Probabilistic Principles, Misc

The Dempster–Shafer approach to expressing beliefabout a parameter in a statistical model is notconsistent with the likelihood principle. Thisinconsistency has been recognized for some time, andmanifests itself as a non-commutativity, in which theorder of operations (combining belief, combininglikelihood) makes a difference. It is proposed herethat requiring the expression of belief to be committed to the model (and to certain of itssubmodels) makes likelihood inference very nearly aspecial case of the Dempster–Shafer theory.

According to the Carter-Leslie Doomsday Argument, we should assign a high probability to the hypothesis that the human species will go extinct very soon. The argument is based on the application of Bayes’s theo-rem and a certain indifference principle with respect to the temporal location of our observed birth rank within the totality of birth ranks of all humans who will ever have lived. According to Bostrom’s Simulation Argument, which appeals to a weaker indifference principle than the Doomsday Argument, at (...) least one of the following three propositions must be true: (1) the human species is very likely to go extinct before reaching a posthuman stage, (2) it is very unlikely that some posthuman civili-zation will run a significant number of ancestor simula-tions, (3) it is almost sure that we are living in a com-puter simulation. According to my Doomsday Simulation Argument, both of the following propositions must be true: (1) it is almost sure that the human species will not go extinct before reaching a posthuman stage, (2) it is almost sure that we are not living in a computer simulation. (shrink)

We pose and resolve several vexing decision theoretic puzzles. Some are variants of existing puzzles, such as 'Trumped' (Arntzenius and McCarthy 1997), 'Rouble trouble' (Arntzenius and Barrett 1999), 'The airtight Dutch book' (McGee 1999), and 'The two envelopes puzzle' (Broome 1995). Others are new. A unified resolution of the puzzles shows that Dutch book arguments have no force in infinite cases. It thereby provides evidence that reasonable utility functions may be unbounded and that reasonable credence functions need not be countably (...) additive. The resolution also shows that when infinitely many decisions are involved, the difference between making the decisions simultaneously and making them sequentially can be the difference between riches and ruin. Finally, the resolution reveals a new way in which the ability to make binding commitments can save perfectly rational agents from sure losses. (shrink)

Extracting predictions from cosmological theories that describe a multiverse, for what we are likely to observe in our domain, is crucial to establishing the validity of these theories. One way to extract such predictions is from theory-generated probability distributions that allow for selection effects---generally expressed in terms of assumptions about anthropic conditionalization and how typical we are. In this paper, I urge three lessons about typicality in multiverse settings. Because it is difficult to characterize our observational situation in the multiverse, (...) we cannot assume that we are typical : nor can we ignore the issue of typicality, for it has a measurable impact on predictions for our observations. There are spectra of assumptions about both conditionalization and typicality, which lead to coincident predictions for our observations, leading to problems of confirmation in multiverse cosmology. And moreover, when one has the freedom to consider competing theories of the multiverse, the assumption of typicality may not lead to the highest likelihoods for our observations. These three entwined aspects of typicality imply that positive assertions about our typicality, such as the 'principle of mediocrity', are more questionable than has been recently claimed. (shrink)

In extracting predictions from theories that describe a multiverse, we face the difficulty that we must assess probability distributions over possible observations, prescribed not just by an underlying theory, but by a theory together with a conditionalization scheme that allows for selection effects. This means we usually need to compare distributions that are consistent with a broad range of possible observations, with actual experimental data. One controversial means of making this comparison is by invoking the 'principle of mediocrity': that is, (...) the principle that we are typical of the reference class implicit in the conjunction of the theory and the conditionalization scheme. In this paper, I quantitatively assess the principle of mediocrity in a range of cosmological settings, employing 'xerographic distributions' to impose a variety of assumptions regarding typicality. I find that for a fixed theory, the assumption that we are typical gives rise to higher likelihoods for our observations. If, however, one allows both the underlying theory and the assumption of typicality to vary, then the assumption of typicality does not always provide the highest likelihoods. Interpreted from a Bayesian perspective, these results support the claim that when one has the freedom to consider different combinations of theories and xerographic distributions, one should favor the framework that has the highest posterior probability; and then from this framework one can infer, in particular, how typical we are. In this way, the invocation of the principle of mediocrity is more questionable than has been recently claimed. (shrink)

