Subjective probability plays an increasingly important role in many fields concerned with human cognition and behavior. Yet there have been significant criticisms of the idea that probabilities could actually be represented in the mind. This paper presents and elaborates a view of subjective probability as a kind of sampling propensity associated with internally represented generative models. The resulting view answers to some of the most well known criticisms of subjective probability, and is also supported (...) by empirical work in neuroscience and behavioral psychology. The repercussions of the view for how we conceive of many ordinary instances of subjective probability, and how it relates to more traditional conceptions of subjective probability, are discussed in some detail. (shrink)

How do we ascribe subjective probability? In decision theory, this question is often addressed by representation theorems, going back to Ramsey (1926), which tell us how to define or measure subjective probability by observable preferences. However, standard representation theorems make strong rationality assumptions, in particular expected utility maximization. How do we ascribe subjective probability to agents which do not satisfy these strong rationality assumptions? I present a representation theorem with weak rationality assumptions which can (...) be used to define or measure subjective probability for partly irrational agents. (shrink)

It is well known that classical, aka ‘sharp’, Bayesian decision theory, which models belief states as single probability functions, faces a number of serious difficulties with respect to its handling of agnosticism. These difficulties have led to the increasing popularity of so-called ‘imprecise’ models of decision-making, which represent belief states as sets of probability functions. In a recent paper, however, Adam Elga has argued in favour of a putative normative principle of sequential choice that he claims to be (...) borne out by the sharp model but not by any promising incarnation of its imprecise counterpart. After first pointing out that Elga has fallen short of establishing that his principle is indeed uniquely borne out by the sharp model, I cast aspersions on its plausibility. I show that a slight weakening of the principle is satisfied by at least one, but interestingly not all, varieties of the imprecise model and point out that Elga has failed to motivate his stronger commitment. (shrink)

In the Bayesian approach to quantum mechanics, probabilities—and thus quantum states—represent an agent’s degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that (...) probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent’s prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent’s prior beliefs. Quantum certainty is therefore always some agent’s certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [(1983). Quantum logic, realism, and value-definiteness. Philosophy of Science, 50, 578], which applies the Kochen–Specker theorem to an entangled bipartite system. (shrink)

Bayesianism is the position that scientific reasoning is probabilistic and that probabilities are adequately interpreted as an agent's actual subjective degrees of belief, measured by her betting behaviour. Confirmation is one important aspect of scientific reasoning. The thesis of this paper is the following: if scientific reasoning is at all probabilistic, the subjective interpretation has to be given up in order to get right confirmation—and thus scientific reasoning in general. The Bayesian approach to scientific reasoning Bayesian confirmation theory (...) The example The less reliable the source of information, the higher the degree of Bayesian confirmation Measure sensitivity A more general version of the problem of old evidence Conditioning on the entailment relation The counterfactual strategy Generalizing the counterfactual strategy The desired result, and a necessary and sufficient condition for it Actual degrees of belief The common knock-down feature, or ‘anything goes’ The problem of prior probabilities. (shrink)

This article analyzes the role of entropy in Bayesian statistics, focusing on its use as a tool for detection, recognition and validation of eigen-solutions. “Objects as eigen-solutions” is a key metaphor of the cognitive constructivism epistemological framework developed by the philosopher Heinz von Foerster. Special attention is given to some objections to the concepts of probability, statistics and randomization posed by George Spencer-Brown, a figure of great influence in the field of radical constructivism.

In his classic book “the Foundations of Statistics” Savage developed a formal system of rational decision making. The system is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation that (...) satisfies certain arguably acceptable postulates determines a (finitely additive) probability distribution over the states and a utility assignment to the consequences, such that the preferences among acts are determined by their expected utilities. Additional problematic assumptions are however required in Savage's proofs. First, there is a Boolean algebra of events (sets of states) which determines the richness of the set of acts. The probabilities are assigned to members of this algebra. Savage's proof requires that this be a σ-algebra (i.e., closed under infinite countable unions and intersections), which makes for an extremely rich preference relation. On Savage's view we should not require subjective probabilities to be σ-additive. He therefore finds the insistence on a σ-algebra peculiar and is unhappy with it. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every consequence there is a constant act, which produces that consequence in every state. This assumption is known to be highly counterintuitive. The present work contains two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the more plausible and much weaker assumption that there are at least two non-equivalent constant acts. The second result also employs a novel way of deriving utilities in Savage-style systems -- without appealing to von Neumann-Morgenstern lotteries. The paper discusses the notion of “idealized agent" that underlies Savage's approach, and argues that the simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent. (shrink)

Credences are beliefs about evidential probabilities. We give the view an assessment-sensitive formulation, show how it evades the standard objections, and give several arguments in support.

