Knowledge,certainty and probability

In this essay, I discuss some of the important logical principles governing the concepts of knowledge, certainty and probability. In the first section, I suggest a series of definitions of epistemic terms, employing as primitive the locution ‘p is epistemi‐cally possible to S’ In the second section, I develop an epistemic concept of probability and compare it to the concepts of certainty and knowledge. In the third section, I relate the epistemic concepts of certainty and probability to the quantifiers of traditional logic and to a non‐episteznic concept of probability. I conclude by noting similarities and differences between the two concepts of probability.     

Logical probability,mathematical statistics,and the problem of induction

Summary In this paper I want to discuss some basic problems of inductive logic, i.e. of the attempt to solve the problem of induction by means of a calculus of logical probability. I shall try to throw some light upon these problems by contrasting inductive logic, based on logical probability, and working with undefined samples of observations, with mathematical statistics, based on statistical probability, and working with representative random samples.     

Truthier Than Thou: Truth,Supertruth and Probability of Truth

Different formal tools are useful for different purposes. For example, when it comes to modelling degrees of belief, probability theory is a better tool than classical logic; when it comes to modelling the truth of mathematical claims, classical logic is a better tool than probability theory. In this paper I focus on a widely used formal tool and argue that it does not provide a good model of a phenomenon of which many think it does provide a good model: I shall argue that while supervaluationism may provide a model of probability of truth, or of assertability, it cannot provide a good model of truth—supertruth cannot be truth. The core of the argument is that an adequate model of truth must render certain connectives truth‐functional (at least in certain circumstances)—and supervaluationism does not do so (in those circumstances).     

Time-dependent gambling: odds now,money later

Four experiments investigated temporal changes in the influence of probability and payoffs on gambling. Using urn draws, the authors found in Experiment I that temporal distance increased the influence of payoffs and decreased the influence of probability on preferences. The authors found in Experiment 2 that in choosing among the more distant gambles, participants offered more reasons dealing with payoffs and fewer reasons dealing with probability. In Experiments 3 and 4, the authors extended the scope of these findings using a card game and a raffle. The results were interpreted in terms of a temporal construal process that highlights the desirability of outcomes in the distant future and the feasibility of attaining the outcomes in the near future.     

Four Keys to Success (Theory, Implementation, Evaluation, and Resource/System Support): High Hopes and Challenges in Participation

In this article, I attempt to merge two themes. First, there is often a large gap between high hopes about impacts of policies or programs and the demonstrated results. I describe four keys/threats to success in any social problem area: theory, implementation, evaluation, and resource/system support. Second, I present theory and research from over 30 years of work on participation, conducted by my colleagues and myself that can illuminate and be illuminated by theory, implementation, evaluation, and resource/system support. I offer ideas for solutions that increase the probability of success. I conclude with the need to have high hopes tempered by theory and research to develop realistically ambitious solutions to social problems.     

Lotteries,Quasi-Lotteries,and Scepticism

I seem to know that I won't experience spaceflight but also that if I win the lottery, then I will take a flight into space. Suppose I competently deduce from these propositions that I won't win the lottery. Competent deduction from known premises seems to yield knowledge of the deduced conclusion. So it seems that I know that I won't win the lottery; but it also seems clear that I don't know this, despite the minuscule probability of my winning (if I have a lottery ticket). So we have a puzzle. It seems to generalize, for analogues of the lottery-proposition threaten almost all ordinary knowledge attributions. For example, my apparent knowledge that my bike is parked outside seems threatened by the possibility that it's been stolen since I parked it, a proposition with a low but non-zero probability; and it seems that I don't know this proposition to be false. Familiar solutions to this family of puzzles incur unacceptable costs—either by rejecting deductive closure for knowledge, or by yielding untenable consequences for ordinary attributions of knowledge or of ignorance. After canvassing and criticizing these solutions, I offer a new solution free of these costs.

Knowledge that p requires an explanatory link between the fact that p and the belief that p. This necessary but insufficient condition on knowledge distinguishes actual lottery cases from typical, apparently analogous ‘quasi-lottery’ cases. It does yield scepticism about my not winning the lottery and not experiencing spaceflight, but the scepticism doesn't generalize to quasi-lottery cases such as that involving my bike.     