Bayesians often assume, suppose, or conjecture that for any reasonable explication of the notion of simplicity a prior can be designed that will enforce a preference for hypotheses simpler in just that sense. Further, it is often claimed that the Bayesian framework automatically implements Occam's razor—that conditionalizing on data consistent with both a simple theory and a complex theory more or less inevitably favours the simpler theory. But it is shown here that there are simplicity-driven approaches to curve-fitting problems that (...) cannot be captured within the orthodox Bayesian framework and that the automatic razor does not function for such problems. (shrink)

Cosmology raises novel philosophical questions regarding the use of probabilities in inference. This work aims at identifying and assessing lines of arguments and problematic principles in probabilistic reasoning in cosmology. -/- The first, second, and third papers deal with the intersection of two distinct problems: accounting for selection effects, and representing ignorance or indifference in probabilistic inferences. These two problems meet in the cosmology literature when anthropic considerations are used to predict cosmological parameters by conditionalizing the distribution of, e.g., the (...) cosmological constant on the number of observers it allows for. However, uniform probability distributions usually appealed to in such arguments are an inadequate representation of indifference, and lead to unfounded predictions. It has been argued that this inability to represent ignorance is a fundamental flaw of any inductive framework using additive measures. In the first paper, I examine how imprecise probabilities fare as an inductive framework and avoid such unwarranted inferences. In the second paper, I detail how this framework allows us to successfully avoid the conclusions of Doomsday arguments in a way no Bayesian approach that represents credal states by single credence functions could. -/- There are in the cosmology literature several kinds of arguments referring to self- locating uncertainty. In the multiverse framework, different "pocket-universes" may have different fundamental physical parameters. We don’t know if we are typical observers and if we can safely assume that the physical laws we draw from our observations hold elsewhere. The third paper examines the validity of the appeal to the "Sleeping Beauty problem" and assesses the nature and role of typicality assumptions often endorsed to handle such questions. -/- A more general issue for the use of probabilities in cosmology concerns the inadequacy of Bayesian and statistical model selection criteria in the absence of well-motivated measures for different cosmological models. The criteria for model selection commonly used tend to focus on optimizing the number of free parameters, but they can select physically implausible models. The fourth paper examines the possibility for Bayesian model selection to circumvent the lack of well-motivated priors. (shrink)

We argue that quantum theory does not allow for a generalization of the notion of classical conditional probability by showing that the probability defined by the Lüders rule, standardly interpreted in the literature as the quantum-mechanical conditionalization rule, cannot be interpreted as such.

The Doomsday argument purports to show that the risk of the human species going extinct soon has been systematically underestimated. This argument has something in common with controversial forms of reasoning in other areas, including: game theoretic problems with imperfect recall, the methodology of cosmology, the epistomology of indexical belief, and the debate over so-called fine-tuning arguments for the design hypothesis. The common denominator is a certain premiss: the Self-Sampling Assumption. We present two strands of argument in favor of this (...) assumption. Through a series of throught experiments we then investigate some bizarre _prima facie_ consequences - backward causation, psychic powers, and an apparent conflict with the Principal Principle. (shrink)

If we receive information from multiple independent and partially reliable information sources, then whether we are justified to believe these information items is affected by how reliable the sources are, by how well the information coheres with our background beliefs and by how internally coherent the information is. We consider the following question. Is coherence a separable determinant of our degree of belief, i.e. is it the case that the more coherent the new information is, the more justified we are (...) in believing the new information, ceteris paribus? We show that if we consider sets of information items of any size (Holism), and if we assume that there exists a coherence Ordering over such sets and that coherence is a function of the probability distribution over the propositions in such sets (Probabilism), then Separability fails to hold. (shrink)

This article defends the Doomsday Argument, the Halfer Position in Sleeping Beauty, the Fine-Tuning Argument, and the applicability of Bayesian confirmation theory to the Everett interpretation of quantum mechanics. It will argue that all four problems have the same structure, and it gives a unified treatment that uses simple models of the cases and no controversial assumptions about confirmation or self-locating evidence. The article will argue that the troublesome feature of all these cases is not self-location but selection effects.

