We offer an alternative approach to the existing methods for intuitionistic formalization of reasoning about probability. In terms of Kripke models, each possible world is equipped with a structure of the form $$\langle H, \mu \rangle $$ that needs not be a probability space. More precisely, though H needs not be a Boolean algebra, the corresponding monotone function (we call it measure) $$\mu : H \longrightarrow [0,1]_{\mathbb {Q}}$$ satisfies the following condition: if $$\alpha $$, $$\beta $$, $$\alpha \wedge (...) \beta $$, $$\alpha \vee \beta \in H$$, then $$\mu (\alpha \vee \beta ) = \mu (\alpha ) + \mu (\beta ) - \mu (\alpha \wedge \beta )$$. Since the range of $$\mu $$ is the set $$[0,1]_{\mathbb {Q}}$$ of rational numbers from the real unit interval, our logic is not compact. In order to obtain a strong complete axiomatization, we introduce an infinitary inference rule with a countable set of premises. The main technical results are the proofs of strong completeness and decidability. (shrink)

Research on preference reversals has demonstrated a disproportionate influence of outcome probability on choices between monetary gambles. The aim was to investigate the hypothesis that this is a prominence effect originally demonstrated for riskless choice. Another aim was to test the structure compatibility hypothesis as an explanation of the effect. The hypothesis implies that probability should be the prominent attribute when compared with value attributes both in a choice and a preference rating procedure. In Experiment 1, two groups of undergraduates (...) were presented with medical treatments described by two value attributes (effectiveness and pain-relief). All participants performed both a matching task and made preference ratings. In the latter task, outcome probabilities were added to the descriptions of the medical treatments for one of the groups. In line with the hypothesis, this reduced the prominence effect on the preference ratings observed for effectiveness. In Experiment 2, a matching task was used to demonstrate that probability was considered more important by a group of participating undergraduates than the value attributes. Furthermore, in both choices and preference ratings the expected prominence effect was found for probability. (shrink)

In concrete applications of probability, statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want is prob(P/Q& R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with the (...) probability calculus for prob(P/Q& R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q& R) 1 A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability—nomic probability—there are principles of "probable probabilities" that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r, s, a) such that if prob(P/Q) = r, prob(P/R) = s, andprob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&R) = Y(r, s, a). Numerous other defeasible inferences are licensed by similar principles of probable probabilities. This has the potential to greatly enhance the usefulness of probabilities in practical application. (shrink)

A symposium on Michael Strevens' book "Tychomancy", concerning the psychological roots and historical significance of physical intuition about probability in physics, biology, and elsewhere.

In this paper we present two families of probability logics (denoted _QLP_ and \(QLP^{ORT}\) ) suitable for reasoning about quantum observations. Assume that \(\alpha \) means “O = a”. The notion of measuring of an observable _O_ can be expressed using formulas of the form \(\square \lozenge \alpha \) which intuitively means “if we measure _O_ we obtain \(\alpha \) ”. In that way, instead of non-distributive structures (i.e., non-distributive lattices), it is possible to relay on classical logic (...) extended with the corresponding modal laws for the modal logic \({\textbf{B}}\). We consider probability formulas of the form \(CS_{z_{1},\rho _{1}; \ldots ; z_{m},\rho _{m}} \square \lozenge \alpha \) related to an observable _O_ and a possible world (vector) _w_: if _a_ is an eigenvalue of _O_, \(w_{1}\),..., \(w_{m}\) form a base of a closed subspace of the considered Hilbert space which corresponds to eigenvalue _a_, and if _w_ is a linear combination of the basis vectors such that \(w=c_{1}\cdot w_{1}+ \cdots + c_{m}\cdot w_{m}\) for some \(c_{i}\in {\mathbb {C}}\), then \(\Vert c_{1}-z_{1}\Vert \le \rho _{1}\),..., \(\Vert c_{m}-z_{m}\Vert \le \rho _{m}\), and the probability of obtaining _a_ while measuring _O_ in the state _w_ is equal to \(\Sigma _{i=1}^{m}\Vert c_{i}\Vert ^{2}\). Formulas are interpreted in reflexive and symmetric Kripke models equipped with probability distributions over families of subsets of possible worlds that are orthocomplemented lattices, while for \(QLP^{ORT}\) also satisfy ortomodularity. We give infinitary axiomatizations, prove the corresponding soundness and strong completeness theorems, and also decidability for _QLP_-logics. (shrink)

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

Using formal systems to represent and reason about uncertainty.

