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

Many approaches to quantum gravity -the theory that should account for quantum and gravitational phenomena under the same theoretical umbrella- seem to point at some form of spacetime emergence, i.e., the fact that spacetime is not a fundamental entity of our physical world. This tenet has sparked many philosophical discussions: from the so-called empirical incoherence problem to different accounts of emergence and mechanisms thereof. In this contribution, I focus on the partial order relation of causal set theory and argue that (...) causation can be characterized as an a-temporal constraint over the kinematic space defined by the theory. The relation constrains the growth of a new element/event with respect to the other elements/events of a given set. Therefrom, the flow of time emerges from the collection of the possible growths of the given set, where each possibility is characterized by a classical probability assigned by the dynamics of the theory. (shrink)

In the quest and search for a physical theory of everything from the macroscopic large body matter to the microscopic elementary particles, with strange and weird concepts springing from quantum physics discovery, irreconcilable positions and inconvenient facts complicated physics – from Newtonian physics to quantum science, the question is- how do we close the gap? Indeed, there is a scientific and mathematical fireworks when the issue of quantum uncertainties and entanglements cannot be explained with classical physics. The Copenhagen interpretation is (...) an expression of few wise men on quantum physics that was largely formulated from 1925 to 1927 namely by Niels Bohr and Werner Heisenberg. From this point on, there is a divergence of quantum science into the realms of indeterminacy, complementarity and entanglement which are principles expounded in Yijing, an ancient Chinese knowledge constructed on symbols, with a vintage of at least 3 millennia, with broken and unbroken lines to form stacked 6-line structure called the hexagram. It is premised on probability development of the hexagram in a space-time continuum. The discovery of the quantization of action meant that quantum physics could not convincingly explain the principles of classical physics. This paper will draw the great departure from classical physics into the realm of probabilistic realities. The probabilistic nature and reality interpretation had a significant influence on Bohr’s line of thought. Apparently, Bohr realized that speaking of disturbance seemed to indicate that atomic objects were classical particles with definite inherent kinematic and dynamic properties (Hanson, 1959). Disturbances, energy excitation and entanglements are processual evolutionary phases in Yijing. This paper will explore the similarities in quantum physics and the methodological ways where Yijing is pivoted to interpret observable realities involving interactions which are uncontrollable and probabilistic and forms an inseparable unity due to the entanglement, superposition Transgressing disciplinary boundaries in the discussion of Yijing, originally from the Western Zhou period (1000-750 BC), over a period of warring states and the early imperial period (500-200 BC) which was compiled, transcribed and transformed into a cosmological texts with philosophical commentaries known as the “Ten Wings” and closely associated with Confucius (551- 479 BC) with the Copenhagen Interpretation (1925-1927) by the few wise men including Niel Bohr and Werner Heisensberg would seem like a subversive undertaking. Subversive as the interpretations from Yijing is based on wisdom derived from thousands of years from ancient China to recently discovered quantum concepts. The subversive undertaking does seem to violate the sanctuaries of accepted ways in looking at Yijing principles, classical physics and quantum science because of the fortified boundaries that have been erected between Yijing and the sciences. Subversive as this paper may be, it is an attempt to re-cast an ancient framework where indeterminism, complementarity, non-linearity entanglement, superposition and probability interpretation is seen in today quantum’s realities. (shrink)

