This book is meant to be a primer, that is, an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premisses. This is a clear well-written text on the subject of probability logic, suitable for (...) advanced undergraduates or graduates, but also of interest to professional philosophers. There are well-thought-out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means, it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature. (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)

The decision-theoretic account of probability in the Everett or many-worlds interpretation, advanced by David Deutsch and David Wallace, is shown to be circular. Talk of probability in Everett presumes the existence of a preferred basis to identify measurement outcomes for the probabilities to range over. But the existence of a preferred basis can only be established by the process of decoherence, which is itself probabilistic.

Another title in the reissued Oxford Classic Texts in the Physical Sciences series, Jeffrey's Theory of Probability, first published in 1939, was the first to develop a fundamental theory of scientific inference based on the ideas of Bayesian statistics. His ideas were way ahead of their time and it is only in the past ten years that the subject of Bayes' factors has been significantly developed and extended. Until recently the two schools of statistics were distinctly different and set (...) apart. Recent work has changed all that and today's graduate students and researchers all require an understanding of Bayesian ideas. This book is their starting point. (shrink)

We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s (...) axiomatization of probability is replaced by a different type of infinite additivity. (shrink)

We take coherence based probability logic as the basic reference theory to model human deductive reasoning. The conditional and probabilistic argument forms are explored. We give a brief overview of recent developments of combining logic and probability in psychology. A study on conditional inferences illustrates our approach. First steps towards a process model of conditional inferences conclude the paper.

Evidentialists say that a necessary condition of sound epistemic reasoning is that our beliefs reflect only our evidence. This thesis arguably conflicts with standard Bayesianism, due to the importance of prior probabilities in the latter. Some evidentialists have responded by modelling belief-states using imprecise probabilities (Joyce 2005). However, Roger White (2010) and Aron Vallinder (2018) argue that this Imprecise Bayesianism is incompatible with evidentialism due to “inertia”, where Imprecise Bayesian agents become stuck in a state of ambivalence towards hypotheses. Additionally, (...) escapes from inertia apparently only create further conflicts with evidentialism. This dilemma gives a reason for evidentialist imprecise probabilists to look for alternatives without inertia. I shall argue that Henry E. Kyburg’s approach offers an evidentialist-friendly imprecise probability theory without inertia, and that its relevant anti-inertia features are independently justified. I also connect the traditional epistemological debates concerning the “ethics of belief” more systematically with formal epistemology than has been hitherto done. (shrink)

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Moreover, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases. Finally, we apply our results to study (...) selected probabilistic versions of classical categorical syllogisms and construct a new version of the square of opposition in terms of defaults and negated defaults. (shrink)

We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of "essentially Kolmogorovian" probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of (...) belief are those representable by probability functions from the class appropriate to that logic. Classical Bayesianism, which fixes the logic as classical logic, is only one version of this general approach. Another, which we call Intuitionistic Bayesianism, selects intuitionistic logic as the preferred logic and the associated class of probability functions as the right class of candidate representions of epistemic states (rational allocations of degrees of belief). Various objections to classical Bayesianism are, we argue, best met by passing to intuitionistic Bayesianism—in which the probability functions are taken relative to intuitionistic logic—rather than by adopting a radically non-Kolmogorovian, for example, nonadditive, conception of (or substitute for) probability functions, in spite of the popularity of the latter response among those who have raised these objections. The interest of intuitionistic Bayesianism is further enhanced by the availability of a Dutch Book argument justifying the selection of intuitionistic probability functions as guides to rational betting behavior when due consideration is paid to the fact that bets are settled only when/if the outcome bet on becomes known. (shrink)

According to principles of probability coordination, such as Miller's Principle or Lewis's Principal Principle, you ought to set your subjective probability for an event equal to what you take to be the objective probability of the event. For example, you should expect events with a very high probability to occur and those with a very low probability not to occur. This paper examines the grounds of such principles. It is argued that any attempt to justify (...) a principle of probability coordination encounters the same difficulties as attempts to justify induction. As a result, no justification can be found. (shrink)

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

Order of information plays a crucial role in the process of updating beliefs across time. In fact, the presence of order effects makes a classical or Bayesian approach to inference difficult. As a result, the existing models of inference, such as the belief-adjustment model, merely provide an ad hoc explanation for these effects. We postulate a quantum inference model for order effects based on the axiomatic principles of quantum probability theory. The quantum inference model explains order effects by transforming (...) a state vector with different sequences of operators for different orderings of information. We demonstrate this process by fitting the quantum model to data collected in a medical diagnostic task and a jury decision-making task. To further test the quantum inference model, a new jury decision-making experiment is developed. Using the results of this experiment, we compare the quantum inference model with two versions of the belief-adjustment model, the adding model and the averaging model. We show that both the quantum model and the adding model provide good fits to the data. To distinguish the quantum model from the adding model, we develop a new experiment involving extreme evidence. The results from this new experiment suggest that the adding model faces limitations when accounting for tasks involving extreme evidence, whereas the quantum inference model does not. Ultimately, we argue that the quantum model provides a more coherent account for order effects that was not possible before. (shrink)

Peter Achinstein has argued at length and on many occasions that the view according to which evidential support is defined in terms of probability-raising faces serious counterexamples and, hence, should be abandoned. Proponents of the positive probabilistic relevance view have remained unconvinced. The debate seems to be in a deadlock. This paper is an attempt to move the debate forward and revisit some of the central claims within this debate. My conclusion here will be that while Achinstein may be (...) right that his counterexamples undermine probabilistic relevance views of what it is for e to be evidence that h, there is still room for a defence of a related probabilistic view about an increase in being supported, according to which, if p > p, then h is more supported given e than it is without e. My argument relies crucially on an insight from recent work on the linguistics of gradable adjectives. (shrink)