Consider someone whom you know to be an expert about some issue. She knows at least as much as you do and reasons impeccably. The issue is a straightforward case of statistical inference that raises no deep problems of epistemology. You happen to know the expert’s opinion on this issue. Should you defer to her by adopting her opinion as your own? An affirmative answer may appear mandatory. But this paper argues that a crucial factor in answering this question is (...) the description under which you identify the expert. Experts may be identified under various descriptions . Section 2 provides cases that illustrate how deference to a known expert may be appropriate under some descriptions, but not others. Section 3 proposes that, in these cases, the differentiating factor is self-identification: deference to an expert under a given description is appropriate when the expert self-identifies under that description. Section 4 presents a formal framework that demonstrates the sufficiency of self-identification for appropriate deference within a significant class of cases. Section 5 notes a way in which this phenomenon allows for a form of “agreeing to disagree” in Robert Aumann’s sense. Section 6 illustrates the practical application of these results to the case of testimonials. (shrink)

Il s'agit ici de la dimension existentielle de la theorie morale de S. Thomas d'Aquin. Pour lui, le domaine de l'incertain est generalement coextensif a celui de la contingence, de ce qui peut etre autre qu'il n'est. En general, S. Thomas envisage la contingence de la meme maniere qu'Aristote, mais dans une perspective totalement differente. Theologien, il s'interesse au monde physique surtout comme manifestation de la sagesse divine vers laquelle il desire monter. Il ne dedaigne pas pour autant les outils (...) de la science humaine, mais il les voit au service d'un dessein sacre. C'est dans cette perspective qu'il s'autorise a "calculer" le contingent. Car, d'apres lui, quel que soit son ideal, l'homme en quete de Dieu doit se frayer un chemin dans un univers qui, a ses yeux du moins, est expose a l'intervention du hasard. (shrink)

The view that decision methods can only be justified by appeal to pragmatic considerations is defended. Pragmatic considerations are viewed as providing the underlying subject matter of decision theories. It is argued that other approaches fail to provide grounds for a normative decision theory.It is argued that preferences that can lead to pragmatically adverse outcomes in a relevantly similar possible decision situation are pragmatically unsound, even if the decision situation never arises. This rebuts several standard objections to money-pump and Dutch (...) book arguments. However, because one can only appeal to relevantly similar decision situations in pragmatic arguments, these will have a less general scope than is often imagined. A conclusion is that pragmatic arguments for strong unconditional principles such as ‘always maximise expected utility!’ do not work. Pragmatic considerations can however be used to argue for conditional principles of the form ‘if conditions X, Y and Z are satisfied, then one ought to satisfy W’, where W need not follow logically from X, Y and Z.The notion of a sound pragmatic argument is defined in terms of particular notion of coherence, it is shown how this can be applied and how it handles problematic cases such as van Fraassen’s Dutch book for the principle of Reflection. (shrink)

This is a discussion of Richard Pettigrew's book "Accuracy and the Laws of Credence". I target Pettigrew's application of the accuracy framework to derive chance-credence principles. My principal contention is that Pettigrew's preferred version of the argument might in one sense be circular and, moreover, that Pettigrew's premises have content that go beyond that of standard chance-credence principles.

One of the main goals of Bayesian epistemology is to justify the rational norms credence functions ought to obey. Accuracy arguments attempt to justify these norms from the assumption that the source of value for credences relevant to their epistemic status is their accuracy. This assumption and some standard decision-theoretic principles are used to argue for norms like Probabilism, the thesis that an agent’s credence function is rational only if it obeys the probability axioms. We introduce an example that shows (...) that the accuracy arguments for Probabilism given by Joyce and Pettigrew fail, and that Probabilism in fact turns out to be false given Pettigrew’s way of conceiving of the goal of having accurate credences. Finally, we use our discussion of Pettigrew’s framework to draw an important general lesson about normative theorizing that relies on the positing of ideal agents. (shrink)

I outline four competing probabilistic accounts of contrastive evidential support and consider various considerations that might help arbitrate between these. The upshot of the discussion is that the so-called ‘Law of Likelihood’ is to be preferred to any of the alternatives considered.