The problem of probabilities of conditionals is one of the long-standing puzzles in philosophy of language. We defend and update Adams' solution to the puzzle: the probability of an epistemic conditional is not the probability of a proposition, but a probability under a supposition. -/- Close inspection of how a triviality result unfolds in a concrete scenario does not provide counterexamples to the view that probabilities of conditionals are conditional probabilities: instead, it supports the conclusion that probabilities (...) of conditionals violate standard probability theory. -/- This does not call into question probability theory per se; rather, it calls for a more careful understanding of its role: probability theory is a theory of probabilities of propositions; but as conditionals do not express propositions, their probabilities are not subject to the standard laws. -/- We argue that both conditional probabilities and probabilities of conditionals are best understood in terms of the dynamics of supposing, modeled as a restriction operation on a probability space. This version of the suppositionalist view allows us to connect Adams' Thesis to the widely held restrictor view of the semantics of conditionals. -/- We address two common objections to Adams' view: that the relevant probabilities are ‘probabilities only in name’, and that giving up conditional propositions puts us at a disadvantage when it comes to interpreting compounds. -/- Finally, we argue that some putative counterexamples to Adams' Thesis can be diagnosed as fallacies of probabilistic reasoning: they arise from applying to conditionals laws of standard probability theory which are invalid for them. (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided (...) that (ii) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

When probability discounting (or probability weighting), one multiplies the value of an outcome by one's subjective probability that the outcome will obtain in decision-making. The broader import of defending probability discounting is to help justify cost-benefit analyses in contexts such as climate change. This chapter defends probability discounting under risk both negatively, from arguments by Simon Caney (2008, 2009), and with a new positive argument. First, in responding to Caney, I argue that small costs (...) and benefits need to be evaluated, and that viewing practices at the social level is too coarse-grained. Second, I argue for probability discounting using a distinction between causal responsibility and moral responsibility. Moral responsibility can be cashed out in terms of blameworthiness and praiseworthiness, while causal responsibility obtains in full for any effect which is part of a causal chain linked to one's act. With this distinction in hand, unlike causal responsibility, moral responsibility can be seen as coming in degrees. My argument is, given that we can limit our deliberation and consideration to that which we are morally responsible for and that our moral responsibility for outcomes is limited by our subjective probabilities, our subjective probabilities can ground probability discounting. (shrink)

There is a trade-off between specificity and accuracy in existing models of belief. Descriptions of agents in the tripartite model, which recognizes only three doxastic attitudes—belief, disbelief, and suspension of judgment—are typically accurate, but not sufficiently specific. The orthodox Bayesian model, which requires real-valued credences, is perfectly specific, but often inaccurate: we often lack precise credences. I argue, first, that a popular attempt to fix the Bayesian model by using sets of functions is also inaccurate, since it requires us to (...) have interval-valued credences with perfectly precise endpoints. We can see this problem as analogous to the problem of higher order vagueness. Ultimately, I argue, the only way to avoid these problems is to endorse Insurmountable Unclassifiability. This principle has some surprising and radical consequences. For example, it entails that the trade-off between accuracy and specificity is in-principle unavoidable: sometimes it is simply impossible to characterize an agent’s doxastic state in a way that is both fully accurate and maximally specific. What we can do, however, is improve on both the tripartite and existing Bayesian models. I construct a new model of belief—the minimal model—that allows us to characterize agents with much greater specificity than the tripartite model, and yet which remains, unlike existing Bayesian models, perfectly accurate. (shrink)

Probability can be used to measure degree of belief in two ways: objectively and subjectively. The objective measure is a measure of the rational degree of belief in a proposition given a set of evidential propositions. The subjective measure is the measure of a particular subject’s dispositions to decide between options. In both measures, certainty is a degree of belief 1. I will show, however, that there can be cases where one belief is stronger than another yet both (...) beliefs are plausibly measurable as objectively and subjectively certain. In ordinary language, we can say that while both beliefs are certain, one belief is more certain than the other. I will then propose second, non probabilistic dimension of measurement, which tracks this variation in certainty in such cases where the probability is 1. A general principle of rationality is that one’s subjective degree of belief should match the rational degree of belief given the evidence available. In this paper I hope to show that it is also a rational principle that the maximum stake size at which one should remain certain should match the rational weight of certainty given the evidence available. Neither objective nor subjective measures of certainty conform to the axioms of probability, but instead are measured in utility. This has the consequence that, although it is often rational to be certain to some degree, there is no such thing as absolute certainty. (shrink)