Introduction Conatus is a specific concept within Descartes’s physics. In particular, it assumes a crucial importance in the purely mechanistic description of the nature of light – an issue that Des- cartes considered one of the most crucial challenges, and major achievements, of his natural phil- osophy. According to Descartes’s cosmology, the universe – understood as a material continuum in which there is no vacuum – is composed of a number of separate yet interconnected vortices. Each of these vortices consists (...) in a set of bands rotating around their centres. The bands are com- posed of corpuscles of the three elements, each distinguished on the basis of their different shapes and sizes. The small globules of the second element, although impeded by the other parts of heaven, strive to move away from the centre of the vortex around which they revolve, thus exerting a certain force against the surrounding bodies. This striving or conatus, though a mere force rather than a genuine motion, is transmitted instan- taneously and along straight lines from body to body. According to Descartes, then, the nature of light consists in this striving alone. This account must be understood as strictly connected to the fundamental laws that regulate the Cartesian world. It is particularly important to recall that for Descartes the centrifugal force exerted by a rotating body is understood to be a consequence of the intrinsic rectilinearity of every motion. As the second of the three Laws of Nature set forth in Book II of the Principles of Philosophy establishes, “all motion is in itself rectilinear; and hence any body moving in a circle tends to move away from the centre of the circle which it describes as proved by the fact that a stone rotating in a sling tends to move centrifugally along the tangent of each point described by a circle.1 Similarly, each of the globules of the second element “strives to recede with a great force from the centre of the vortex in which it rotates; it is in fact prevented by the other globules placed all around, not differently than a stone in a sling”. Within this framework, as I show later, conatus occurs when a body’s intrinsic tendency to rectilinear motion is impeded by an external constraint. As Stephen Gaukroger notes, the use of the term conatus in the Principles – like “vis” and “ action ” – shows that Descartes “cannot avoid dynamic terminology”, despite his declared intent “to construe motion in a purely kinematic way”. Indeed, Gaukroger observes that the notions of force action and striving are systematically employed in the Principles – appearing 290, 59 and 8 times, respectively. Also noteworthy, conatus was – together with the more familiar concepts of actio and vis – a significant element of the conceptual apparatus of Scholastic natural philosophy. As will be seen later – and this is the first goal of this paper – here the concept of conatus had a very specific function. Indeed, it was a central part of the Aristotelian- Scholastic account of gravitation. Conatus was used to refer to the striving of a body to move towards its natural place – especially in cases where its natural motion was hindered or impeded by an external mover. This specific use of the concept of conatus can be found in texts on natural philosophy from the end of the sixteenth through the late seventeenth century. Notably, it occurs in some of the late -Scho- lastic texts which held special significance for Descartes’s thought. Therefore, Descartes introduces conatus in a context in which this concept already had a very specific meaning, and one of which he was very probably aware. The possible relations between these two apparently very different con- cepts have therefore to be scrutinized. I thereby propose to undertake a comparison between the Scho- lastic and the Cartesian conceptions of conatus. I hope to show that certain traits of the former are indeed echoed by the latter, although adapted to the much changed physical paradigm of Cartesian physics. In fact, Descartes’s conatus, though sharing some important similarities with the old one, underwent significant transformations in terms of both its meaning and its application. However, I hope to show that these concepts are both used to describe the behaviour of bodies in relation to their intrinsic motive tendency, and in particular when this tendency is impeded or prevented. Such a reconstruction offers two main areas of interest. On an immediate level, it furthers the general efforts of scholars over the past decades to reconstruct the full extent of the relations between Descartes’s thought and Scholastic philosophy. However, there is another significant reason for taking an interest in the ties between the Scholastic and the Cartesian conceptions of conatus. Indeed, the Cartesian conatus must be seen as crucial to a proper understanding of the broader, diverse conceptualization that this idea enjoyed in the thought of many of the most important philosophers of the early modern period, namely Spinoza, Hobbes and Leibniz. In particular, Spinoza, were directly influenced by Descartes in their unique formulations of conatus. Thus a reconstruction of the context in which Descartes formulated his conatus and of its relations to a pre-existing Scholastic conception will enable us to better reconstruct the history of this idea in the early modern period. The paper is structured as follows: in the first part, I provide a thorough review of the Scho- lastic employment of the concept conatus – both its meaning and the extent of its usage. I then show that it occurs in some of the most prominent Cartesian sources. Finally, I shall provide an account of Descartes’s theory of circular motion and conatus with the specific aim of empha- sizing the elements of both continuity and discontinuity that justify the claim that Descartes’s con- ception of conatus is reminiscent of the Scholastic one. (shrink)