This unique book presents a new interpretation of probability, rooted in the traditional interpretation that was current in the 17th and 18th centuries.

The book contains the transcription of a course on the foundations of probability given by the Italian mathematician Bruno de Finetti in 1979 at the a oeNational Institute of Advanced Mathematicsa in Rome.

This paper is a response to Forster and Sober's [1994] solution to the curve-fitting problem. If their solution is correct, it will provide us with a solution to the New Riddle of Induction as well as provide a basis for choosing realism over conventionalism. Examining this solution is also important as Forster and Sober incorporate it in much of their other philosophical work (see Forster [1995a, b, 1994] and Sober [1996, 1995, 1993]). I argue that Forster and Sober's solution is (...) subject precisely to the problem they seek to solve. They provide a method of choosing among hypotheses but only at the cost of requiring that we have a method of choosing between different ways of conceptualizing the world. Thus the solution raises a new problem--the world-fitting problem. (shrink)

In his "From classical to constructive probability," Weatherson offers a generalization of Kolmogorov's axioms of classical probability that is neutral regarding the logic for the object-language. Weatherson's generalized notion of probability can hardly be regarded as adequate, as the example of supervaluationist logic shows. At least, if we model credences as betting rates, the Dutch-Book argument strategy does not support Weatherson's notion of supervaluationist probability, but various alternatives. Depending on whether supervaluationist bets are specified as (a) conditional bets (Cantwell), (b) (...) unconditional bets with graded payoffs (Milne), or (c) unconditional bets with ungraded payoffs(Dietz), supervaluationist probability amounts to (a) conditional probability of truth given a truth-value, (b) the expected truth-value, or (c) the probability of truth, respectively. It is suggested that for supervaluationist logic, the third option is the most attractive one, for (unlike the other options) it preserves respect for single-premise entailment. (shrink)

There is no set Δ of probability axioms that meets the following three desiderata: (1) Δ is vindicated by a Dutch book theorem; (2) Δ does not imply regularity (and thus allows, among other things, updating by conditionalization); (3) Δ constrains the conditional probability q(·,z) even when the unconditional probability p(z) (=q(z,T)) equals 0. This has significant consequences for Bayesian epistemology, some of which are discussed.

KK is the thesis that if you can know p, you can know that you can know p. Though it’s unpopular, a flurry of considerations have recently emerged in its favor. Here we add fuel to the fire: standard resources allow us to show that any failure of KK will lead to the knowability and assertability of abominable indicative conditionals of the form, ‘If I don’t know it, p.’ Such conditionals are manifestly not assertable—a fact that KK defenders can easily (...) explain. I survey a variety of KK-denying responses and find them wanting. Those who object to the knowability of such conditionals must either (i) deny the possibility of harmony between knowledge and belief, or (ii) deny well-supported connections between conditional and unconditional attitudes. Meanwhile, those who grant knowability owe us an explanation of such conditionals’ unassertability—yet no successful explanations are on offer. Upshot: we have new evidence for KK. (shrink)

You have higher-order uncertainty iff you are uncertain of what opinions you should have. I defend three claims about it. First, the higher-order evidence debate can be helpfully reframed in terms of higher-order uncertainty. The central question becomes how your first- and higher-order opinions should relate—a precise question that can be embedded within a general, tractable framework. Second, this question is nontrivial. Rational higher-order uncertainty is pervasive, and lies at the foundations of the epistemology of disagreement. Third, the answer is (...) not obvious. The Enkratic Intuition---that your first-order opinions must “line up” with your higher-order opinions---is incorrect; epistemic akrasia can be rational. If all this is right, then it leaves us without answers---but with a clear picture of the question, and a fruitful strategy for pursuing it. (shrink)

Van Fraassen (1989) argues that Inference to the Best Explanation is incoherent in the sense that adopting it as a rule for belief change will make one susceptible to a dynamic Dutch book. The present paper argues against this. A strategy is described that allows us to infer to the best explanation free of charge.

