Reflection) and Diachronic DBAs that we should find compelling (including the DBA for Conditionalization). I argue (...) that Brigg’s suppositional test is wrong: it sets the bar for coherence too high and places certain cases of self-doubt on the wrong side of the divide. Given that the suppositional test is unsatisfactory, we are left without any justification for discriminating between Diachronic DBAs and ought to reject them all—including the DBA for Conditionalization. (shrink)

Recently a number of authors have tried to avoid the failures of traditional Dutch book arguments by separating them from pragmatic concerns of avoiding a sure loss. In this paper I examine defenses of this kind by Howson and Urbach, Hellman, and Christensen. I construct rigorous explications of their arguments and show that they are not cogent. I advocate abandoning Dutch book arguments in favor of a representation theorem.

Dutch Book arguments have been presented for static belief systems and for belief change by conditionalization. An argument is given here that a rule for belief change which under certain conditions violates probability kinematics will leave the agent open to a Dutch Book.

in The Oxford Handbook of Corporate Social Responsibility, ed. Paul Anand, Prasanta Pattanaik, and Clemens Puppe, forthcoming 2007.

I consider Dutch Book arguments for three principles of classical Bayesianism: (i) agents' belief-probabilities are consistent only if they obey the probability axioms. (ii) beliefs are updated by Bayesian conditionalisation. (iii) that the so-called Principal Principle connects statistical and belief probabilities. I argue that while there is a sound Dutch Book argument for (i), the standard ones for (ii) based on the Lewis-Teller strategy are unsound, for reasons pointed out by Christensen. I consider a type (...) of Dutch Book argument for (iii), where the statistical probability is a von Mises one. (shrink)

“Dutch Book” arguments and references to gambling theorems are typical in the debate between Bayesians and scientists committed to “classical” statistical methods. These arguments have rarely convinced non-Bayesian scientists to abandon certain conventional practices, partially because many scientists feel that gambling theorems have little relevance to their research activities. In other words, scientists “don’t bet.” This article examines one attempt, by Schervish, Seidenfeld, and Kadane, to progress beyond such apparent stalemates by connecting “Dutch Book”–type (...) mathematical results with principles actually endorsed by practicing experimentalists. (shrink)

Cílem této práce je rozebrat možnosti argumentu holandské sázky ve prospěch probabilismu a stanovit jeho meze. Existuje mnoho podob argumentu, proto se nejprve budu věnovat popisu argumentu v jeho klasické podobě. Následně je podroben kritice ve třech oblastech: (1) problémy behaviorismu, (2) užitku peněz, (3) vztah koherence, jisté ztráty a racionality. Zásadní je dle mého především třetí oblast týkající se neschopnosti jednoduše propojit probabilistickou nekoherenci stupňů přesvědčení s jistou ztrátou a pragmatickou racionalitou. Věnuji se třem současným reinterpretacím argumentu ve prospěch (...) probabilistických norem (Armendt, Howson, Christensen), přičemž v textu docházím k závěru, že ani jedna z nich není uspokojivá. Selhávají v řešení problémů (Armendt, Christensen), nebo neříkají nic zajímavého ve prospěch probabilismu (Howson). Proto se domnívám, že rozbor argumentu především ukazuje, že pragmatická obhajoba epistemických norem racionality, jako je ta v argumentu holandské sázky, není pro probabilismus schůdnou cestou. Závěr se snaží nastínit, proč tomu tak je. (shrink)

The Dutch Book Argument for Probabilism assumes Ramsey's Thesis (RT), which purports to determine the prices an agent is rationally required to pay for a bet. Recently, a new objection to Ramsey's Thesis has emerged (Hedden 2013, Wronski & Godziszewski 2017, Wronski 2018)--I call this the Expected Utility Objection. According to this objection, it is Maximise Subjective Expected Utility (MSEU) that determines the prices an agent is required to pay for a bet, and this often disagrees with Ramsey's (...) Thesis. I suggest two responses to Hedden's objection. First, we might be permissive: agents are permitted to pay any price that is required or permitted by RT, and they are permitted to pay any price that is required or permitted by MSEU. This allows us to give a revised version of the Dutch Book Argument for Probabilism, which I call the Permissive Dutch Book Argument. Second, I suggest that even the proponent of the Expected Utility Objection should admit that RT gives the correct answer in certain very limited cases, and I show that, together with MSEU, this very restricted version of RT gives a new pragmatic argument for Probabilism, which I call the Bookless Pragmatic Argument. (shrink)