There is a long tradition in formal epistemology and in the psychology of reasoning to investigate indicative conditionals. In psychology, the propositional calculus was taken for granted to be the normative standard of reference. Experimental tasks, evaluation of the participants’ responses and psychological model building, were inspired by the semantics of the material conditional. Recent empirical work on indicative conditionals focuses on uncertainty. Consequently, the normative standard of reference has changed. I argue why neither logic nor standard probability theory (...) provide appropriate rationality norms for uncertain conditionals. I advocate coherence based probability logic as an appropriate framework for investigating uncertain conditionals. Detailed proofs of the probabilistic non-informativeness of a paradox of the material conditional illustrate the approach from a formal point of view. I survey selected data on human reasoning about uncertain conditionals which additionally support the plausibility of the approach from an empirical point of view. (shrink)

According to the Doomsday Argument we have to rethink the probabilities we assign to a soon or not so soon extinction of mankind when we realize that we are living now, rather early in the history of mankind. Sleeping Beauty finds herself in a similar predicament: on learning the date of her first awakening, she is asked to re-evaluate the probabilities of her two possible future scenarios. In connection with Doom, I argue that it is wrong to assume (...) that our ordinary probability judgements do not already reflect our place in history: we justify the predictive use we make of the probabilities yielded by science by our knowledge of the fact that we live now, a certain time before the possible occurrence of the events the probabilities refer to. Our degrees of belief should change drastically when we forget the date—importantly, this follows without invoking the “Self Indication Assumption”. Subsequent conditionalization on information about which year it is cancels this probability shift again. The Doomsday Argument is about such probability shifts, but tells us nothing about the concrete values of the probabilities —for these, experience provides the only basis. Essentially the same analysis applies to the Sleeping Beauty problem. I argue that Sleeping Beauty “thirders” should be committed to thinking that the Doomsday Argument is ineffective; whereas “halfers” should agree that doom is imminent—but they are wrong. (shrink)

The present paper deals with the tools that can be used to represent causation and to reason about it and, specifically, with their diversity. It focuses on so-called “causal probabilities”—that is, probabilities of effects given one of their causes—and critically surveys a recent paper in which Joyce argues that the values of these probabilities do not depend on one’s conception of causation. I first establish a stronger independence claim: I show that the very definition of causal (...) probabilities is independent of one’s conception of causation. Second, I investigate whether causal probabilities indeed take the same values under their different possible definitions. (shrink)

This paper attempts to reconstruct Hume's argument in Treatise 1.4.1, 'Of Scepticism with Regard to Reason'.

When reasoning about self-locating belief, one should reason as if one were a randomly selected bit of information. This principle can be considered to be an application of Bostrom's Strong Self-Sampling Assumption\cite{Bostrom} according to which one should reason as if one were a randomly selected element of some suitable reference class of observer-moments. The reference class is the class of all observer-moments. In order to randomly select an observer-moment from the reference class, one first randomly chooses a possible (...) world $w$ and then selects an observer-moment $z$ from world $w$. The probability that one selects $z$ given that one has chosen $w$ should be proportional to the amount of information that $z$ is capable of representing. There are both wagering arguments and relative frequency arguments that support our theory of anthropic reasoning. Our theory works best when the amount of information represented is finite. The infinite case is represented as a limit of a finite cases. We can learn from experience how best to represent the infinite case as a limit of finite cases and also learn from experience whether our theory or some other theory is the superior theory of anthropic reasoning. In order to test which theory is best, we use standard Bayesian methodology: We just need prior probabilities for the theories that are being tested and then we only have to use Bayes' rule. (shrink)