Contextualism is the view that knowledge ascriptions – utterances of sentences containing the word “knows” - express different propositions in different contexts of utterance. But what features of context determine the propositions expressed by knowledge ascriptions? According to a version of contextualism I call conversational contextualism, the conversational dynamics or kinematics determine the propositions expressed by knowledge ascriptions. In this paper I argue that the most sophisticated version of conversational contextualism, which is the view defended by Michael Blome-Tillmann (2009; (...) 2014), faces serious problems. My (very) tentative conclusion will be that conversational contextualism should be abandoned because of these problems. (shrink)

In response to cases that involve tiny probabilities of huge payoffs, some argue that we ought to discount small probabilities down to zero. However, this paper shows that doing so violates Independence and Continuity, and as a result of these violations, those who discount small probabilities can be exploited by money pumps. Various possible ways of avoiding exploitation will be discussed. This paper concludes that the money pump for Independence undermines the plausibility of discounting small probabilities. Much of the discussion (...) on Independence generalizes to other views that also violate Independence. (shrink)

The work is an attempt to transfer a structure from Euclidean plane (pure geometrical) under the physical observation limit (resolving power) to a physical space (observable space). The transformation from the mathematical space to physical space passes through the observation condition. The mathematical modelling is adopted. The project is based on two stapes: (1) Looking for a simple mathematical model satisfies the definition of Euclidian plane; (2)That model is examined against three observation resolution conditions (resolved, unresolved and partially resolved). The (...) simplest mechanical model satisfies the definition of Euclidian plane is a planetary gear. The interesting examination of the mechanical model is that is under partial resolution. That examination shows analogous equation for Euler’s formula. The derived complex formula contains the resolved (observable) quantities of the mechanical system and satisfies the linear wave equation. The interpretation of this complex formula is: it is a function related to the position vector of a point in the small wheel of the partially resolved planetary gear system. The function is in terms of the observable quantities only. The work shows the possibility of transformation from real to complex space. The work is purely classical but the result of the partial resolution shows a function similar to the Quantum mechanics wave function. (shrink)

This book explores a question central to philosophy--namely, what does it take for a belief to be justified or rational? According to a widespread view, whether one has justification for believing a proposition is determined by how probable that proposition is, given one's evidence. In this book this view is rejected and replaced with another: in order for one to have justification for believing a proposition, one's evidence must normically support it--roughly, one's evidence must make the falsity of that proposition (...) abnormal in the sense of calling for special, independent explanation. This conception of justification bears upon a range of topics in epistemology and beyond. Ultimately, this way of looking at justification guides us to a new, unfamiliar picture of how we should respond to our evidence and manage our own fallibility. This picture is developed here. (shrink)

This paper develops an information-sensitive theory of the semantics and probability of conditionals and statements involving epistemic modals. The theory validates a number of principles linking probability and modality, including the principle that the probability of a conditional If A, then C equals the probability of C, updated with A. The theory avoids so-called triviality results, which are standardly taken to show that principles of this sort cannot be validated. To achieve this, we deny that rational (...) agents update their credences via conditionalization. We offer a new rule of update, Hyperconditionalization, which agrees with Conditionalization whenever nonmodal statements are at stake but differs for modal and conditional sentences. (shrink)

In a recent paper, Tuomas Tahko has argued for a hybrid view of the laws of nature, according to which some physical laws are metaphysically necessary, while others are metaphysically contingent. In this paper, we show that his criterion for distinguishing between these two kinds of laws — which crucially relies on the essences of natural kinds — is on its own unsatisfactory. We then propose an alternative way of drawing the metaphysically necessary/contingent distinction for laws of physics based on (...) the central kinematical/dynamical distinction used in physical theorising, and argue that the criterion can be used to amend Tahko’s own account, but also that it can be combined with different metaphysical views about the source of necessity. (shrink)

I conjecture that the probability amplitudes of a quantum state are fractions of the Planck frequency, stemming from the rich dynamics at the Planck scale. This offers a means to indirectly measure the fundamental properties of quantum spacetime and potentially resolves the measurement problem.