_Philosophy of Probability: Contemporary Readings_ is the first anthology to collect essential readings in this important area of philosophy. Featuring the work of leading philosophers in the field such as Carnap, Hájek, Jeffrey, Joyce, Lewis, Loewer, Popper, Ramsey, van Fraassen, von Mises, and many others, the book looks in depth at the following key topics: subjective probability and credence probability updating: conditionalization and reflection Bayesian confirmation theory classical, logical, and evidential probability frequentism physical probability: propensities and objective chances. The book (...) features a useful primer on the mathematics of probability, and each section includes an introduction by the editor, as well as a guide to further reading. A broad-ranging and highly accessible exploration of the subject, _Philosophy of Probability_ is ideal for any student of formal epistemology, philosophy of science, metaphysics, or philosophy of mathematics. (shrink)

Expected accuracy arguments have been used by several authors (Leitgeb and Pettigrew, and Greaves and Wallace) to support the diachronic principle of conditionalization, in updates where there are only finitely many possible propositions to learn. I show that these arguments can be extended to infinite cases, giving an argument not just for conditionalization but also for principles known as ‘conglomerability’ and ‘reflection’. This shows that the expected accuracy approach is stronger than has been realized. I also argue that we should (...) be careful to distinguish diachronic update principles from related synchronic principles for conditional probability. (shrink)

In the first paper, I discussed the basic claims of Bayesianism (that degrees of belief are important, that they obey the axioms of probability theory, and that they are rationally updated by either standard or Jeffrey conditionalization) and the arguments that are often used to support them. In this paper, I will discuss some applications these ideas have had in confirmation theory, epistemol- ogy, and statistics, and criticisms of these applications.

We introduce a family of rules for adjusting one's credences in response to learning the credences of others. These rules have a number of desirable features. 1. They yield the posterior credences that would result from updating by standard Bayesian conditionalization on one's peers' reported credences if one's likelihood function takes a particular simple form. 2. In the simplest form, they are symmetric among the agents in the group. 3. They map neatly onto the familiar Condorcet voting results. 4. They (...) preserve shared agreement about independence in a wide range of cases. 5. They commute with conditionalization and with multiple peer updates. Importantly, these rules have a surprising property that we call synergy - peer testimony of credences can provide mutually supporting evidence raising an individual's credence higher than any peer's initial prior report. At first, this may seem to be a strike against them. We argue, however, that synergy is actually a desirable feature and the failure of other updating rules to yield synergy is a strike against them. (shrink)

The “puzzle of the unmarked clock” derives from a conflict between the following: (1) a plausible principle of epistemic modesty, and (2) “Rational Reflection”, a principle saying how one’s beliefs about what it is rational to believe constrain the rest of one’s beliefs. An independently motivated improvement to Rational Reflection preserves its spirit while resolving the conflict.

According to the Bayesian paradigm in the psychology of reasoning, the norms by which everyday human cognition is best evaluated are probabilistic rather than logical in character. Recently, the Bayesian paradigm has been applied to the domain of argumentation, where the fundamental norms are traditionally assumed to be logical. Here, we present a major generalisation of extant Bayesian approaches to argumentation that utilizes a new class of Bayesian learning methods that are better suited to modelling dynamic and conditional inferences than (...) standard Bayesian conditionalization, is able to characterise the special value of logically valid argument schemes in uncertain reasoning contexts, greatly extends the range of inferences and argumentative phenomena that can be adequately described in a Bayesian framework, and undermines some influential theoretical motivations for dual function models of human cognition. We conclude that the probabilistic norms given by the Bayesian approach to rationality are not necessarily at odds with the norms given by classical logic. Rather, the Bayesian theory of argumentation can be seen as justifying and enriching the argumentative norms of classical logic. (shrink)