An aspect of Peirce’s thought that may still be underappreciated is his resistance to what Levi calls _pedigree epistemology_, to the idea that a central focus in epistemology should be the justification of current beliefs. Somewhat more widely appreciated is his rejection of the subjective view of probability. We argue that Peirce’s criticisms of subjectivism, to the extent they grant such a conception of probability is viable at all, revert back to pedigree epistemology. A thoroughgoing rejection of (...) pedigree in the context of probabilistic epistemology, however, _does_ challenge prominent subjectivist responses to the problem of the priors. (shrink)

The major competing statistical paradigms share a common remarkable but unremarked thread: in many of their inferential applications, different probability interpretations are combined. How this plays out in different theories of inference depends on the type of question asked. We distinguish four question types: confirmation, evidence, decision, and prediction. We show that Bayesian confirmation theory mixes what are intuitively “subjective” and “objective” interpretations of probability, whereas the likelihood-based account of evidence melds three conceptions of what constitutes an (...) “objective” probability. (shrink)

Many philosophers accept that, as well as having a right that others not harm us, we also have a right that others not subject us to a risk of harm. And yet, when we attempt to spell out precisely what this ‘right against risk imposition’ involves, we encounter a series of notorious puzzles. Existing attempts to deal with these puzzles have tended to focus on the nature of rights – but I propose an approach that focusses instead on the nature (...) of risk. The key move is to distinguish two different ways in which to conceptualise the risk that a given action presents – one of which is linked to the notion of probability and the other to the notion of normalcy. (shrink)

Although written in Japanese, 確率としての信念（Belief as Probability） pursues a good explanation of Ramsey's subjective theory of probability.

In this paper, we use PCR5 in order to fusion the information of two sources providing subjective probabilities of an event A to occur in the following form: chance that A occurs, indeterminate chance of occurrence of A, chance that A does not occur. -/- .

We find a simple counterfactual acceptable, it is argued, to the extent that (i) our probability of the consequent under the thought experiment of counterfactually supposing the antecedent is high, (ii) provided the latter is on-topic with respect to the former. Counterfactual supposition is represented by Lewisian imaging. Topicality, by an algebra of subject matters. A topic-sensitive probabilistic logic is then provided, to reason about the acceptability of simple counterfactuals.

Simulation theory, the modern incarnation of skeptic philosophy, suggests our reality may be the result of a detailed computer simulation. Various prominent researchers and intellectuals have deduced a probability that our reality is a computer simulation, including Nick Bostrom, David Kipping, Elon Musk, and Neil deGrasse Tyson. Herein we argue that no sensible probability can be quantified.

The notion of comparative probability defined in Bayesian subjectivist theory stems from an intuitive idea that, for a given pair of events, one event may be considered “more probable” than the other. Yet it is conceivable that there are cases where it is indeterminate as to which event is more probable, due to, e.g., lack of robust statistical information. We take that these cases involve indeterminate comparative probabilities. This paper provides a Savage-style decision-theoretic foundation for indeterminate comparative probabilities.

The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can (...) be difficult to apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. -/- A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’. (shrink)

IBE ('Inference to the best explanation' or abduction) is a popular and highly plausible theory of how we should judge the evidence for claims of past events based on present evidence. It has been notably developed and supported recently by Meyer following Lipton. I believe this theory is essentially correct. This paper supports IBE from a probability perspective, and argues that the retrodictive probabilities involved in such inferences should be analysed in terms of predictive probabilities and a priori (...) class='Hi'>probability ratios of initial events. The key point is to separate these two features. Disagreements over evidence can be traced to disagreements over either the a priori probability ratios or predictive conditional ratios. In many cases, in real science, judgements of the former are necessarily subjective. The principles of iterated evidence are also discussed. The Sceptic's position is criticised as ignoring iteration of evidence, and characteristically failing to adjust a priori probability ratios in response to empirical evidence. (shrink)