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)

Abstract. In this article is presented a model of the change of the probability of the global catastrophic risks in the world with exponentially evolving technologies. Increasingly cheaper technologies become accessible to a larger number of agents. Also, the technologies become more capable to cause a global catastrophe. Examples of such dangerous technologies are artificial viruses constructed by the means of synthetic biology, non-aligned AI and, to less extent, nanotech and nuclear proliferation. The model shows at least double exponential (...) growth of the probability of the global catastrophe which means that the accumulated probability of the catastrophe will grow from negligible to overwhelming at the period of a few doubling times of the technological capabilities. For biotech and AI, such doubling time roughly corresponds to the doubling period of Moore’s law and its analogues in other technologies and is around 2 years. Thus the global catastrophe in the exponential technologies world will most likely happen during a “dangerous decade”. Such a dangerous decade could start as early as in the 2020s. We also found that the double exponential growth in the model makes the model less sensitive to its initial conditions, like the number of dangerous agents or initial probabilities constants. The model also shows that smaller catastrophes could happen earlier than larger ones, and such a smaller catastrophe may be able to stop the technological growth before larger ones become possible, thus global risks will be self-limiting. However, if the growth of the number of dangerous agents will be very quick, the multiple smaller catastrophes could happen simultaneously and will be equal to a global catastrophe. (shrink)

I argue that when we use ‘probability’ language in epistemic contexts—e.g., when we ask how probable some hypothesis is, given the evidence available to us—we are talking about degrees of support, rather than degrees of belief. The epistemic probability of A given B is the mind-independent degree to which B supports A, not the degree to which someone with B as their evidence believes A, or the degree to which someone would or should believe A if they had (...) B as their evidence. My central argument is that the degree-of-support interpretation lets us better model good reasoning in certain cases involving old evidence. Degree-of-belief interpretations make the wrong predictions not only about whether old evidence confirms new hypotheses, but about the values of the probabilities that enter into Bayes’ Theorem when we calculate the probability of hypotheses conditional on old evidence and new background information. (shrink)

Many philosophers argue that Keynes’s concept of the “weight of arguments” is an important aspect of argument appraisal. The weight of an argument is the quantity of relevant evidence cited in the premises. However, this dimension of argumentation does not have a received method for formalisation. Kyburg has suggested a measure of weight that uses the degree of imprecision in his system of “Evidential Probability” to quantify weight. I develop and defend this approach to measuring weight. I illustrate the (...) usefulness of this measure by employing it to develop an answer to Popper’s Paradox of Ideal Evidence. (shrink)

This chapter will review selected aspects of the terrain of discussions about probabilities in statistical mechanics (with no pretensions to exhaustiveness, though the major issues will be touched upon), and will argue for a number of claims. None of the claims to be defended is entirely original, but all deserve emphasis. The first, and least controversial, is that probabilistic notions are needed to make sense of statistical mechanics. The reason for this is the same reason that convinced Maxwell, Gibbs, and (...) Boltzmann that probabilities would be needed, namely, that the second law of thermodynamics, which in its original formulation says that certain processes are impossible, must, on the kinetic theory, be replaced by a weaker formulation according to which what the original version deems impossible is merely improbable. Second is that we ought not take the standard measures invoked in equilibrium statistical mechanics as giving, in any sense, the correct probabilities about microstates of the system. We can settle for a much weaker claim: that the probabilities for outcomes of experiments yielded by the standard distributions are effectively the same as those yielded by any distribution that we should take as a representing probabilities over microstates. Lastly, (and most controversially): in asking about the status of probabilities in statistical mechanics, the familiar dichotomy between epistemic probabilities (credences, or degrees of belief) and ontic (physical) probabilities is insufficient; the concept of probability that is best suited to the needs of statistical mechanics is one that combines epistemic and physical considerations. (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)

Many have argued that a rational agent's attitude towards a proposition may be better represented by a probability range than by a single number. I show that in such cases an agent will have unstable betting behaviour, and so will behave in an unpredictable way. I use this point to argue against a range of responses to the ‘two bets’ argument for sharp probabilities.