This (brief) note is about the (evidential) “favoring” relation. Pre-theoretically, favoring is a three-place (epistemic) relation, between an evidential proposition E and two hypotheses H1 and H2. Favoring relations are expressed via locutions of the form: E favors H1 over H2. Strictly speaking, favoring should really be thought of as a four-place relation, between E, H1, H2, and a corpus of background evidence K. But, for present purposes (which won't address issues involving K), I will suppress the background corpus, so (...) as to simplify our discussion. Moreover, the favoring relation is meant to be a propositional epistemic relation, as opposed to a doxastic epistemic relation. That is, the favoring relation is not meant to be restricted to bodies of evidence that are possessed (as evidence) by some actual agent(s), or to hypotheses that are (in fact) entertained by some actual agent(s). In this sense, favoring is analogous to the relation of propositional justification — as opposed to doxastic justification (Conee 1980). In order to facilitate a comparison of Likelihoodist vs Bayesian explications of favoring, I will presuppose the following bridge principle, linking favoring and evidential support: • E favors H1 over H2 iff E supports H1 more strongly than E supports H2.1 Finally, I will only be discussing instances of the favoring relation involving contingent, empirical claims. So, it is to be understood that “favoring” will not apply if any of E, H1, or H2 are non-contingent (and/or non-empirical). With this background in place, we're ready to begin. (shrink)

Likelihoodists and Bayesians seem to have a fundamental disagreement about the proper probabilistic explication of relational (or contrastive) conceptions of evidential support (or confirmation). In this paper, I will survey some recent arguments and results in this area, with an eye toward pinpointing the nexus of the dispute. This will lead, first, to an important shift in the way the debate has been couched, and, second, to an alternative explication of relational support, which is in some sense a "middle way" (...) between Likelihoodism and Bayesianism. In the process, I will propose some new work for an old probability puzzle: the "Monty Hall" problem. (shrink)

Recent solutions to the curve-fitting problem, described in Forster and Sober ([1995]), trade off the simplicity and fit of hypotheses by defining simplicity as the paucity of adjustable parameters. Scott De Vito ([1997]) charges that these solutions are 'conventional' because he thinks that the number of adjustable parameters may change when the hypotheses are described differently. This he believes is exactly what is illustrated in Goodman's new riddle of induction, otherwise known as the grue problem. However, the 'number of adjustable (...) parameters' is actually a loose way of referring to a quantity that is not language dependent. The quantity arises out of Akaike's theorem in a way that ensures its language invariance. (shrink)

Traditional analyses of the curve fitting problem maintain that the data do not indicate what form the fitted curve should take. Rather, this issue is said to be settled by prior probabilities, by simplicity, or by a background theory. In this paper, we describe a result due to Akaike [1973], which shows how the data can underwrite an inference concerning the curve's form based on an estimate of how predictively accurate it will be. We argue that this approach throws light (...) on the theoretical virtues of parsimoniousness, unification, and non ad hocness, on the dispute about Bayesianism, and on empiricism and scientific realism. * Both of us gratefully acknowledge support from the Graduate School at the University of Wisconsin-Madison, and NSF grant DIR-8822278 (M.F.) and NSF grant SBE-9212294 (E.S.). Special thanks go to A. W. F. Edwards.William Harper. Martin Leckey. Brian Skyrms, and especially Peter Turney for helpful comments on an earlier draft. (shrink)

We resolve a useful formulation of the question how a statistician can coherently incorporate the information in a consulted expert’s probability assessment for an event into a personal posterior probability assertion. Using a framework that recognises the total information available as composed of units available only to each of them along with units available to both, we show: that a sufficient statistic for all the information available to both the expert and the statistician is the product of their odds ratios (...) in favour of the event; that the geometric mean of their two probabilities specifies a contour of pairs of assertions in the unit-square that yield the same posterior probability; that the information-combining function is parameterised by an unknown probability for the event conditioned only on the unspecified information common to both the statistician and the expert; and that an assessable mixing distribution over this unspecified probability allows an integrable mixture distribution to represent a computable posterior probability. The exact results allow the identification of the subclass of coherent probabilities that are externally Bayesian operators. This subclass is equivalent to the class of combining functions that honour the principles of uniformity and compromise. (shrink)