Impurism,Practical Reasoning,and the Threshold Problem

I consider but reject one broad strategy for answering the threshold problem for fallibilist accounts of knowledge, namely what fixes the degree of probability required for one to know? According to the impurist strategy to be considered, the required degree of probability is fixed by one's practical reasoning situation. I distinguish two different ways to implement the suggested impurist strategy. According to the Relevance Approach, the threshold for a subject to know a proposition at a time is determined by the practical reasoning situations she is then in to which that particular proposition is relevant. According to the Unity Approach, the threshold for a subject to know any proposition whatsoever at a time is determined by a privileged practical reasoning situation she then faces, most plausibly the highest stakes practical reasoning situation she is then in. I argue that neither way of implementing the impurist strategy succeeds and so impurism does not offer a satisfactory response to the threshold problem.     

Triviality Results for Probabilistic Modals

In recent years, a number of theorists have claimed that beliefs about probability are transparent. To believe probably p is simply to have a high credence that p. In this paper, I prove a variety of triviality results for theses like the above. I show that such claims are inconsistent with the thesis that probabilistic modal sentences have propositions or sets of worlds as their meaning. Then I consider the extent to which a dynamic semantics for probabilistic modals can capture theses connecting belief, certainty, credence, and probability. I show that although a dynamic semantics for probabilistic modals does allow one to validate such theses, it can only do so at a cost. I prove that such theses can only be valid if probabilistic modals do not satisfy the axioms of the probability calculus.     

Origins of the logical theory of probability: Von Kries,Wittgenstein, Waismann

The physiologist and neo-Kantian philosopher Johannes von Kries (1853-1928) wrote one of the most philosophically important works on the foundation of probability after P.S. Laplace and before the First World War, his Principien der Wohrscheinlich-keitsrechnung (1886, repr. 1927). In this book, von Kries developed a highly original interpretation of probability, which maintains it to be both logical and objectively physical. After presenting his approach I shall pursue the influence it had on Ludwig Wittgenstein and Friedrich Waismann. It seems that von Kries's approach had more potential than recognized in his time and that putting Waismann's and Wittgenstein's early work in a von Kries perspective is able to shed light on the notion of an elementary proposition.     

Epistemology and probability

Probability is sometimes regarded as a universal panacea for epistemology. It has been supposed that the rationality of belief is almost entirely a matter of probabilities. Unfortunately, those philosophers who have thought about this most extensively have tended to be probability theorists first, and epistemologists only secondarily. In my estimation, this has tended to make them insensitive to the complexities exhibited by epistemic justification. In this paper I propose to turn the tables. I begin by laying out some rather simple and uncontroversial features of the structure of epistemic justification, and then go on to ask what we can conclude about the connection between epistemology and probability in the light of those features. My conclusion is that probability plays no central role in epistemology. This is not to say that probability plays no role at all. In the course of the investigation, I defend a pair of probabilistic acceptance rules which enable us, under some circumstances, to arrive at justified belief on the basis of high probability. But these rules are of quite limited scope. The effect of there being such rules is merely that probability provides one source for justified belief, on a par with perception, memory, etc. There is no way probability can provide a universal cure for all our epistemological ills.     

Power comparisons of significance tests of location using scores,ranks, and modular ranks

The Type I error probability and the power of the independent samples t test, performed directly on the ranks of scores in combined samples in place of the original scores, are known to be the same as those of the non‐parametric Wilcoxon–Mann–Whitney (WMW) test. In the present study, simulations revealed that these probabilities remain essentially unchanged when the number of ranks is reduced by assigning the same rank to multiple ordered scores. For example, if 200 ranks are reduced to as few as 20, or 10, or 5 ranks by replacing sequences of consecutive ranks by a single number, the Type I error probability and power stay about the same. Significance tests performed on these modular ranks consistently reproduce familiar findings about the comparative power of the t test and the WMW tests for normal and various non‐normal distributions. Similar results are obtained for modular ranks used in comparing the one‐sample t test and the Wilcoxon signed ranks test.     

Stimulus probability and same-different classification

Three experiments investigated the effect of stimulus probability on same-different classification time. In Experiments I and II, subjects made same responses on the basis of name matches of simultaneously presented letters. Half of the same trials involved letters that were also physically identical. Experiment I showed that the presentation probability of specific letters affected name matches and different responses, but not physical matches. Experiment II varied stimulus contrast as well as probability. Contrast had a main effect but did not interact with probability at any level of processing. In Experiment III, subjects were switched to the physical level of processing. Stimuli that now had the same name but differed in case were called different. In this condition, the probability effects obs(irved in Experiment II disappeared. These results are interpreted as demonstrating that stimulus probability has its effect during the process that derives the name of the stimulus from the visual representation. This process takes place before the name comparison is made, and the name comparison process precedes the determination of the different response.     