The main objective of the suspension of a vehicle is to maximize the contact between the vehicle tires and the road surface, provide steering stability and provide safe vehicle control in all conditions, evenly support the weight of the vehicle, transfer the loads to springs, and guaranteeing the comfort of the driver by absorbing and dampening shock. This paper discusses the kinematic design of a double a-arm Suspension system for ABAY Vehicle. The hardpoint's location can be determined using this procedure (...) to simulate motion in any kinematic simulation software. Here, Optimum Kinematics is used as kinematic simulation software. The method illustrated deals with the basics of Kinematics which helps to predict the characteristics of the suspension even before simulating it in the kinematic simulation software. (shrink)

Conditional probability is often used to represent the probability of the conditional. However, triviality results suggest that the thesis that the probability of the conditional always equals conditional probability leads to untenable conclusions. In this paper, I offer an interpretation of this thesis in a possible worlds framework, arguing that the triviality results make assumptions at odds with the use of conditional probability. I argue that these assumptions come from a theory called the operator theory (...) and that the rival restrictor theory can avoid these problematic assumptions. In doing so, I argue that recent extensions of the triviality arguments to restrictor conditionals fail, making assumptions which are only justified on the operator theory. (shrink)

We will flip a fair coin infinitely many times. Al calls the first flip, claiming it will land heads. Betty calls every odd numbered flip, claiming they will all land heads. Carl calls every flip bar none, claiming they will all land heads. Pre-theoretically, it seems that Al's claim is infinitely more likely than Betty's, and that Betty's claim is infinitely more likely than Carl's. But standard, real-valued probability theory says that, while Al's claim is infinitely more likely than (...) Betty's, Betty's claim is exactly as likely as Carl's. I use John Conway's surreal numbers to capture these pre-theoretic judgements about the relative probabilities of Al's, Betty's, and Carl's claims. Surreal-valued probabilities also allow us to calculate precise numerical values for those claims. For instance, we will see that Betty's claim has a probability equal to the reciprocal of the product of the square root of cardinality of the continuum and the fourth root of 2. The article provides a 'user's guide' to surreal-valued probabilities and raises questions for further research. (shrink)

In De gravitatione, Newton contends that Descartes' physics is fundamentally untenable since the "fixed" spatial landmarks required to ground the concept of inertial motion cannot be secured in the constantly changing Cartesian plenum. Likewise, it is has often been alleged that the collision rules in Descartes' Principles of Philosophy undermine the "relational" view of space and motion advanced in this text. This paper attempts to meet these challenges by investigating the theory of connected gears (or "kinematics of mechanisms") for (...) a potential Cartesian method of positing the permanent reference frames necessary to uphold Descartes' conservation law for the quantity of motion. In particular, the insights gained from an examination of the kinematics of mechanisms will provide the Cartesian with much needed resources to counter the two threats posed above. (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)

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)

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)

Enjoying great popularity in decision theory, epistemology, and philosophy of science, Bayesianism as understood here is fundamentally concerned with epistemically ideal rationality. It assumes a tight connection between evidential probability and ideally rational credence, and usually interprets evidential probability in terms of such credence. Timothy Williamson challenges Bayesianism by arguing that evidential probabilities cannot be adequately interpreted as the credences of an ideal agent. From this and his assumption that evidential probabilities cannot be interpreted as the actual credences (...) of human agents either, he concludes that no interpretation of evidential probabilities in terms of credence is adequate. I argue to the contrary. My overarching aim is to show on behalf of Bayesians how one can still interpret evidential probabilities in terms of ideally rational credence and how one can maintain a tight connection between evidential probabilities and ideally rational credence even if the former cannot be interpreted in terms of the latter. By achieving this aim I illuminate the limits and prospects of Bayesianism. (shrink)

John Maynard Keynes’s A Treatise on Probability is the seminal text for the logical interpretation of probability. According to his analysis, probabilities are evidential relations between a hypothesis and some evidence, just like the relations of deductive logic. While some philosophers had suggested similar ideas prior to Keynes, it was not until his Treatise that the logical interpretation of probability was advocated in a clear, systematic and rigorous way. I trace Keynes’s influence in the philosophy of (...) class='Hi'>probability through a heterogeneous sample of thinkers who adopted his interpretation. This sample consists of Frederick C. Benenson, Roy Harrod, Donald C. Williams, Henry E. Kyburg and David Stove. The ideas of Keynes prove to be adaptable to their diverse theories of probability. My discussion indicates both the robustness of Keynes’s probability theory and the importance of its influence on the philosophers whom I describe. I also discuss the Problem of the Priors. I argue that none of those I discuss have obviously improved on Keynes’s theory with respect to this issue. (shrink)

