Alan Hajek

The thesis that probabilities of conditionals are conditional probabilities has putatively been refuted many times by so-called ‘triviality results’, although it has also enjoyed a number of resurrections. In this paper I assault it yet again with a new such result. I begin by motivating the thesis and discussing some of the philosophical ramifications of its fluctuating fortunes. I will canvas various reasons, old and new, why the thesis seems plausible, and why we should care about its fate. I will look at some objections to Lewis’s famous triviality results, and thus some reasons for the pursuit of further triviality results. I will generalize Lewis’s results in ways that meet the objections. I will conclude with some reflections on the demise of the thesis—or otherwise

This paper is a response to Paul Bartha’s ‘Making Do Without Expectations’. We provide an assessment of the strengths and limitations of two notable extensions of standard decision theory: relative expectation theory and Paul Bartha’s relative utility theory. These extensions are designed to provide intuitive answers to some well-known problems in decision theory involving gaps in expectations. We argue that both RET and RUT go some way towards providing solutions to the problems in question but neither extension solves all the relevant problems.

This paper revisits the Pasadena game (Nover and Háyek 2004), a St Petersburg-like game whose expectation is undefined. We discuss serveral respects in which the Pasadena game is even more troublesome for decision theory than the St Petersburg game. Colyvan (2006) argues that the decision problem of whether or not to play the Pasadena game is ‘ill-posed’. He goes on to advocate a ‘pluralism’ regarding decision rules, which embraces dominance reasoning as well as maximizing expected utility. We rebut Colyvan’s argument, offering several considerations in favour of the Pasadena decision problem being well posed. To be sure, current decision theory, which is underpinned by various preference axioms, leaves indeterminate how one should value the Pasadena game. But we suggest that determinacy might be achieved by adding further preference axioms. We conclude by opening the door to a far greater plurality of decision rules. We suggest how the goal of unifying these rules might guide future research.

I came to philosophy as a refugee from mathematics and statistics. I was impressed by their power at codifying and precisifying antecedently understood but rather nebulous concepts, and at clarifying and exploring their interrelations. I enjoyed learning many of the great theorems of probability theory—equations rich in ‘P’s of this and of that. But I wondered what is this ‘P’? What do statements of probability mean? When I asked one of my professors, he looked at me like I needed medication. That medication was provided by philosophy, and I found it first during my Masters at the University of Western Ontario, working with Bill Harper, and then during my Ph.D. at Princeton, working with Bas van Fraassen, David Lewis, and Richard Jeffrey—all deft practitioners of formal methods. I found that philosophers had been asking my question about ‘P’ since about 1650, but they were still struggling to find definitive answers. I was also introduced to a host of other philosophical problems, and it became clear to me within nanoseconds of arriving at U.W.O. that I wanted to spend my life pursuing some of them. But I kept being drawn back to the formal methods of mathematics, and in particular of probability theory. It may be worthwhile to pause for a moment and to ask “What are formal methods?” Of course, it’s easy to come up with examples: the use of various logical systems, computational algorithms, causal graphs, information theory, probability theory and mathematics more generally. What do they have in common? They are all abstract representational systems. Sometimes the systems are studied in their own..

Much is asked of the concept of chance. It has been thought to play various roles, some in tension with or even incompatible with others. Chance has been characterized negatively, as the absence of causation; yet also positively—the ancient Greek τυχη´ reifies it—as a cause of events that are not governed by laws of nature, or as a feature of the laws themselves. Chance events have been understood epistemically as those whose causes are unknown; yet also objectively as a distinct ontological kind, sometimes called ‘pure’ chance events. Chance gives rise to individual unpredictability and disorder; yet it yields collective predictability and order—stable long-run statistics, and in the limit, aggregate behavior susceptible to precise mathematical theorems. Some authors believe that to posit chances is to abjure explanation; yet others think that chances are themselves explanatory. During the Enlightenment, talk of ‘chance’ was regarded as unscientific, unphilosophical, the stuff of superstition or ignorance; yet today it is often taken to be a fundamental notion of our most successful scientific theory, quantum mechanics, and a central concept of contemporary metaphysics. Chance has both negative and positive associations in daily life. The old word in English for it, hazard, which derives from French and originally from Arabic, still has unwelcome connotations of risk; ‘chance’ evokes uncertainty, uncontrollability, and chaos. Yet chance is also allied with luck, fortune, freedom from constraint, and diversity. And it apparently has various practical uses and benefits. It forms the basis of randomized trials in statistics, and of mixed strategies in decision theory and game theory; it is appealed to in order to resolve problems of fair division and other ethical..

We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required . Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins with our describing a world that plausibly has indeterminate chances. Rationality requires a certain alignment of your credences with corresponding hypotheses about the chances. Thus, if you hypothesize the chances to be indeterminate, your will inherit their indeterminacy in your corresponding credences. Our third argument is motivated by a dilemma. Epistemic rationality requires you to stay open-minded about contingent matters about which your evidence has not definitively legislated. Practical rationality requires you to be able to act decisively at least sometimes. These requirements can conflict with each other-for thanks to your open-mindedness, some of your options may have undefined expected utility, and if you are choosing among them, decision theory has no advice to give you. Such an option is playing Nover and Hájek’s Pasadena Game , and indeed any option for which there is a positive probability of playing the Pasadena Game. You can serve both masters, epistemic rationality and practical rationality, with an indeterminate credence to the prospect of playing the Pasadena game. You serve epistemic rationality by making your upper probability positive-it ensures that you are open-minded. You serve practical rationality by making your lower probability 0-it provides guidance to your decision-making. No sharp credence could do both.

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)