It was Hume's concept of knowledge that marked Masaryk's philosophy more than influential Kant. Masaryk devoted Hume his inaugural lecture The Probability Calculus and Hume's Scepticism at Prague University in 1882. He tried to face his scepticism concerning causation and induction by a formula P = n: or P = : where P means the increasing probability that an event that had happened n times in the past will happen again. Hume stresses that there is an essential difference between probability (...) and certainty, and we have no logical reason to presuppose that the past experience is legitimate for expectation that the future will resemble to the past. Only our belief, not knowledge, leads us to that. Although Masaryk did not accept Hume's scepticism, he had deep understanding for his way of thinking. Reflection of Hume's influence on Masaryk in Czech philosophy is unsatisfactory, which is stated in this article. (shrink)

Nonmonotonic conditionals (A |∼ B) are formalizations of common sense expressions of the form “if A, normally B”. The nonmonotonic conditional is interpreted by a “high” coherent conditional probability, P(B|A) > .5. Two important properties are closely related to the nonmonotonic conditional: First, A |∼ B allows for exceptions. Second, the rules of the nonmonotonic system p guiding A |∼ B allow for withdrawing conclusions in the light of new premises. This study reports a series of three experiments on (...) class='Hi'>reasoning with inference rules about nonmonotonic conditionals in the framework of coherence. We investigated the cut, and the right weakening rule of system p. As a critical condition, we investigated basic monotonic properties of classical (monotone) logic, namely monotonicity, transitivity, and contraposition. The results suggest that people reason nonmonotonically rather than monotonically. We propose nonmonotonic reasoning as a competence model of human reasoning. (shrink)

The naïve see causal connections everywhere. Consider the fact that Evelyn Marie Adams won the New Jersey lottery twice. The naïve find it irresistible to think that this cannot be a coincidence. Maybe the lottery was rigged or perhaps some uncanny higher power placed its hand upon her brow. Sophisticates respond with an indulgent smile and ask the naïve to view Adams’ double win within a larger perspective. Given all the lotteries there have been, it isn’t at all surprising that (...) someone would win one of them twice. No need to invent conspiracy theories or invoke the paranormal – the double win was a mere coincidence. (shrink)

In this paper we present two types of logics and \ ) where certain p-adic functions are associated to propositional formulas. Logics of the former type are p-adic valued probability logics. In each of these logics we use probability formulas K r,ρ α and D ρ α,β which enable us to make sentences of the form “the probability of α belongs to the p-adic ball with the center r and the radius ρ”, and “the p-adic distance between the probabilities (...) of α and β is less than or equal to ρ”, respectively. Logics of the later type formalize processes of thinking where information are coded by p-adic numbers. We use the same operators as above, but in this formalism K r,ρ α means “the p-adic code of the information α belongs to the p-adic ball with the center r and the radius ρ”, while D ρ α,β means “the p-adic distance between codes of α and β are less than or equal to ρ”. The corresponding strongly complete axiom systems are presented and decidability of the satisfiability problem for each logic is proved. (shrink)

Abstract Finding the western just war criterion of reasonable chance of success to be a contribution to ethical decision making about armed conflict requires dealing with a number of critiques. Specifying ?probability? rather than the alternatives ?hope? or ?chance?, and raising standards of evidence involved, makes the term less vague. Expanding the concept of ?success? to include morally defensible aims that can be achieved without military victory enriches the understanding of the moral relationship between ends and means in armed (...) conflict. Asking decision makers to accept moral responsibility for the costs of possible failure is a unique contribution to the just war criteria. The enriched concept of reasonable probability of success thus offers morally significant insights to prewar jus ad bellum decisions, and can benefit ethical decision-making about whether to continue once fighting has begun. (shrink)

This book review describes and evaluates Joseph Halpern's ideas about reasoning, in particular, reasoning involving conditions.