We consider here an important family of conditional bets, those that proceed to settlement if and only if some agreed evidence is received that a condition has been met. Despite an opinion widespread in the literature, we observe that when the evidence is strong enough to generate certainty as to whether the condition has been met or not, using traditional conditional probabilities for such bets will NOT preserve a gambler from having a synchronic Dutch Book imposed upon him. (...) On the contrary (I show) the gambler can be Dutch-Booked if his betting ratios ever depart from a rather different probability, one that involves the probability of the agreed evidence being provided. We note furthermore that this same alternative probability assessment is necessary if the evidence is weaker (i.e. if it fails to provide knowledge whether or not the condition has been met.) By contrast, some of the (rather different) probability assessments proposed by Jeffrey, precisely for such situations, still expose the gambler to a Dutch-Book. (shrink)

Probabilistic theories of rationality claim that degrees of belief have to satisfy the probability axioms in order to be rational. A standard argument to support this claim is the Dutch Book argument. This paper tries to show that, in spite of its popularity, the Dutch Book argument does not provide a foundation for normative theories of rationality. After a presentation of the argument and some of its criticisms a problem is pointed out: the Dutch (...) class='Hi'>Book argument applies only to situations with a specific formal structure. Several attempts to justify the argument for more general situations are considered and rejected. The only way to remedy the shortcoming, it is argued, seems to be the acceptance of a far-reaching and highly implausible empirical hypothesis. (shrink)

In this essay, I repair the group Dutch Book argument presented by Donald Gillies. I then examine what additional assumptions would be needed for the argument to generate genuinely normative prescriptions for groups of inquirers. Although the resulting norms will apply to fewer groups than Gillies originally intended, they are still an important addition to the normative landscape in social epistemology.

Consistent application of coherece arguments shows that fair betting quotients are subject to constraints that are too stringent to allow their identification with either degrees of belief or probabilities. The pivotal role of fair betting quotients in the Dutch Book Argument, which is said to demonstrate that a rational agent's degrees of belief are probabilities, is thus undermined from both sides.

A starting point for this paper is that there is at least one concept of probability, call it epistemic probability, which can be identified with belief or some sort of idealised belief. If this identification is to be of any significance, then it needs to be shown that epistemic probability is a ‘true’ probability concept and is subject to those restrictions and requirements which relate and govern probabilities, which we call the probability calculus.The most rehearsed argument to establish the probability (...) calculus for epistemic probabilities is the Dutch Book Argument. There are two intuitions behind the DBA. The first is that if we can find some fine-grained behavioural measure of epistemic probability, then we may be able to show that epistemic probabilities obey the probability calculus by showing that the behaviour is of a kind which is, as a matter of necessity, subject to certain limitations and restrictions. (shrink)

It is a common view that the axioms of probability can be derived from the following assumptions: probabilities reflect degrees of belief, degrees of belief can be measured as betting quotients; and a rational agent must select betting quotients that are coherent. In this paper, I argue that a consideration of reasonable betting behaviour, with respect to the alleged derivation of the first axiom of probability, suggests that and are incorrect. In particular, I show how a rational agent might assign (...) a ‘probability’ of zero to an event which she is sure will occur. (shrink)

If an agent believes that the probability of E being true is 1/2, should she accept a bet on E at even odds or better? Yes, but only given certain conditions. This paper is about what those conditions are. In particular, we think that there is a condition that has been overlooked so far in the literature. We discovered it in response to a paper by Hitchcock (2004) in which he argues for the 1/3 answer to the Sleeping Beauty problem. (...) Hitchcock argues that this credence follows from calculating her fair betting odds, plus the assumption that Sleeping Beauty’s credences should track her fair betting odds. We will show that this last assumption is false. Sleeping Beauty’s credences should not follow her fair betting odds due to a peculiar feature of her epistemic situation. (shrink)

If an agent believes that the probability of E being true is 1/2, should she accept a bet on E at even odds or better? Yes, but only given certain conditions. This paper is about what those conditions are. In particular, we think that there is a condition that has been overlooked so far in the literature. We discovered it in response to a paper by Hitchcock (2004) in which he argues for the 1/3 answer to the Sleeping Beauty problem. (...) Hitchcock argues that this credence follows from calculating her fair betting odds, plus the assumption that Sleeping Beauty’s credences should track her fair betting odds. We will show that this last assumption is false. Sleeping Beauty’s credences should not follow her fair betting odds due to a peculiar feature of her epistemic situation. (shrink)

We point out a yet unnoticed flaw in Dutch Book arguments that relates to a link between degrees of belief and betting quotients. We offer a set of precise conditions governing when a nonprobabilist is immune to the classical Dutch Book argument. We suggest that diachronic Dutch Book arguments are also affected.