In this dissertation, I construct scientifically and practically adequate moral analogs of cognitive heuristics and biases. Cognitive heuristics are reasoning “shortcuts” that are efficient but flawed. Such flaws yield systematic judgment errors—i.e., cognitive biases. For example, the availability heuristic infers an event’s probability by seeing how easy it is to recall similar events. Since dramatic events, such as airplane crashes, are disproportionately easy to recall, this heuristic explains systematic overestimations of their probability (availability bias). The research program on (...) cognitive heuristics and biases (e.g., Daniel Kahneman’s work) has been scientifically successful and has yielded useful error-prevention techniques—i.e., cognitive debiasing. I attempt to apply this framework to moral reasoning to yield moral heuristics and biases. For instance, a moral bias of unjustified differences in the treatment of particular animal species might be partially explained by a moral heuristic that dubiously infers animals’ moral status from their aesthetic features. While the basis for identifying judgments as cognitive errors is often unassailable (e.g., per violating laws of logic), identifying moral errors seemingly requires appealing to moral truth, which, I argue, is problematic within science. Such appeals can be avoided by repackaging moral theories as mere “standards-of-interest” (a la non-normative metrics of purportedly right-making features/properties). However, standards-of-interest do not provide authority, which is needed for effective debiasing. Nevertheless, since each person deems their own subjective morality authoritative, subjective morality (qua standard-of-interest and not moral subjectivism) satisfies both scientific and practical concerns. As such, (idealized) subjective morality grounds a moral analog of cognitive biases—namely, subjective moral biases (e.g., committed anti-racists unconsciously discriminating). I also argue that "cognitive heuristic" is defined by its contrast with rationality. Consequently, heuristics explain biases, which are also so defined. However, such contrasting with rationality is causally irrelevant to cognition. This frustrates the presumed usefulness of the kind, heuristic, in causal explanation. As such, in the moral case, I jettison the role of causal explanation and tailor categories solely for contrastive explanation. As such, “moral heuristic” is replaced with "subjective moral fallacy," which is defined by its contrast with subjective morality and explains subjective moral biases. The resultant subjective moral biases and fallacies framework can undergird future empirical research. (shrink)

I argue that existing objectivist accounts of subjective reasons face systematic problems with cases involving probability and possibility. I then offer a diagnosis of why objectivists face these problems, and recommend that objectivists seek to provide indirect analyses of subjective reasons.

This note discusses three issues that Allen and Pardo believe to be especially problematic for a probabilistic interpretation of standards of proof: (1) the subjectivity of probability assignments; (2) the conjunction paradox; and (3) the non-comparative nature of probabilistic standards. I offer a reading of probabilistic standards that avoids these criticisms.

Probability plays a crucial role regarding the understanding of the relationship which exists between mathematics and physics. It will be the point of departure of this brief reflection concerning this subject, as well as about the placement of Poincaré’s thought in the scenario offered by some contemporary perspectives.

The idea of a common currency underlying our choice behaviour has played an important role in sciences of behaviour, from neurobiology to psychology and economics. However, while it has been mainly investigated in terms of values, with a common scale on which goods would be evaluated and compared, the question of a common scale for subjective probabilities and confidence in particular has received only little empirical investigation so far. The present study extends previous work addressing this question, by showing (...) that confidence can be compared across visual and auditory decisions, with the same precision as for the comparison of two trials within the same task. We discuss the possibility that confidence could serve as a common currency when describing our choices to ourselves and to others. others. (shrink)

*This work is no longer under development* Two major themes in the literature on indicative conditionals are that the content of indicative conditionals typically depends on what is known;1 that conditionals are intimately related to conditional probabilities.2 In possible world semantics for counterfactual conditionals, a standard assumption is that conditionals whose antecedents are metaphysically impossible are vacuously true.3 This aspect has recently been brought to the fore, and defended by Tim Williamson, who uses it in to characterize alethic necessity by (...) exploiting such equivalences as: A⇔¬A A. One might wish to postulate an analogous connection for indicative conditionals, with indicatives whose antecedents are epistemically impossible being vacuously true: and indeed, the modal account of indicative conditionals of Brian Weatherson has exactly this feature.4 This allows one to characterize an epistemic modal by the equivalence A⇔¬A→A. For simplicity, in what follows we write A as KA and think of it as expressing that subject S knows that A.5 The connection to probability has received much attention. Stalnaker suggested, as a way of articulating the ‘Ramsey Test’, the following very general schema for indicative conditionals relative to some probability function P: P = P 1For example, Nolan ; Weatherson ; Gillies. 2For example Stalnaker ; McGee ; Adams. 3Lewis. See Nolan for criticism. 4‘epistemically possible’ here means incompatible with what is known. 5This idea was suggested to me in conversation by John Hawthorne. I do not know of it being explored in print. The plausibility of this characterization will depend on the exact sense of ‘epistemically possible’ in play—if it is compatibility with what a single subject knows, then can be read ‘the relevant subject knows that p’. If it is more delicately formulated, we might be able to read as the epistemic modal ‘must’. (shrink)