This article outlines a theory of naive probability. According to the theory, individuals who are unfamiliar with the probability calculus can infer the probabilities of events in an extensional way: They construct mental models of what is true in the various possibilities. Each model represents an equiprobable alternative unless individuals have beliefs to the contrary, in which case some models will have higher probabilities than others. The probability of an event depends on the proportion of models in which (...) it occurs. The theory predicts several phenomena of reasoning about absolute probabilities, including typical biases. It correctly predicts certain cognitive illusions in inferences about relative probabilities. It accommodates reasoning based on numerical premises, and it explains how naive reasoners can infer posterior probabilities without relying on Bayes's theorem. Finally, it dispels some common misconceptions of probabilistic reasoning. (shrink)

The Power of Critical Thinking: Effective Reasoning About Ordinary and Extraordinary Claims explores the essentials of critical reasoning, argumentation, logic, and argumentative essay writing while also incorporating important topics that most other texts leave out, such as "inference to the best explanation," scientific reasoning, evidence and authority, visual reasoning, and obstacles to critical thinking.The text integrates many pedagogical features, including hundreds of diverse exercises, examples, and illustrations; text boxes that apply critical thinking to student experience; (...) step-by-step guidelines for evaluating claims, arguments, and explanations; a glossary of important terms; and many reminders, summaries, and review notes. New to this editionNew sections on topics suggested by reviewers - The coverage now includes more discussion of legal reasoning, rhetorical ploys, informal fallacies, probability and statistics, and necessary and sufficient conditions.New Essays for Analysis - Nine essays, several of them by women authors, have been added to the collection of "Essays for Evaluation" in Appendix A, each article accompanied by writing prompts and linked to writing assignments in every chapter. Four pairs of essays are arranged in a pro/con format, each pair debating a single issue. New topics include homosexuality, feminists and pornography, adultery, airport security screenings, women in Afghanistan, Islamic extremists and free speech, and fear of vaccines.New text boxes on current topics. The division of labor for the text boxes is the same, but some of the content is new. The three types of boxes are "Newsmakers" ; "From the Web" ; and "Further Thought".The chapter objectives now correspond to the point-by-point summary at the end of each chapter.Important Clarifications - The following discussions are now even clearer: scientific conservatism in theory choice, the relationship between enumerative induction and statistical syllogisms, and informal fallacies.Some new discussions: the straw man fallacy, biased opinion polls, dishonest political discourse, and gut reactions and intuitions. (shrink)

This paper presents a formalism that combines useful properties of both logic and probabilities. Like logic, the formalism admits qualitative sentences and provides symbolic machinery for deriving deductively closed beliefs and, like probability, it permits us to express if-then rules with different levels of firmness and to retract beliefs in response to changing observations. Rules are interpreted as order-of-magnitude approximations of conditional probabilities which impose constraints over the rankings of worlds. Inferences are supported by a unique priority ordering (...) on rules which is syntactically derived from the knowledge base. This ordering accounts for rule interactions, respects specificity considerations and facilitates the construction of coherent states of beliefs. Practical algorithms are developed and analyzed for testing consistency, computing rule ordering, and answering queries. Imprecise observations are incorporated using qualitative versions of Jeffrey's rule and Bayesian updating, with the result that coherent belief revision is embodied naturally and tractably. Finally, causal rules are interpreted as imposing Markovian conditions that further constrain world rankings to reflect the modularity of causal organizations. These constraints are shown to facilitate reasoning about causal projections, explanations, actions and change. (shrink)