Once upon a time, some thought that indicative conditionals could be effectively analyzed as material conditionals. Later on, an alternative theoretical construct has prevailed and received wide acceptance, namely, the conditional probability of the consequent given the antecedent. Partly following critical remarks recently ap- peared in the literature, we suggest that evidential support—rather than conditional probability alone—is key to understand indicative conditionals. There have been motivated concerns that a theory of evidential conditionals (unlike their more tra- ditional counterparts) (...) cannot generate a sufficiently interesting logical system. Here, we will describe results dispelling these worries. Happily, and perhaps surprisingly, appropriate technical variations of Ernst Adams’s classical approach allow for the construction of a logic of evidential conditionals with distinctive fea- tures, which is also well-behaved and reasonably strong. (shrink)

I present new counterexamples to the asymmetry of grounding: we have prima facie reason to think that some conditional probabilities partially ground their inverse conditional probabilities, and vice versa. These new counterexamples may require that we reject the asymmetry of grounding, or alternatively may require that we reject one or more of the assumptions which enable the counterexamples. Either way, by reflecting on these purported counterexamples to grounding asymmetry we learn something important, either about the formal properties of grounding, or (...) about the nature of probability. (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)

In this paper, we illustrate some serious difficulties involved in conveying information about uncertain risks and securing informed consent for risky interventions in a clinical setting. We argue that in order to secure informed consent for a medical intervention, physicians often need to do more than report a bare, numerical probability value. When probabilities are given, securing informed consent generally requires communicating how probability expressions are to be interpreted and communicating something about the quality and quantity of the (...) evidence for the probabilities reported. Patients may also require guidance on how probability claims may or may not be relevant to their decisions, and physicians should be ready to help patients understand these issues. (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 article is a plea for ethicists to regard probability as one of their most important concerns. It outlines a series of topics of central importance in ethical theory in which probability is implicated, often in a surprisingly deep way, and lists a number of open problems. Topics covered include: interpretations of probability in ethical contexts; the evaluative and normative significance of risk or uncertainty; uses and abuses of expected utility theory; veils of ignorance; Harsanyi’s aggregation theorem; (...) population size problems; equality; fairness; giving priority to the worse off; continuity; incommensurability; nonexpected utility theory; evaluative measurement; aggregation; causal and evidential decision theory; act consequentialism; rule consequentialism; and deontology. (shrink)

Modern scientific cosmology pushes the boundaries of knowledge and the knowable. This is prompting questions on the nature of scientific knowledge. A central issue is what defines a 'good' model. When addressing global properties of the Universe or its initial state this becomes a particularly pressing issue. How to assess the probability of the Universe as a whole is empirically ambiguous, since we can examine only part of a single realisation of the system under investigation: at some point, data (...) will run out. We review the basics of applying Bayesian statistical explanation to the Universe as a whole. We argue that a conventional Bayesian approach to model inference generally fails in such circumstances, and cannot resolve, e.g., the so-called 'measure problem' in inflationary cosmology. Implicit and non-empirical valuations inevitably enter model assessment in these cases. This undermines the possibility to perform Bayesian model comparison. One must therefore either stay silent, or pursue a more general form of systematic and rational model assessment. We outline a generalised axiological Bayesian model inference framework, based on mathematical lattices. This extends inference based on empirical data (evidence) to additionally consider the properties of model structure (elegance) and model possibility space (beneficence). We propose this as a natural and theoretically well-motivated framework for introducing an explicit, rational approach to theoretical model prejudice and inference beyond data. (shrink)

A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson (...) is speaking of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (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)