Reports of the Intergovernmental Panel on Climate Change (IPCC) employ an evolving framework of calibrated language for assessing and communicating degrees of certainty in findings. A persistent challenge for this framework has been ambiguity in the relationship between multiple degree-of-certainty metrics. We aim to clarify the relationship between the likelihood and confidence metrics used in the Fifth Assessment Report (2013), with benefits for mathematical consistency among multiple findings and for usability in downstream modeling and decision analysis. We discuss how our (...) proposal meshes with current and proposed practice in IPCC uncertainty assessment. (shrink)

In this chapter we draw connections between two seemingly opposing approaches to probability and statistics: evidential probability on the one hand and objective Bayesian epistemology on the other.

Many philosophers think that part of what makes an event lucky concerns how probable that event is. In this paper, I argue that an epistemic probability account of luck successfully resists recent arguments that all theories of luck, including probability theories, are subject to counterexample (Hales 2016). I argue that an event is lucky if and only if it is significant and sufficiently improbable. An event is significant when, given some reflection, the subject would regard the event as (...) significant, and the event’s epistemic probability, determined by the subject’s evidence, is the only kind of probability that directly bears on whether or not the event is lucky. I conclude with some lessons that are applicable to probability theorists of luck generally, including those defending non-epistemic probability theories. (shrink)

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

The Everett interpretation of quantum mechanics divides naturally into two parts: first, the interpretation of the structure of the quantum state, in terms of branching, and second, the interpretation of this branching structure in terms of probability. This is the second of two reviews of the Everett interpretation, and focuses on probability. Branching processes are identified as chance processes, and the squares of branch amplitudes are chances. Since branching is emergent, physical probability is emergent as well.

The question of whether humans represent grammatical knowledge as a binary condition on membership in a set of well-formed sentences, or as a probabilistic property has been the subject of debate among linguists, psychologists, and cognitive scientists for many decades. Acceptability judgments present a serious problem for both classical binary and probabilistic theories of grammaticality. These judgements are gradient in nature, and so cannot be directly accommodated in a binary formal grammar. However, it is also not possible to simply reduce (...) acceptability to probability. The acceptability of a sentence is not the same as the likelihood of its occurrence, which is, in part, determined by factors like sentence length and lexical frequency. In this paper, we present the results of a set of large-scale experiments using crowd-sourced acceptability judgments that demonstrate gradience to be a pervasive feature in acceptability judgments. We then show how one can predict acceptability judgments on the basis of probability by augmenting probabilistic language models with an acceptability measure. This is a function that normalizes probability values to eliminate the confounding factors of length and lexical frequency. We describe a sequence of modeling experiments with unsupervised language models drawn from state-of-the-art machine learning methods in natural language processing. Several of these models achieve very encouraging levels of accuracy in the acceptability prediction task, as measured by the correlation between the acceptability measure scores and mean human acceptability values. We consider the relevance of these results to the debate on the nature of grammatical competence, and we argue that they support the view that linguistic knowledge can be intrinsically probabilistic. (shrink)

It is shown that the Kolmogorovian Censorship Hypothesis, according to which quantum probabilities are interpretable as conditional probabilities in a classical probability measure space, holds not only for Hilbert space quantum mechanics but for general quantum probability theories based on the theory of von Neumann algebras.

Background on probability and evolution -- Laying the foundation. Population-environment systems ; Causal probability and empirical practice ; Irrelevance of fitness as a causal property of token organisms ; Roles of environmental variation in selection -- Reconstructing evolution and chance. Populations in biological practice: Pragmatic yet real ; Real causation in pragmatic population-environment systems ; Fitness concepts in measurement and modeling ; Chance in population-environment systems ; The input measure problem for MM-CCS chance -- Conclusion.

Philosophical Investigations, Volume 45, Issue 2, Page 130-145, April 2022.

I examine what the mathematical theory of random structures can teach us about the probability of Plenitude, a thesis closely related to David Lewis's modal realism. Given some natural assumptions, Plenitude is reasonably probable a priori, but in principle it can be (and plausibly it has been) empirically disconfirmed—not by any general qualitative evidence, but rather by our de re evidence.

In this paper, I propose an assessment of the interpretation of the mathematical notion of probability that Wittgenstein presents in TLP (1963: 5.15 – 5.156). I start by presenting his definition of probability as a relation between propositions. I claim that this definition qualifies as a logical interpretation of probability, of the kind defended in the same years by J. M. Keynes. However, Wittgenstein’s interpretation seems prima facie to be safe from two standard objections moved to logical (...) probability, i. e. the mystic nature of the postulated relation and the reliance on Laplace’s principle of indifference. I then proceed to evaluate Wittgenstein’s idea against three criteria for the adequacy of an interpretation of probability: admissibility, ascertainability, and applicability. If the interpretation is admissible on Kolmogorov’s classical axiomatisation, the problem of ascertainability brings up a difficult dilemma. Finally, I test the interpretation in the application to three main contexts of use of probabilities. While the application to frequencies rests ungrounded, the application to induction requires some elaboration, and the application to rational belief depends on ascertainability. (shrink)

I defend a simple argument for why considerations of epistemic probability should lead us away from Open Future views according to which claims about the future are never true.