Bayesians often appeal to “merging of opinions” to rebut charges of excessive subjectivity. But what happens in the short run is often of greater interest than what happens in the limit. Seidenfeld and coauthors use this observation as motivation for investigating the counterintuitive short run phenomenon of dilation, since, they allege, dilation is “the opposite” of asymptotic merging of opinions. The measure of uncertainty relevant for dilation, however, is not the one relevant for merging of opinions. We explicitly investigate the (...) short run behavior of the metric relevant for merging, and show that dilation is independent of the opposite of merging. (shrink)

This study aimed to assess the retention of grade 12 students in statistics and probability, along with a comparative analysis of these retentions across the distinct topics of the subject. Statistics and probability subjects were taken by the students when they were in grade 11. Employing a quantitative approach, the research used descriptive-comparative design to describe the level of retention of the students and compare the retention of the students in each topic. These encompass random variables and (...) class='Hi'>probability distribution, normal distribution, sampling and sampling distribution, and estimation parameters. The retention of the students studying at Nueva Vizcaya General Comprehensive High School, a public school in Nueva Vizcaya, Philippines was analyzed through the validated retention test administered to them. Using frequency count and percent, the result revealed that the grade 12 students’ retention was described as “poor retention rate”. Repeated-measures ANOVA confirmed that the retention of the students in all the topics are all poor. Hence, teachers should consider making an intervention to improve the retention of the students in statistics and probability. They may consider innovative ways to engage the students in learning. Also, the factors that influence the students in solving problems in designing the interventions. (shrink)

Objective: In this essay, I will try to track some historical and modern stages of the discussion on the Gettier problem, and point out the interrelations of the questions that this problem raises for epistemologists, with sceptical arguments, and a so-called problem of relevance. Methods: historical analysis, induction, generalization, deduction, discourse, intuition results: Albeit the contextual theories of knowledge, the use of different definitions of knowledge, and the different ways of the uses of knowledge do not resolve all the issues (...) that the sceptic can put forward, but they can be productive in giving clarity to a concept of knowledge for us. On the other hand, our knowledge will always have an element of intuition and subjectivity, however not equating to epistemic luck and probability. Significance novelty: the approach to the context in general, not giving up being a Subject may give us a clarity about the sense of what it means to say – “I know”. (shrink)

Unlike other classical arguments for the existence of God, Pascal’s Wager provides a pragmatic rationale for theistic belief. Its most popular version says that it is rationally mandatory to choose a way of life that seeks to cultivate belief in God because this is the option of maximum expected utility. Despite its initial attractiveness, this long-standing argument has been subject to various criticisms by many philosophers. What is less discussed, however, is the rationality of this choice in situations where the (...) decision-makers are confronted with greater uncertainty. In this paper, I examine the imprecise version of Pascal’s Wager: those scenarios where an agent’s credence that God exists is imprecise or vague rather than precise. After introducing some technical background on imprecise probabilities, I apply five different principles for decision-making to two cases of state uncertainty. In the final part of the paper, I argue that it is not rationally permitted to include zero as the lower probability of God’s existence. Although the conditions for what makes an act uniquely optimal vary significantly across those principles, I also show how the option of wagering for God can defeat any mixed strategy under two distinct interpretations of salvation. (shrink)

Contains a description of a generalized and constructive formal model for the processes of subjective and creative thinking. According to the author, the algorithm presented in the article is capable of real and arbitrarily complex thinking and is potentially able to report on the presence of consciousness.

A complete probability guide of Hold'em Poker, this guide covers all possible gaming situations. The author focuses on the practical side of the presentation and use of the probabilities involved in Hold'em, while taking into account the subjective side of the probability-based criteria of each player's strategy.

We introduce a ranking of multidimensional alternatives, including uncertain prospects as a particular case, when these objects can be given a matrix form. This ranking is separable in terms of rows and columns, and continuous and monotonic in the basic quantities. Owing to the theory of additive separability developed here, we derive very precise numerical representations over a large class of domains (i.e., typically notof the Cartesian product form). We apply these representationsto (1)streams of commodity baskets through time, (2)uncertain social (...) prospects, (3)uncertain individual prospects. Concerning(1), we propose a finite horizon variant of Koopmans’s (1960) axiomatization of infinite discounted utility sums. The main results concern(2). We push the classic comparison between the exanteand expostsocial welfare criteria one step further by avoiding any expected utility assumptions, and as a consequence obtain what appears to be the strongest existing form of Harsanyi’s (1955) Aggregation Theorem. Concerning(3), we derive a subjective probability for Anscombe and Aumann’s (1963) finite case by merely assuming that there are two epistemically independent sources of uncertainty. (shrink)

Following a long-standing philosophical tradition, impartiality is a distinctive and determining feature of moral judgments, especially in matters of distributive justice. This broad ethical tradition was revived in welfare economics by Vickrey, and above all, Harsanyi, under the form of the so-called Impartial Observer Theorem. The paper offers an analytical reconstruction of this argument and a step-wise philosophical critique of its premisses. It eventually provides a new formal version of the theorem based on subjective probability.