Stalnaker's Thesis about indicative conditionals is, roughly, that the probability one ought to assign to an indicative conditional equals the probability that one ought to assign to its consequent conditional on its antecedent. The thesis seems right. If you draw a card from a standard 52-card deck, how confident are you that the card is a diamond if it's a red card? To answer this, you calculate the proportion of red cards that are diamonds -- that is, you (...) calculate the probability of drawing a diamond conditional on drawing a red card. Skyrms' Thesis about counterfactual conditionals is, roughly, that the probability that one ought to assign to a counterfactual equals one's rational expectation of the chance, at a relevant past time, of its consequent conditional on its antecedent. This thesis also seems right. If you decide not to enter a 100-ticket lottery, how confident are you that you would have won had you bought a ticket? To answer this, you calculate the prior chance--that is, the chance just before your decision not to buy a ticket---of winning conditional on entering the lottery. The central project of this article is to develop a new uniform theory of conditionals that allows us to derive a version of Skyrms' Thesis from a version of Stalnaker's Thesis, together with a chance-deference norm relating rational credence to beliefs about objective chance. (shrink)

In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the (...) density matrices induced by the experiments or `measurements' is the Lüders mixture operation as in QM. And finally by moving the machinery into the n-dimensional vector space over ℤ₂, different basis sets become different outcome sets. That `non-commutative' extension of finite probability theory yields the pedagogical model of quantum mechanics over ℤ₂ that can model many characteristic non-classical results of QM. (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)

The topic of this thesis is how we should treat tiny probabilities of vast value. This thesis consists of six independent papers. Chapter 1 discusses the idea that utilities are bounded. It shows that bounded decision theories prescribe prospects that are better for no one and worse for some if combined with an additive axiology. Chapter 2, in turn, points out that standard axiomatizations of Expected Utility Theory violate dominance in cases that involve possible states of zero probability. Chapters (...) 3–6 discuss the idea that we should ignore tiny probabilities in practical decision-making. Chapter 3 argues that discounting small probabilities solves the ‘Intrapersonal Addition Paradox’ and thus helps avoid the Repugnant Conclusion. Chapter 4 explores what the most plausible version of this view might look like and what problems the different versions have. Chapter 5 focuses on one type of problem, namely, money pumps. The Independence Money Pump, in particular, presents a difficult challenge for those who wish to discount small probabilities. Finally, Chapter 6 discusses the implications of discounting small probabilities for the value of the far future. (shrink)

The role of probability is one of the most contested issues in the interpretation of contemporary physics. In this paper, I’ll be reevaluating some widely held assumptions about where and how probabilities arise. Larry Sklar voices the conventional wisdom about probability in classical physics in a piece in the Stanford Online Encyclopedia of Philosophy, when he writes that “Statistical mechanics was the first foundational physical theory in which probabilistic concepts and probabilistic explanation played a fundamental role.” And the (...) conventional wisdom about quantum probabilities is that they are basic, not reducible to the types of probabilities we see in statistical mechanics. In the first section of this paper, I’ll argue that in fact classical physics was steeped in probability long before statistical mechanics came on the scene, specifically, that an objective measure over phase space is an indispensable component of any informative physical theory. In the next section, I’ll argue that this objective measure is the fundamental form of physical probability and that quantum probabilities can be defined in terms of it. In the last, I’ll raise some questions about the metaphysical status of the fundamental measure. (shrink)

This paper argues that the technical notion of conditional probability, as given by the ratio analysis, is unsuitable for dealing with our pretheoretical and intuitive understanding of both conditionality and probability. This is an ontological account of conditionals that include an irreducible dispositional connection between the antecedent and consequent conditions and where the conditional has to be treated as an indivisible whole rather than compositional. The relevant type of conditionality is found in some well-defined group of conditional statements. (...) As an alternative, therefore, we briefly offer grounds for what we would call an ontological reading: for both conditionality and conditional probability in general. It is not offered as a fully developed theory of conditionality but can be used, we claim, to explain why calculations according to the RATIO scheme does not coincide with our intuitive notion of conditional probability. What it shows us is that for an understanding of the whole range of conditionals we will need what John Heil (2003), in response to Quine (1953), calls an ontological point of view. (shrink)