Children’s reasoning on food properties and health relationships can contribute to healthier food choices. Food properties can either be positive (“gives strength”) or negative (“gives nausea”). One of the main challenges in public health is to foster children’s dietary variety, which contributes to a normal and healthy development. To face this challenge, it is essential to investigate how children generalize these positive and negative properties to other foods, including familiar and unfamiliar ones. In the present experiment, we hypothesized that (...) children might rely on cues of food processing (e.g., signs of human intervention such as slicing) to convey information about item edibility. Furthermore, capitalizing on previous results showing that food rejections (i.e., food neophobia and picky eating) are a significant source of inter-individual variability to children’s inferences in the food domain, we followed an individual approach. We expected that children would generalize the positive properties to familiar foods and, in contrast, that they would generalize more often the negative properties to unfamiliar foods. However, we expected that children would generalize more positive and less negative properties to unfamiliar sliced foods than to whole unfamiliar foods. Finally, we expected that children displaying higher levels of food rejections would generalize more negative properties than children displaying lower levels of food rejections. One-hundred and twenty-six children, aged 3–6 years, performed an induction task in which they had to generalize positive or negative health-related properties to familiar or unfamiliar foods, whole or sliced. We measured children’s probability of generalization for positive and negative properties. The children’s food rejection score was assessed on a standardized scale. Results indicated that children evaluated positively familiar foods (regardless of processing), whereas they tend to view unfamiliar food negatively. In contrast, children were at chance for processed unfamiliar foods. Furthermore, children displaying higher levels of food rejections were more likely to generalize the negative properties to all kinds of foods than children displaying lower levels of food rejections. These findings entitle us to hypothesize that knowledge-based food education programs should take into account the valence of the properties taught to children, as well as the state of processing of the food presented. Furthermore, one should take children’s interindividual differences into account because they influence how the knowledge gained through these programs may be generalized. (shrink)

Reviews Joseph Halpern's book with special attention to his points about conditionals.

To qualify the truth of a proposition probabilistically is to place it within the scope of a special type of alethic modality. We expect that, as in other modal contexts, the merely probabilistic truth of an assumption in a valid inference must carry over to whatever conclusions are derived from the assumption. That expectation, however, is not always fulfilled in ordinary reasoning about conditional probabilities. There are simpler ways of illustrating what I shall call the paradoxes of (...) conditional probabilistic reasoning in ordinary language, but the following argument is a colorful example. Consider an apparently deductively valid inference, by an imaginary inmate of a penal institution supervised by a bitterly hated warden who is surrounded by dangerous criminals, including the argument’s author. (shrink)

Research into the cognition of conditionals has predominantly focused on conditional reasoning, producing a range of theories which explain associated phenomena with considerable success. However, such theories have been less successful in accommodating experimental data concerning how agents assess the probability of indicative conditionals. Since an acceptable account of conditional reasoning should be compatible with evidence regarding how we evaluate conditionals’ likelihoods, this constitutes a failing of such theories. Section 1 introduces the most dominant established approach to conditional (...) reasoning: mental models theory. Surveying a range of experimental results, I show that mental models theory (along with competing theories) is incapable of fully accounting for findings regarding judgements about conditionals’ probabilities. Section 2 introduces an alternative account of deductive reasoning, the erotetic theory, recently proposed by Koralus and Mascarenhas (2013). Section 3 argues that, given a natural extension, this theory is able to explain the otherwise unaccounted for data. (shrink)

Most legal evidence scholars agree that proof of guilt beyond a reasonable doubt requires the belief that the defendant probably committed the alleged acts. However, they also agree that this is not a sufficient condition, as this belief may be unreasonable. I focus on two popular proposals for additional conditions: (i) that the degree of belief should be robust and (ii) that it should be reasonable given the available evidence (should be an evidential probability). Both criteria face difficulties regarding their (...) meaning and utility. I propose an interpretation which overcomes these difficulties and which combines the two conditions into a single criterion on which both conditions are about the possibility of overlooked, exculpatory information. Proof of guilt beyond a reasonable doubt requires that we are sufficiently certain that we did not miss such information for the sake of achieving accurate beliefs and a just error distribution. How certain we can be that we did not miss anything relevant depends on our higher-order evidence. One important type of higher-order evidence is whether the hypotheses under consideration are sufficiently detailed. This point leads to a novel way of combining the Bayesian and explanation-based accounts of rational legal proof. (shrink)