The typical habitat of overt nominative subjects is in finite clauses. But infinitival complements and infinitival adjuncts are also known to have overt nominative subjects, e.g. in Italian (Rizzi 1982), European Portuguese (Raposo 1987), and Spanish (Torrego 1998, Mensching 2000). The analyses make crucial reference to the movement of Aux or Infl to Comp, and to overt or covert infinitival inflection. This working paper is concerned with a novel set of data that appear to be of a different sort, in (...) that they probably do not depend on either rich infinitival inflection or on movement to C. (shrink)

Ramsey (1926) sketches a proposal for measuring the subjective probabilities of an agent by their observable preferences, assuming that the agent is an expected utility maximizer. I show how to extend the spirit of Ramsey's method to a strictly wider class of agents: risk-weighted expected utility maximizers (Buchak 2013). In particular, I show how we can measure the risk attitudes of an agent by their observable preferences, assuming that the agent is a risk-weighted expected utility maximizer. Further, we can (...) leverage this method to measure the subjective probabilities of a risk-weighted expected utility maximizer. (shrink)

My dissertation is about Bayesian rationality for non-ideal agents. I show how to derive subjective probabilities from preferences using much weaker rationality assumptions than other standard representation theorems. I argue that non-ideal agents might be uncertain about how they will update on new information and consider two consequences of this uncertainty: such agents should sometimes reject free information and make choices which, taken together, yield sure loss. The upshot is that Bayesian rationality for non-ideal agents makes very different normative (...) demands than ideal Bayesian rationality. (shrink)

The difficulty of the students in Statistics and Probability subject, and the pedagogical approaches used by the teachers, were the challenges encountered by both students and teachers due to the restrictions during the CoViD-19 pandemic. Hence, this study aimed to determine the pedagogical approaches used in teaching statistics and probability during the pandemic. The study used a qualitative approach, particularly document analysis. The main source of the data was the module in statistics and probability specifically the learning (...) activity sheets in the third quarter because this is the most convenient approach that can be used by the teachers during the pandemic. The pedagogical approaches were identified and validated by three inter-raters. Based on the findings, communication, and study skills, technology-aided instruction, problem-based learning, manipulatives, modules, models and multiple representations, and direct instruction were the identified pedagogical approaches used in the statistics and probability module. However, the effectiveness of the delivery of these approaches cannot be measured because the modality was limited to a modular approach. The teachers were able to use different pedagogical approaches, however, cooperative learning was not used because the CoViD-19pandemic restricted interactions between the teacher and students or students with their classmates. (shrink)

DNA evidence is one of the most significant modern advances in the search for truth since the cross examination, but its format as a random-match-probability makes it difficult for people to assign an appropriate probative value (Koehler, 2001). While Frequentist theories propose that the presentation of the match as a frequency rather than a probability facilitates more accurate assessment (e.g., Slovic et al., 2000), Exemplar-Cueing Theory predicts that the subjective weight assigned may be affected by the frequency (...) or probability format, and how easily examples of the event, i.e., ‘exemplars’, are generated from linguistic cues that frame the match in light of further evidence (Koehler & Macchi, 2004). This paper presents two juror research studies to examine the difficulties that jurors have in assigning appropriate probative value to DNA evidence when contradictory evidence is presented. Study 1 showed that refuting evidence significantly reduced guilt judgments when exemplars were linguistically cued, even when the probability match and the refuting evidence had the same objective probative value. Moreover, qualitative reason for judgment responses revealed that interpreting refuting evidence was found to be complex and not necessarily reductive; refutation was found indicative of innocence or guilt depending on whether exemplars have been cued or not. Study 2 showed that the introduction of judges’ directions to linguistically cue exemplars, did not increase the impact of refuting evidence beyond its objective probative value, but less guilty verdicts were returned when jurors were instructed to consider all possible explanations of the evidence. The results are discussed in light of contradictory frequentist and exemplar-cueing theoretical positions, and their real-world consequences. (shrink)