Choice under uncertainty is treated in economics by different approaches. We can distinguish three of them, two of which concern individual choice, while the third frames individual choices within the analysis of the social system. The first approach can determine how a rational decision-maker must choose; the second one how a real decision-maker behaves; and the third one how decision-makers are represented in the general economic theory. The main theories that result from these approaches are briefly presented. This paper considers, (...) in particular, the third approach, which is the most general since it represents preferences by means of a continuous utility function of the possible outcomes, without any specification with regard to uncertainty. The link between the reservation prices of bets on events and their subjective probabilities is examined. It is shown as the additivity condition for these price-probabilities is not required by the Dutch book argument if preferences are represented by a continuous utility function that is not differentiable. (shrink)

One guide to an argument's significance is the number and variety of refutations it attracts. By this measure, the Dutch book argument has considerable importance.2 Of course this measure alone is not a sure guide to locating arguments deserving of our attention—if a decisive refutation has really been given, we are better off pursuing other topics. But the presence of many and varied counterarguments at least suggests that either the refutations are controversial, or that their target admits (...) of more than one interpretation, or both. The main point of this paper is to focus on a way of understanding the Dutch Book argument (DBA) that avoids many of the well-known criticisms, and to consider how it fares against an important criticism that still remains: the objection that the DBA presupposes value-independence of bets. (shrink)

My focus is on pragmatic arguments for various ‘rationality constraints’ on a decision maker’s state of mind: on his beliefs or preferences. An argument of this kind purports to show that a violator of a given constraint can be exposed to a decision problem in which he will act to his guaranteed disadvantage. Dramatically put, he can be exploited by a clever bookie who doesn’t know more than the agent himself. Examples of pragmatic arguments of this kind are (...) synchronic Dutch Books, for the standard probability axioms, diachronic Dutch Books, for the more controversial principles of reflection and conditionalization, and Money Pumps, for the transitivity requirement on preferences. The proposed exploitation set-ups share a common feature. If the violator of a given constraint is logically and mathematically competent, he can be exploited only if he is disunified in his decision-making. Exploitation is possible only if the agent makes decisions on various issues he confronts one by one, rather than on all of them together. Unity in decision making may be quite costly and is often inconvenient, especially when it concerns opportunity packages that are spread over time. On my view, therefore, pragmatic arguments should be seen as delivering conditional conclusions: “If you want to afford being disunified as a decision maker, then you’d better satisfy these constraints.” The arguments of this kind fail to establish the inherent rationality of the constraints under consideration. Levi’s view of the status of pragmatic arguments is opposed to mine. According to him, only synchronic pragmatic arguments are valid. The diachronic ones, he argues, lack any validity at all. This line of reasoning is questioned in my paper. (shrink)

The Dutch Book argument, like Route 66, is about to turn 80. It is arguably the most celebrated argument for subjective Bayesianism. Start by rejecting the Cartesian idea that doxastic attitudes are ‘all-or-nothing’; rather, they are far more nuanced degrees of belief, for short credences, susceptible to fine-grained numerical measurement. Add a coherentist assumption that the rationality of a doxastic state consists in its internal consistency. The remaining problem is to determine what consistency of credences amounts to. The (...) Dutch Book argument, in a nutshell, says that if your credences do not obey the probability calculus, you are ‘incoherent’—susceptible to sure losses at the hands of a ‘Dutch Bookie’—and thus irrational. Conclusion: rationality requires your credences to obey the probability calculus. And like Route 66, the fortunes of the Dutch Book argument have been mixed. Opinions on the argument are sharply divided. The list of its proponents is quite a ‘who’s who’ of philosophers of probability; they include de Finetti (1937, 1980), Carnap (1950, 1962, and more fully, 1955), Kemeny (1955), Lehman (1955), Shimony (1955), Adams (1962), Mellor (1971), Rosenkrantz (1981), van Fraassen (1989), Jeffrey (1983, 1992). (shrink)

A handful of well-known arguments (the 'diachronic Dutch book arguments') rely upon theorems establishing that, in certain circumstances, you are immune from sure monetary loss (you are not 'diachronically Dutch bookable') if and only if you adopt the strategy of conditionalizing (or Jeffrey conditionalizing) on whatever evidence you happen to receive. These theorems require non-trivial assumptions about which evidence you might acquire---in the case of conditionalization, the assumption is that, if you might learn that e, (...) then it is not the case that you might learn something else that is consistent with e. These assumptions may not be relaxed. When they are, not only will non-(Jeffrey) conditionalizers be immune from diachronic Dutch bookability, but (Jeffrey) conditionalizers will themselves be diachronically Dutch bookable. I argue: 1) that there are epistemic situations in which these assumptions are violated; 2) that this reveals a conflict between the premise that susceptibility to sure monetary loss is irrational, on the one hand, and the view that rational belief revision is a function of your prior beliefs and the acquired evidence alone, on the other; and 3) that this inconsistency demonstrates that diachronic Dutch book arguments for (Jeffrey) conditionalization are invalid. (shrink)