Illuminates legal reasoning -- and its justification At least since plato and Aristotle, thinkers have pondered the relationship between philosophical arguments and the "sophistical" arguments offered by the Sophists -- who were the first professional lawyers. Judges wield substantial political power, and the justifications they offer for their decisions are a vital means by which citizens can assess the legitimacy of how that power is exercised. However, to evaluate judicial justifications requires close attention to the method of reasoning (...) behind decisions. This new collection illuminates and explains the political and moral importance in justifying the exercise of judicial power. Explores enduring questions Focusing on work over the past century, the essays address important recurring questions, such as: When a judge or a lawyer reasons to a conclusion about what is legally required in a given case, must he also ask what is morally required? To what extent do a judge's personal, political, or moral biases affect his legal reasoning? What is the impact of such biases? Can all such biases be avoided? Is legal reasoning similar to reasoning in mathematics, logic, and linguistics, the physical sciences, the social sciences, or literature and history? Do formal logical modes of argument play any roles in legal reasoning? Solid coverage, well organized The articles were chosen to present some of the most influential works on the topic, as well as less familiar works that are thought provoking and informative. Each volume also offers a representative range of theoretical approaches to its topic, contains an introduction that locates the subject within the larger framework oftheories of legal reasoning and jurisprudence, and includes bibliographical notes on further readings. Many 19th century legal theorists argued that deduction is the central mode of legal reasoning, and that legal argument is like a deductive proof in mathematics or logic. They were attracted to this "deductivist" concept because it suggested that legal reasoning could be politically and morally neutral. This volume brings together some of the most thought-provoking articles on both sides of the debate. It also contains several leading articles that explain the role of probabilistic judgments and presumptions in various types of legal arguments, including the laws of evidence and criminal procedure. The collection is a solid introduction to the basic modes of logical argument in legal reasoning. (shrink)

In recent years, a number of works have appeared with important implications for the age-old question of the existence of a god. These writings, many of which are not by theologians, strengthen the rational case for the existence of a god, even as this god may not be exactly the Christian God of history. This book brings together for the first time such recent diverse contributions from fields such as physics, the philosophy of human consciousness, evolutionary biology, mathematics, the history (...) of religion, and theology. Based on such new materials as well as older ones from the twentieth century, it develops five rational arguments that point strongly to the (very probable) existence of a god. They do not make use of the scientific method, which is inapplicable to the question of a god. Rather, they are in an older tradition of rational argument dating back at least to the ancient Greeks. For those who are already believers, the book will offer additional rational reasons that may strengthen their belief. Those who do not believe in the existence of a god at present will encounter new rational arguments that may cause them to reconsider their opinion. --back cover. (shrink)

Evidential reasoning is hard, and errors can lead to miscarriages of justice with serious consequences. Analytic methods for the correct handling of evidence come in different styles, typically focusing on one of three tools: arguments, scenarios or probabilities. Recent research used Bayesian networks for connecting arguments, scenarios, and probabilities. Well-known issues with Bayesian networks were encountered: More numbers are needed than are available, and there is a risk of misinterpretation of the graph underlying the Bayesian network, for (...) instance as a causal model. The formalism presented here models presumptive arguments about coherent hypotheses that are compared in terms of their strength. No choice is needed between qualitative or quantitative analytic styles, since the formalism can be interpreted with and without numbers. The formalism is applied to key concepts in argumentative, scenario and probabilistic analyses of evidential reasoning, and is illustrated with a fictional crime investigation example based on Alfred Hitchcock’s film ‘To Catch A Thief’. (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)