The Borel-Kolmogorov Paradox is typically taken to highlight a tension between our intuition that certain conditional probabilities with respect to probability zero conditioning events are well defined and the mathematical definition of conditional probability by Bayes' formula, which looses its meaning when the conditioning event has probability zero. We argue in this paper that the theory of conditional expectations is the proper mathematical device to conditionalize, and this theory allows conditionalization with respect to probability zero events. The conditional probabilities on (...) probability zero events in the Borel-Kolmogorov Paradox also can be calculated using conditional expectations. The alleged clash arising from the fact that one obtains different values for the conditional probabilities on probability zero events depending on what conditional expectation one uses to calculate them is resolved by showing that the different conditional probabilities obtained using different conditional expectations cannot be interpreted as calculating in different parametrizations of the conditional probabilities of the same event with respect to the same conditioning conditions. We conclude that there is no clash between the correct intuition about what the conditional probabilities with respect to probability zero events are and the technically proper concept of conditionalization via conditional expectations -- the Borel-Kolmogorov Paradox is just a pseudo-paradox. (shrink)

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. (shrink)

We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Rios Insua and Alon and Lehrer is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott, relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature.

A possible event always seems to be more probable than an impossible event. Although this constraint, usually alluded to as regularity , is prima facie very attractive, it cannot hold for standard probabilities. Moreover, in a recent paper Timothy Williamson has challenged even the idea that regularity can be integrated into a comparative conception of probability by showing that the standard comparative axioms conflict with certain cases if regularity is assumed. In this note, we suggest that there is a natural (...) weakening of the standard comparative axioms. It is shown that these axioms are consistent both with the regularity condition and with the essential feature of Williamson’s example. (shrink)

I advocate Time-Slice Rationality, the thesis that the relationship between two time-slices of the same person is not importantly different, for purposes of rational evaluation, from the relationship between time-slices of distinct persons. The locus of rationality, so to speak, is the time-slice rather than the temporally extended agent. This claim is motivated by consideration of puzzle cases for personal identity over time and by a very moderate form of internalism about rationality. Time-Slice Rationality conflicts with two proposed principles of (...) rationality, Conditionalization and Reflection. Conditionalization is a diachronic norm saying how your current degrees of belief should fit with your old ones, while Reflection is a norm enjoining you to defer to the degrees of belief that you expect to have in the future. But they are independently problematic and should be replaced by improved, time-slice-centric principles. Conditionalization should be replaced by a synchronic norm saying what degrees of belief you ought to have given your current evidence and Reflection should be replaced by a norm which instructs you to defer to the degrees of belief of agents you take to be experts. These replacement principles do all the work that the old principles were supposed to do while avoiding their problems. In this way, Time-Slice Rationality puts the theory of rationality on firmer foundations and yields better norms than alternative, non-time-slice-centric approaches. (shrink)

We propose and investigate an Analogy Principle in the context of Unary Inductive Logic based on a notion of support by structural similarity which is often employed to motivate scientific conjectures.

We propose an Analogy Principle in the context of Unary Inductive Logic and characterize the probability functions which satisfy it. In particular in the case of a language with just two predicates the probability functions satisfying this principle correspond to solutions of Skyrmsʼ ‘Wheel of Fortune’.

Fine-tuning arguments attempt to infer God’s existence from the empirical fact that life would not be possible if any of approximately two-dozen fundamental laws and properties of the universe had been even slightly different. In this essay, I consider a version that relies on the following principle: if an observation O is more likely to occur under hypothesis H1 than under hypothesis H2, then O supports accepting H1 over H2. I argue that this particular application of this principle is vulnerable (...) to straightforward counterexamples and attempt to explain the proper application conditions for this principle. (shrink)