The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object’s weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what probabilities (...) are not determined in this way—these are the basic probabilities which determine values for all other probabilities. The substantive question asks how the values of these basic probabilities are determined. I defend an answer to the structural question on which basic probabilities are the probabilities of atomic propositions conditional on potential direct explanations. I defend this against the view, implicit in orthodox mathematical treatments of probability, that basic probabilities are the unconditional probabilities of complete worlds. I then apply my answer to the structural question to clear up common confusions in expositions of Bayesianism and shed light on the “problem of the priors.”. (shrink)

This Open Access book addresses the age-old problem of infinite regresses in epistemology. How can we ever come to know something if knowing requires having good reasons, and reasons can only be good if they are backed by good reasons in turn? The problem has puzzled philosophers ever since antiquity, giving rise to what is often called Agrippa's Trilemma. The current volume approaches the old problem in a provocative and thoroughly contemporary way. Taking seriously the idea that good reasons are (...) typically probabilistic in character, it develops and defends a new solution that challenges venerable philosophical intuitions and explains why they were mistakenly held. Key to the new solution is the phenomenon of fading foundations, according to which distant reasons are less important than those that are nearby. The phenomenon takes the sting out of Agrippa's Trilemma; moreover, since the theory that describes it is general and abstract, it is readily applicable outside epistemology, notably to debates on infinite regresses in metaphysics. (shrink)

This dissertation is a contribution to formal and computational philosophy. -/- In the first part, we show that by exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. Case 1: Infinite lotteries. We discuss how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. The solution boils down to the (...) introduction of infinitesimal probability values, which can be achieved using non-standard analysis. Our solution can be generalized to uncountable sample spaces, giving rise to a Non-Archimedean Probability (NAP) theory. Case 2: Large but finite lotteries. We propose application of the language of relative analysis (a type of non-standard analysis) to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’. -/- The second part presents a case study in social epistemology. We model a group of agents who update their opinions by averaging the opinions of other agents. Our main goal is to calculate the probability for an agent to end up in an inconsistent belief state due to updating. To that end, an analytical expression is given and evaluated numerically, both exactly and using statistical sampling. The probability of ending up in an inconsistent belief state turns out to be always smaller than 2%. (shrink)

This paper is about teaching probability to students of philosophy who don’t aim to do primarily formal work in their research. These students are unlikely to seek out classes about probability or formal epistemology for various reasons, for example because they don’t realize that this knowledge would be useful for them or because they are intimidated by the material. However, most areas of philosophy now contain debates that incorporate probability, and basic knowledge of it is essential even (...) for philosophers whose work isn’t primarily formal. In this paper, I explain how to teach probability to students who are not already enthusiastic about formal philosophy, taking into account the common phenomena of math anxiety and the lack of reading skills for formal texts. I address course design, lesson design, and assignment design. Most of my recommendations also apply to teaching formal methods other than probability theory. (shrink)

Recent years have witnessed a proliferation of attempts to apply the mathematical theory of probability to the semantics of natural language probability talk. These sorts of “probabilistic” semantics are often motivated by their ability to explain intuitions about inferences involving “likely” and “probably”—intuitions that Angelika Kratzer’s canonical semantics fails to accommodate through a semantics based solely on an ordering of worlds and a qualitative ranking of propositions. However, recent work by Wesley Holliday and Thomas Icard has been widely (...) thought to undercut this motivation: they present a world-ordering semantics that yields essentially the same logic as probabilistic semantics. In this paper, I argue that the challenge remains: defenders of world-ordering semantics have yet to offer a plausible semantics that captures the logic of comparative likelihood. Holliday & Icard’s semantics yields an adequate logic only if models are restricted to Noetherian pre-orders. But I argue that the Noetherian restriction faces problems in cases involving infinitely large domains of epistemic possibilities. As a result, probabilistic semantics remains the better explanation of the data. (shrink)

I defend the thesis that legal standards of proof are reducible to thresholds of probability. Many reject this thesis because it appears to permit finding defendants liable solely on the basis of statistical evidence. To the contrary, I argue – by combining Thomson's (1986) causal analysis of legal evidence with formal methods of causal inference – that legal standards of proof can be reduced to probabilities, but that deriving these probabilities involves more than just statistics.