This paper defends two theses about probabilistic reasoning. First, although modus ponens has a probabilistic analog, modus tollens does not – the fact that a hypothesis says that an observation is very improbable does not entail that the hypothesis is improbable. Second, the evidence relation is essentially comparative; with respect to hypotheses that confer probabilities on observation statements but do not entail them, an observation O may favor one hypothesis H1 over another hypothesis H2 , but O (...) cannot be said to confirm or disconfirm H1 without such relativization. These points have serious consequences for the Intelligent Design movement. Even if evolutionary theory entailed that various complex adaptations are very improbable, that would neither disconfirm the theory nor support the hypothesis of intelligent design. For either of these conclusions to follow, an additional question must be answered: With respect to the adaptive features that evolutionary theory allegedly says are very improbable, what is their probability of arising if they were produced by intelligent design? This crucial question has not been addressed by the ID movement. (shrink)

As Kant claimed in the Groundwork, and as the idea has been developed by Korsgaard 1997, Bratman 1987, and Broome 2002. This formulation is agnostic on whether reasons for ends derive from our desiring those ends, or from the relation of those ends to things of independent value. However, desire-based theorists may deny, against Hubin 1999, that their theory is a combination of a principle of instrumental transmission and the principle that reasons for ends are provided by desires. Instead, they (...) may say, there is just one principle, a principle of, if you will, instrumental transmutation: if one desires the end, then one has reason to take the means. See the discussion of General Production, in section 8, for a doubt about this. (shrink)

Probability logics have been an active topic of investigation of beliefs in type spaces in game theoretical economics. Beliefs are expressed as subjective probability measures. Savage’s postulates in decision theory imply that subjective probability measures are not necessarily countably additive but finitely additive. In this paper, we formulate a probability logic Σ + that is strongly complete with respect to this class of type spaces with finitely additive probability measures, i.e. a set of formulas is consistent in Σ + iff (...) it is satisfied in a finitely additive type space. Although we can characterize Σ + -theories satisfiable in the class as maximally consistent sets of formulas, we prove that any canonical model of maximally consistent sets is not universal in the class of type spaces with finitely additive measures, and, moreover, it is not a type space. At the end of this paper, we show that even a minimal use of probability indices causes the failure of compactness in probability logics. (shrink)

Many psychologists currently assume that there is a psychologically real distinction to be made between concepts that are abstract and concepts that are concrete. It is for example largely agreed that concepts and words are more easily processed if they are concrete. Moreover, it is assumed that this isbecausethese words and concepts are concrete. It is thought that interesting generalizations can be made about certain conceptsbecausethey are concrete. I argue that we have surprisingly little reason to believe that the (...) abstract‐concrete distinction is psychologically real. (shrink)

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each (...) having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

Why is understanding causation so important in philosophy and the sciences? Should causation be defined in terms of probability? Whilst causation plays a major role in theories and concepts of medicine, little attempt has been made to connect causation and probability with medicine itself. Causality, Probability, and Medicine is one of the first books to apply philosophical reasoning about causality to important topics and debates in medicine. Donald Gillies provides a thorough introduction to and assessment of competing theories (...) of causality in philosophy, including action-related theories, causality and mechanisms, and causality and probability. Throughout the book he applies them to important discoveries and theories within medicine, such as germ theory; tuberculosis and cholera; smoking and heart disease; the first ever randomized controlled trial designed to test the treatment of tuberculosis; the growing area of philosophy of evidence-based medicine; and philosophy of epidemiology. This book will be of great interest to students and researchers in philosophy of science and philosophy of medicine, as well as those working in medicine, nursing and related health disciplines where a working knowledge of causality and probability is required. (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)

