Propositional Reasoning that Tracks Probabilistic Reasoning

This paper concerns the extent to which uncertain propositional reasoning can track probabilistic reasoning, and addresses kinematic problems that extend the familiar Lottery paradox. An acceptance rule assigns to each Bayesian credal state p a propositional belief revision method ${\sf B}_{p}$ , which specifies an initial belief state ${\sf B}_{p}(\top)$ that is revised to the new propositional belief state ${\sf B}(E)$ upon receipt of information E. An acceptance rule tracks Bayesian conditioning when ${\sf B}_{p}(E) = {\sf B}_{p|_{E}}(\top)$ , for every E such that p(E)?>?0; namely, when acceptance by propositional belief revision equals Bayesian conditioning followed by acceptance. Standard proposals for uncertain acceptance and belief revision do not track Bayesian conditioning. The ??Lockean?? rule that accepts propositions above a probability threshold is subject to the familiar lottery paradox (Kyburg 1961), and we show that it is also subject to new and more stubborn paradoxes when the tracking property is taken into account. Moreover, we show that the familiar AGM approach to belief revision (Harper, Synthese 30(1?C2):221?C262, 1975; Alchourrón et al., J Symb Log 50:510?C530, 1985) cannot be realized in a sensible way by any uncertain acceptance rule that tracks Bayesian conditioning. Finally, we present a plausible, alternative approach that tracks Bayesian conditioning and avoids all of the paradoxes. It combines an odds-based acceptance rule proposed originally by Levi (1996) with a non-AGM belief revision method proposed originally by Shoham (1987).     

Knowledge,safety, and Gettierized lottery cases: Why mere statistical evidence is not a (safe) source of knowledge

The lottery problem is the problem of explaining why mere reflection on the long odds that one will lose the lottery does not yield knowledge that one will lose. More generally, it is the problem of explaining why true beliefs merely formed on the basis of statistical evidence do not amount to knowledge. Some have thought that the lottery problem can be solved by appeal to a violation of the safety principle for knowledge, i.e., the principle that if S knows that p, not easily would S have believed that p without p being the case. Against the standard safety‐based solution, I argue that understanding safe belief as belief that directly covaries with the truth of what is believed in a suitably defined set of possible worlds forces safety theorists to make a series of theoretical choices that ultimately prevent a satisfactory solution to the problem. In this way, I analyze several safety principles that result from such choices—the paper thus gives valuable insights into the nature of safety—and explain why none solves the lottery problem, including their inability to explain away Gettierized lottery cases. On a more positive note, I show that there is a viable solution in terms of safety if we get rid of the unquestioned assumption that safe beliefs directly track the truth. The alternative is a conception of safe belief according to which what safe beliefs directly track is the appropriateness of the circumstances and, indirectly, the truth. The resulting safety principle, I argue, explains why mere statistical evidence is not a safe source of knowledge.     

Bridging Ranking Theory and the Stability Theory of Belief

In this paper we compare Leitgeb’s stability theory of belief (Annals of Pure and Applied Logic, 164:1338-1389, 2013; The Philosophical Review, 123:131-171, [2014]) and Spohn’s ranking-theoretic account of belief (Spohn, 1988, 2012). We discuss the two theories as solutions to the lottery paradox. To compare the two theories, we introduce a novel translation between ranking (mass) functions and probability (mass) functions. We draw some crucial consequences from this translation, in particular a new probabilistic belief notion. Based on this, we explore the logical relation between the two belief theories, showing that models of Leitgeb’s theory correspond to certain models of Spohn’s theory. The reverse is not true (or holds only under special constraints on the parameter of the translation). Finally, we discuss how these results raise new questions in belief theory. In particular, we raise the question whether stability (a key ingredient of Leitgeb’s theory) is rightly thought of as a property pertaining to belief (rather than to knowledge).     

Against global method safety

The global method safety account of knowledge states that an agent’s true belief that p is safe and qualifies as knowledge if and only if it is formed by method M, such that her beliefs in p and her beliefs in relevantly similar propositions formed by M in all nearby worlds are true. This paper argues that global method safety is too restrictive. First, the agent may not know relevantly similar propositions via M because the belief that p is the only possible outcome of M. Second, there are cases where there is a fine-grained belief that is unsafe and a relevantly similar coarse-grained belief (with looser truth conditions) that is safe and where both beliefs are based on the same method M. Third, the reliability of conditional reasoning, a basic belief-forming method, seems to be sensitive to fine-grained contents, as suggested by the wide variation in success rates for thematic versions of the Wason selection task.

     

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.     

Two Facets of Belief

Abstract

I begin by contrasting two facets of belief: that belief is a response to a sufficiency of evidence and that belief plays a role in one’s representation of reality. I claim that these conceptions of belief are in tension because whilst the latter – Representationalism – requires Logical Coherence of belief the former – Thresholdism – conflicts with Logical Coherence. Thus we need to choose between conceptions. Many have argued that the Preface Paradox supports Thresholdism. In contrast I argue that Representationalism has a more plausible response to the paradox.     