Dutch Book Arguments (DBAs) have been invoked to support various requirements of rationality. Some are plausible: probabilism and conditionalization. Others are less so: credal transparency and reflection. Anna Mahtani has argued for a new understanding of DBAs which, she claims, allow us to keep the DBAs for probabilism (and perhaps conditionalization) and reject the DBAs for credal transparency and reflection. I argue that Mahtani’s new account fails as (a) it does not support highly plausible requirements of rational (...) coherence and (b) it does not, even setting aside the first objection, succeed in undermining the DBAs for credal transparency or reflection. (shrink)

In this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch (...) class='Hi'>Book Argument for Reflection fails. The question of whether we have a Dutch Book Argument for Conditionalization is left open. (shrink)

Hitchcock advances a diachronic Dutch Book argument (DDB) for a 1/3 answer to the Sleeping Beauty problem. Bradley and Leitgeb argue that Hitchcock’s DDB argument fails. We demonstrate the following: (a) Bradley and Leitgeb’s criticism of Hitchcock is unconvincing; (b) nonetheless, there are serious reasons to worry about the success of Hitchcock’s argument; (c) however, it is possible to construct a new DDB for 1/3 about which such worries cannot be raised.

Dutch book and accuracy arguments are used to justify certain rationality constraints on credence functions. Underlying these Dutch book and accuracy arguments are associated theorems, and I show that the interpretation of these theorems can vary along a range of dimensions. Given that the theorems can be interpreted in a variety of different ways, what is the status of the associated arguments? I consider three possibilities: we could aggregate the results of the differently (...) interpreted theorems in some way, and motivate rationality constraints based on this aggregation; we could be permissive, and accept the conclusions of the Dutch book and accuracy arguments under all interpretations of the associated theorems; or we could select one uniquely correct interpretation of the Dutch book or accuracy theorem, and use that to justify certain rationality constraints. I show that each possibility faces problems, and conclude that Dutch book and accuracy theorems cannot be used to justify any principle of rationality. (shrink)

In the context of the Sleeping Beauty problem, it has been argued that so-called “halfers” can avoid Dutch book arguments by adopting evidential decision theory. I introduce a Dutch book for a variant of the Sleeping Beauty problem and argue that evidential decision theorists fall prey to it, whether they are halfers or thirders. The argument crucially requires that an action can provide evidence for what the agent would do not only at other decision points (...) where she has exactly the same information, but also at decision points where she has different but “symmetric” information. (shrink)

It is often loosely said that Ramsey The foundations of mathematics and other logical essays, Routledge and Kegan Paul, Abingdon, pp 156–198, 1931) and de Finetti Studies in subjective probability, Kreiger Publishing, Huntington, 1937) proved that if your credences are inconsistent, then you will be willing to accept a Dutch Book, a wager portfolio that is sure to result in a loss. Of course, their theorems are true, but the claim about acceptance of Dutch Books assumes a (...) particular method of calculating expected utilities given the inconsistent credences. I will argue that there are better ways of calculating expected utilities given a potentially inconsistent credence assignment, and that for a large class of credences—a class that includes many inconsistent examples—these ways are immune to Dutch Books and single-shot domination failures. The crucial move is to replace Finite Additivity with Monotonicity, then \\le P\)) and then calculate expected utilities for positive U via the formula \\, dy\). This shows that Dutch Book arguments for probabilism, the thesis that one’s credences should be consistent, do not establish their conclusion. Finally, I will consider a modified argument based on multi-step domination failure that does better, but nonetheless is not as compelling as the Dutch Book arguments appeared to be. (shrink)

According to the Dutch Book Argument (DBA), if an agent's subjective probabilities fail to satisfy the axioms of the probability calculus and so make the agent vulnerable to a Dutch Book, the agent's subjective probabilities are incoherent and the agent is therefore irrational. Critics of DBA have argued, however, that probabilistic incoherence is compatible with various kinds of rationality – logico-semantic, epistemic, instrumental and prudential. In this paper, I provide an interpretation of DBA on which it (...) is true that probabilistic incoherence entails agent irrationality. Articulating this interpretation requires the specification of some of the connections among the varieties of rationality. Once this is done, it becomes possible to vindicate a modest version of DBA. (shrink)