Consciousness vehicle externalism is the claim that the material machinery of a subject’s phenomenology partially leaks outside a subject’s brain, encompassing bodily and environmental structures. The DEUTS argument is the most prominent argument for CVE in the sensorimotor enactivists’ arsenal. In a recent series of publications, Kirchhoff and Kiverstein have deployed such an argument to claim that a prominent view of neural processing, namely predictive processing, is fully compatible with CVE. Indeed, in Kirchhoff and Kiverstein’s view, a proper understanding of (...) predictive processing mandates CVE. In this essay, we critically examine Kirchhoff and Kiverstein’s argument. Our aim is to argue in favor of the following three points. First, that Kirchhoff and Kiverstein’s emphasis on cultural practices lends no support to CVE: at best, it vindicates some form of content externalism about phenomenal content. Secondly, the criteria Kirchhoff and Kiverstein propose to identify a subject’s phenomenal machinery greatly overgeneralize, leaving them open to a “consciousness bloat” objection, which is an analog of the cognitive bloat objection against the extended mind. Lastly, we will argue that the “consciousness bloat” problem is inbuilt in the very argumentative structure of the DEUTS argument. We will thus conclude that, contrary to the philosophical mainstream, DEUTS is not the best argument for CVE in the sensorimotor enactivists’ argumentative arsenal. (shrink)

Is it not surprising that we look with so much pleasure and emotion at works of art that were made thousands of years ago? Works depicting people we do not know, people whose backgrounds are usually a mystery to us, who lived in a very different society and time and who, moreover, have been ‘frozen’ by the artist in a very deliberate pose. It was the Classical Greek philosopher Aristotle who observed in his Poetics that people could apparently be moved (...) even by the imitation of a person or an act. And although we are usually well aware that it is a simulacrum, not a real situation, it nevertheless sometimes seems as if we ourselves are standing there on the stage or in the painting, so intense and emotional is our response, even though we are just spectators. Aristotle concludes from this that we have intellectual capacities which allow us put ourselves in another’s place and consequently to react to simulated situations as though they are actually happening to us, here and now. In this process, he contends, observation, memory, imagination and emotions are crucial elements. In the past it was not customary to invoke human mental faculties to explain our response to works of art. The Ancient Greeks, after all, knew little about the human body or brain and usually referred to the extended world of the gods in their endeavours to comprehend the ‘inner world’ of human beings. In our time the situation is completely different—such an allusion to the brain no longer surprises us. Whether it is about the mystery of the consciousness, the question of free will or accounts of bizarre psychological aberrations or disorders, we have become accustomed to references to parts of the brain, to images of brain scans, to reports about neural networks and the like. However, because there are so many factors that play a part in our appreciation of works of art we need a complex explanation for it, and it is not enough to look only at certain properties of the brain that are determined by evolution. Those properties are shared by every human being, and so are rather useless in explaining people’s different reactions to the same work of art. Evidently the brains of individuals differ so much that they make it possible for people to respond differently to one and the same artwork. This, of course, raises questions concerning the painted emotions that can be seen in this exhibition. Virtually everyone, after all, is fascinated by such paintings and usually recognizes the emotions they represent. The reactions to these painted emotions are also often similar. This is probably why artworks like this are generally highly valued, then and now, here and elsewhere: from the enigmatically smiling Egyptian Queen Nefertiti and the startled Rembrandt to a seemingly despairing African mask. Aristotle observed that in the theatre players imitate actions that are associated with emotions in a number of ways and that these emotions are shared in a particular fashion by the playwright, the actors and the audience. The audience may even be carried away by these emotions to such an extent that they are in a sense purged of them and can subsequently leave the theatre relieved. Are such emotional reactions perhaps related to the fact that emotions are universal and that brains respond similarly to them? Is this why we can so readily identify painted emotions? May we therefore also assume that the properties of the brain determined by evolution help us to explain these emotions? In answering these questions we shall discuss a number of insights into emotions in psychology and brain science and explore some theories about the possible function of emotions and their expression. (shrink)

In this paper, we provide a Bayesian analysis of the well-known surprise exam paradox. Central to our analysis is a probabilistic account of what it means for the student to accept the teacher's announcement that he will receive a surprise exam. According to this account, the student can be said to have accepted the teacher's announcement provided he adopts a subjective probability distribution relative to which he expects to receive the exam on a day on which he expects (...) not to receive it. We show that as long as expectation is not equated with subjective certainty there will be contexts in which it is possible for the student to accept the teacher's announcement, in this sense. In addition, we show how a Bayesian modeling of the scenario can yield plausible explanations of the following three intuitive claims: (1) the teacher's announcement becomes easier to accept the more days there are in class; (2) a strict interpretation of the teacher's announcement does not provide the student with any categorical information as to the date of the exam; and (3) the teacher's announcement contains less information about the date of the exam the more days there are in class. To conclude, we show how the surprise exam paradox can be seen as one among the larger class of paradoxes of doxastic fallibilism, foremost among which is the paradox of the preface. (shrink)