《逻辑哲学论》论概率和归纳

《逻辑哲学论》虽然注家甚多,此书对于概率和归纳问题的零星讨论却一直没有受到后人的重视,而本文则试图对这些讨论进行尝试性的重构。本文的写作目的是双重的。首先,本人试图揭示早期维特根斯坦的概率理论与他的整个逻辑原子论架构之间的关系,说明他是如何通过“真值根据”这个概念而将日常语言中的概率表述全部还原为真值函项理论的。在此基础上,笔者将说明为何他的这种概率理论缺乏实际的应用价值。其次,本文将讨论维氏将归纳问题和概率问题相互分离的“分离策略”,并尝试重构出他提出该策略的隐蔽动机。在此基础上．笔者将站在认知科学的立场上为这个策略本身提供一些辩护和批评。笔者认为,早期维氏虽然正确地看到了“思维经济原则”对于归纳推理的重要指导意义,并看到了该原则本身是无法被还原为逻辑真理的,却并没有意识到该原则本身可能是一切具有一定智能水准的问题解决系统所必须依赖的问题解决策略。令人遗憾的是,他对“心理学”和“逻辑”所作的简单的二分法阻止了他更为深入地对智能系统的普遍问题解决逻辑作出探究,尽管这种研究的确是可以相容于他的前述“分离策略”的。总而言之,本文认为早期维氏的概率理论的科学价值相当有限;与之相比较,他的归纳观却可能已指出了如何揭示智能系统的普遍认知机制的正确道路,尽管他在《逻辑哲学论》阶段还缺乏对于这条道路的全面自觉。     

Spontaneous Anomalystic Phenomena,Pragmatic Information and Formal Representations of Uncertainty

I discuss the application of the Model of Pragmatic Information to the study of spontaneous anomalystic mental phenomena like telepathy, precognition, etc. In these phenomena the most important effects are related to anomalous information gain by the subjects. I consider the basic ideas of the Model, as they have been applied to experimental anomalystic phenomena and to spontaneous phenomena that have strong physical effects, like poltergeist cases, highlighting analogies and differences. Moreover, I point out that in such cases we cannot assign a probability of being accepted to every proposition, and so we cannot use standard formulas for pragmatic information and other relevant measures. To overcome the problem, I propose that qualitative possibility theory could be used to describe the situation. In such theory, the confidence in a proposition is expressed using a scale. Basic concepts like epistemic states, belief revision, information gain, pragmatic information etc. are discussed in this frame. Finally an application to some specific cases is sketched.     

Dutch Books,Coherence, and Logical Consistency

In this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff (s)he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch Book Argument for Reflection fails. The question of whether we have a Dutch Book Argument for Conditionalization is left open.     

Natural-born determinists: a new defense of causation as probability-raising

A definition of causation as probability-raising is threatened by two kinds of counterexample: first, when a cause lowers the probability of its effect; and second, when the probability of an effect is raised by a non-cause. In this paper, I present an account that deals successfully with problem cases of both these kinds. In doing so, I also explore some novel implications of incorporating into the metaphysical investigation considerations of causal psychology.     

Belief,Credence, and the Preface Paradox

Many discussions of the ‘preface paradox’ assume that it is more troubling for deductive closure constraints on rational belief if outright belief is reducible to credence. I show that this is an error: we can generate the problem without assuming such reducibility. All that we need are some very weak normative assumptions about rational relationships between belief and credence. The only view that escapes my way of formulating the problem for the deductive closure constraint is in fact itself a reductive view: namely, the view that outright belief is credence 1. However, I argue that this view is unsustainable. Moreover, my version of the problem turns on no particular theory of evidence or evidential probability, and so cannot be avoided by adopting some revisionary such theory. In sum, deductive closure is in more serious, and more general, trouble than some have thought.     

Reward, novelty and spontaneous alternation

It has been suggested that rats' propensity for “win-shift” behaviour in spatial memory and spontaneous alternation tests reflects a species-specific foraging strategy which leads them to avoid places where they have recently found food. An alternative explanation is that they avoid places which are familiar. In three experiments using a T-maze, we evaluated these accounts by comparing the probability of avoiding or re-entering a recently visited arm, as a function of whether food had or had not been found on the previous visit. Each rat received a series of 16 exposure-test trial pairs over 8 days. Neither alternation nor repetition of the previous choice was differentially reinforced. Experiments I and II forced rats to enter a specific arm before a subsequent choice, and differed in the overall probability of reward; in Experiment III all choices were free. In all three experiments the probability of alternating was greater after nonreward than after reward. This effect occurred more reliably on later tests within a day, little difference appearing on earlier tests. It was concluded that there was no evidence for a spontaneous “win-shift” tendency as such, and that these and other results can be adequately accounted for by a combination of exploratory tendencies (spontaneous alternation) and the conventional effects of reward.