"A Little More Logical" is the perfect guide for anyone looking to improve their critical thinking and logical reasoning skills. With chapters on everything from logic basics to fallacies of weak induction to moral reasoning, this book covers all the essential concepts you need to become a more logical thinker. You'll learn about influential figures in the field of logic, such as Rudolph Carnap, Betrrand Russell, and Ada Lovelace, and how to apply your newfound knowledge to real-world (...) situations. Whether you're looking to engage in debates with others, make better decisions in your personal and professional life, or simply want to improve your overall critical thinking skills, "A Little More Logical" has you covered. So why wait? Start learning and become a little more logical today! -/- "A Little More Logical" differs from typical logical textbooks in a number of ways. One key difference is its emphasis on engaging and relatable examples and case studies. Rather than simply presenting dry definitions and concepts, the book uses fables, stories, and real-world situations to illustrate key ideas and make them more relatable for readers. -/- Another unique aspect of "A Little More Logical" is its inclusion of "Minds that Mattered" sections, which highlight the contributions and insights of influential figures in the field of logic and critical thinking. These sections provide readers with a deeper understanding of the history and development of logical principles and offer valuable context for the concepts being discussed. -/- Additionally, "A Little More Logical" covers a wide range of topics beyond the basics of logic and argument evaluation. Chapters on moral reasoning, probability and inductive logic, scientific reasoning, conspiracy theories, statistical reasoning, and the history of formal logic offer a more comprehensive and well-rounded understanding of logic and critical thinking. -/- Overall, "A Little More Logical" stands out as a dynamic and engaging resource for anyone looking to improve their logical reasoning abilities. Its relatable examples, historical context, and broad coverage make it a valuable resource for anyone interested in mastering the principles of logic. -/- This is a free, Creative-Commons-licensed book. (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)

It is a popular view thatpractical deliberation excludes foreknowledge of one's choice. Wolfgang Spohn and Isaac Levi have argued that not even a purely probabilistic self-predictionis available to thedeliberator, if one takes subjective probabilities to be conceptually linked to betting rates. It makes no sense to have a betting rate for an option, for one's willingness to bet on the option depends on the net gain from the bet, in combination with the option's antecedent utility, rather than on the (...) offered odds. And even apart from this consideration, assigning probabilities to the options among which one is choosing is futile since such probabilities could be of no possible use in choice. The paper subjects these arguments to critical examination and suggests that, appearances notwithstanding, practical deliberation need not crowd outself-prediction. (shrink)

Maxwell's deduction of the probability distribution over the velocity of gas molecules—one of the most important passages in physics (Truesdell)—presents a riddle: a physical discovery of the first importance was made in a single inferential leap without any apparent recourse to empirical evidence. -/- Tychomancy proposes that Maxwell's derivation was not made a priori; rather, he inferred his distribution from non-probabilistic facts about the dynamics of intermolecular collisions. Further, the inference is of the same sort as everyday reasoning (...) about the physical probabilities attached to such canonical chance setups as tossed coins or rolled dice. The structure of this reasoning is investigated and some simple rules for inferring physical probabilities from symmetries and other causally relevant properties of physical systems are proposed. -/- Not only physics but evolutionary biology and population ecology, the science of measurement error, and climate modeling have benefited enormously from the same kind of reasoning, the book goes on to argue. Inferences from dynamics to probability are so obvious to us, however, that their methodological importance has been largely overlooked. (shrink)

A. J. Ayer was one of the foremost analytical philosophers of the twentieth century, and was known as a brilliant and engaging speaker. In essays based on his influential Dewey Lectures, Ayer addresses some of the most critical and controversial questions in epistemology and the philosophy of science, examining the nature of inductive reasoning and grappling with the issues that most concerned him as a philosopher. This edition contains revised and expanded versions of the lectures and two additional essays. (...) Ayer begins by considering Hume's formulation of the problem of induction and then explores the inferences on which we base our beliefs in factual matters. In other essays, he defines the three kinds of probability that inform inductive reasoning and examines the various criteria for verifiability and falsifiability. In his extensive introduction, Graham Macdonald discusses the arguments in _Probability and Evidence_, how they relate to Ayer's other works, and their influence in contemporary philosophy. He also provides a brief biographical sketch of Ayer, and includes a bibliography of works about and in response to _Probability and Evidence_. (shrink)