We investigate the conflict between the ex ante and ex post criteria of social welfare in a new framework of individual and social decisions, which distinguishes between two sources of uncertainty, here interpreted as an objective and a subjective source respectively. This framework makes it possible to endow the individuals and society not only with ex ante and ex post preferences, as is usually done, but also with interim preferences of two kinds, and correspondingly, to introduce interim forms of (...) the Pareto principle. After characterizing the ex ante and ex post criteria, we present a first solution to their conflict that extends the former as much possible in the direction of the latter. Then, we present a second solution, which goes in the opposite direction, and is also maximally assertive. Both solutions translate the assumed Pareto conditions into weighted additive utility representations, and both attribute to the individuals common probability values on the objective source of uncertainty, and different probability values on the subjective source. We discuss these solutions in terms of two conceptual arguments, i.e., the by now classic spurious unanimity argument and a novel informational argument labelled complementary ignorance. The paper complies with the standard economic methodology of basing probability and utility representations on preference axioms, but for the sake of completeness, also considers a construal of objective uncertainty based on the assumption of an exogeneously given probability measure. JEL classification: D70; D81. (shrink)

My purpose in this paper is to argue for two separate, but related theses. The first is that contemporary analytic philosophy is incoherent. This is so, I argue, because its methods contain as an essential constituent a conception of intuition that cannot be rendered consistent with a key tenet of analytic philosophy unless we allow a Bayesian-subjectivist epistemology. I argue for this within a discussion of two theories of intuition: a classical account as proposed by Descartes and a modern reliabilist (...) account as proposed by Kornblith, maintaining that reliabilist accounts require a commitment to Bayesian subjectivism about probability. However, and this is the second thesis, Bayesian subjectivism is itself logically incoherent given two simple assumptions: (1) that some empirical propositions are known and (2) any proposition that is known is assigned a degree of subjective credence of 1. I establish this using a formal reductio proof. I argue for the first thesis in section 2 and for the second in section 3. The final section contains a summary and conclusion. (shrink)

This paper explains and defends the idea that metaphysical necessity is the strongest kind of objective necessity. Plausible closure conditions on the family of objective modalities are shown to entail that the logic of metaphysical necessity is S5. Evidence is provided that some objective modalities are studied in the natural sciences. In particular, the modal assumptions implicit in physical applications of dynamical systems theory are made explicit by using such systems to define models of a modal temporal logic. Those assumptions (...) arguably include some necessitist principles. -/- Too often, philosophers have discussed ‘metaphysical’ modality — possibility, contingency, necessity — in isolation. Yet metaphysical modality is just a special case of a broad range of modalities, which we may call ‘objective’ by contrast with epistemic and doxastic modalities, and indeed deontic and teleological ones (compare the distinction between objective probabilities and epistemic or subjective probabilities). Thus metaphysical possibility, physical possibility and immediate practical possibility are all types of objective possibility. We should study the metaphysics and epistemology of metaphysical modality as part of a broader study of the metaphysics and epistemology of the objective modalities, on pain of radical misunderstanding. Since objective modalities are in general open to, and receive, natural scientific investigation, we should not treat the metaphysics and epistemology of metaphysical modality in isolation from the metaphysics and epistemology of the natural sciences. -/- In what follows, Section 1 gives a preliminary sketch of metaphysical modality and its place in the general category of objective modality. Section 2 reviews some familiar forms of scepticism about metaphysical modality in that light. Later sections explore a few of the many ways in which natural science deals with questions of objective modality, including questions of quantified modal logic. (shrink)

Savage's framework of subjective preference among acts provides a paradigmatic derivation of rational subjective probabilities within a more general theory of rational decisions. The system is based on a set of possible states of the world, and on acts, which are functions that assign to each state a consequence€. The representation theorem states that the given preference between acts is determined by their expected utilities, based on uniquely determined probabilities (assigned to sets of states), and numeric utilities assigned (...) to consequences. Savage's derivation, however, is based on a highly problematic well-known assumption not included among his postulates: for any consequence of an act in some state, there is a "constant act" which has that consequence in all states. This ability to transfer consequences from state to state is, in many cases, miraculous -- including simple scenarios suggested by Savage as natural cases for applying his theory. We propose a simplification of the system, which yields the representation theorem without the constant act assumption. We need only postulates P1-P6. This is done at the cost of reducing the set of acts included in the setup. The reduction excludes certain theoretical infinitary scenarios, but includes the scenarios that should be handled by a system that models human decisions. (shrink)