According to the PCCP thesis, the probability of a conditional A → C is the conditional probability P. This claim is undermined by Lewis’ triviality results, which purport to show that apart from trivial cases, PCCP is not true. In the present article we show that the only rational, “Dutch Book-resistant” extension of the agent’s beliefs concerning non-conditional sentences A and C to the conditional A → C is by assuming that P = P. In other cases a (...) diachronic Dutch Book against the agent can be constructed. There is a tension between our findings and Lewis’ results, which needs to be explained. Therefore, we present a probability space which corresponds in a natural way to the diachronic Dutch Book—and which allows the conditional A → C to be interpreted as an event in a mathematically sound way. It also allows to formalize the notion of conditionalizing A → C on ¬C which plays a crucial role in Lewis’ proof. Our conclusion is that Lewis’ proof is circular, so it cannot be considered to be a sound argument against PCCP. (shrink)

In this paper, I present an argument for a rational norm involving a kind of credal attitude called a quantificational credence – the kind of attitude we can report by saying that Lucy thinks that each record in Schroeder’s collection is 5% likely to be scratched. I prove a result called a Dutch Book Theorem, which constitutes conditional support for the norm. Though Dutch Book Theorems exist for norms on ordinary and conditional credences, there is controversy (...) about the epistemic significance of these results. So, my conclusion is that if Dutch Book Theorems do, in general, support norms on credal states, then we have support for the suggested norm on quantificational credences. Providing conditional support for this norm gives us a fuller picture of the normative landscape of credal states. (shrink)

I give a Dutch book argument for CDT thirders in the context of a generalized Sleeping Beauty scenario. In combination with the Briggs (2010) Dutch book for EDT thirders, this amounts to a purely decision-theoretic argument for halfing in Sleeping Beauty. In combination with the Hitchcock (2004) Dutch book for CDT halfers, this amounts to a Dutch book argument against CDT. The combined Dutch book against CDT invites a plausible diagnosis (...) of the reasons for CDT’s failure: CDT is incompatible with the Dutch book requirement because it allows the values of epistemically impossible outcomes influence one’s decision. (shrink)

The Dutch Book Argument shows that an agent will lose surely in a gamble (a Dutch Book is made) if his degrees of belief do not satisfy the laws of the probability. Yet a question arises here: What does the Dutch Book imply? This paper firstly argues that there exists a utility function following Ramsey’s axioms. And then, it explicates the properties of the utility function and degree of belief respectively. The properties show that (...) coherence in partial beliefs for Subjective Bayesianism means that the degree of belief, representing a belief ordering, satisfies the laws of probability, and that coherence in preferences means that the preferences are represented by expected utilities. A coherent belief ordering and a utility scale induce a coherent preference ordering; a coherent preference ordering induces a coherent belief ordering which can be uniquely represented by a degree-of-belief function. The preferences (values) and beliefs are both incoherent or disordered if a Dutch Book is made. (shrink)

Conditionalization governs how to reallocate credence in light of new evidence. One prominent argument in favor of Conditionalization holds that an agent who violates it is vulnerable to a diachronic Dutch book: a series of acceptable bets offered at multiple times that inflict a sure loss. van Fraassen argues that an agent who violates the Principle of Reflection is likewise vulnerable to a diachronic Dutch book. He concludes that agents should conform to both Conditionalization and Reflection. (...) Some authors reply that Reflection is implausible and hence that there must be something wrong with diachronic Dutch book arguments. Other authors try to isolate a principled difference between the Dutch book argument for Conditionalization and the Dutch book argument for Reflection, such that the former argument may succeed even though the latter fails. I pursue a version of this strategy. I contend that, once we properly elucidate the notion of sure loss, non‐reflectors are not vulnerable to a sure loss. An agent who violates Reflection is not thereby subject to a diachronic Dutch book. Appearances to the contrary result from an unmotivated focus upon an overly narrow set of gambling scenarios. (shrink)

On the Lockean thesis one ought to believe a proposition if and only if one assigns it a credence at or above a threshold (Foley in Am Philos Q 29(2):111–124, 1992). The Lockean thesis, thus, provides a way of characterizing sets of all-or-nothing beliefs. Here we give two independent characterizations of the sets of beliefs satisfying the Lockean thesis. One is in terms of betting dispositions associated with full beliefs and one is in terms of an accuracy scoring system for (...) full beliefs. These characterizations are parallel to, but not merely derivative from, the more familiar Dutch Book (de Finetti in Theory of probability, vol 1, Wiley, London, 1974) and accuracy (Joyce in Philos Sci 65(4):575–603, 1998) arguments for probabilism. (shrink)