No Justificatory Closure without Truth

It is well-known that versions of the lottery paradox and of the preface paradox show that the following three principles are jointly inconsistent: (Sufficiency) very probable propositions are justifiably believable; (Conjunction Closure) justified believability is closed under conjunction introduction; (No Contradictions) propositions known to be contradictory are not justifiably believable. This paper shows that there is a hybrid of the lottery and preface paradoxes that does not require Sufficiency to arise, but only Conjunction Closure and No Contradictions; and it argues that, given any plausible solution to this paradox, if one is not ready to deny Conjunction Closure (and analogous consistency principles), then one must endorse the thesis that justified believability is factive.     

A solution to the discursive dilemma

An impossibility result pertaining to the aggregation of individual judgements is thought by many to have significant implications for political theory, social epistemology and metaphysics. When members of a group hold a rational set of judgments on some interconnected questions, the theorem shows, it isn’t always (logically) possible for them to aggregate their judgements into a collective one in conformity with seemingly very plausible constraints. I reject one of the constraints which engender the dilemma. The analogy with the lottery paradox, I argue, shows that rational belief needn’t be consistent. So the alleged implications of the dilemma are dispelled.     

Platonic Knowledge and the Standard Analysis

Abstract

In this paper I explore Plato’s reasons for his rejection of the so‐called standard analysis of knowledge as justified true belief. I argue that Plato held that knowledge is an infallible mental state in which (a) the knowable is present in the knower and (b) the knower is aware of this presence. Accordingly, knowledge (epistēmē) is non‐propositional. Since there are no infallible belief states, the standard analysis, which assumes that knowledge is a type of belief, cannot be correct. In addition, I argue that Plato held that belief (doxa) is only possible for the sort of being capable of knowledge. This is because self‐reflexivity is necessary for infallible knowledge and self‐reflexivity is only possible if the intellect is immaterial. This capacity for self‐reflexivity is also essential for belief, since beliefs are, paradigmatically, not dispositions but self‐reflexive mental states.     

Warrant <Emphasis Type="Italic">does</Emphasis> entail truth

Let ‘warrant’ denote whatever precisely it is that makes the difference between knowledge and mere true belief. A current debate in epistemology asks whether warrant entails truth, i.e., whether (Infallibilism) S’s belief that p is warranted only if p is true. The arguments for infallibilism have come under considerable and, as of yet, unanswered objections. In this paper, I will defend infallibilism. In Part I, I advance a new argument for infallibilism; the basic outline is as follows. Suppose fallibilism is true. An implication of fallibilism is that the property that makes the difference between knowledge and mere belief (which I dub ‘warrant*’) is the conjunctive property being warranted and true. I show that this implication of fallibilism conflicts with an uncontroversial thesis we have learned from reflection on Gettier cases: that nonaccidental truth is a constituent of warrant*. It follows that infallibilism is true. In the second part of the paper, I present and criticize a new argument against infallibilism. The argument states that there are plausible cases where, intuitively, the only thing that is keeping a belief from counting as knowledge is the falsity of that belief. Furthermore, it is plausible that such a belief is warranted and false. So, the argument goes, infallibilism is false. I show that this argument fails.     

Six Levels of Mentality

Abstract

Examination of recent debates about belief shows the need to distinguish:

• (a)?non–linguistic informational states in animal perception;

• (b)?the uncritical use of language, e.g. by children;

• (c)?adult humans' reasoned judgments.

If we also distinguish between mind-directed and object–directed mental states, we have: 1. Perceptual ‘beliefs’ of animals and infants about their material environment.

2. ‘Beliefs’ of animals and infants about the mental states of others.

3. Linguistically-expressible beliefs about the world, resulting from e.g. the uncritical tendency to believe what we are told.

4. Uncritically-formed beliefs about the mental states.

5. Beliefs about the material world arrived at by the weighing of evidence.

6. Beliefs about mental states formed by critical assessment.

     

Measuring Inconsistency

I provide a method of measuring the inconsistency of a set of sentences – from 1-consistency, corresponding to complete consistency, to 0-consistency, corresponding to the explicit presence of a contradiction. Using this notion to analyze the lottery paradox, one can see that the set of sentences capturing the paradox has a high degree of consistency (assuming, of course, a sufficiently large lottery). The measure of consistency, however, is not limited to paradoxes. I also provide results for general sets of sentences.     

The epistemic virtues of consistency

The lottery paradox has been discussed widely. The standard solution to the lottery paradox is that a ticket holder is justified in believing each ticket will lose but the ticket holder is also justified in believing not all of the tickets will lose. If the standard solution is true, then we get the paradoxical result that it is possible for a person to have a justified set of beliefs that she knows is inconsistent. In this paper, I argue that the best solution to the paradox is that a ticket holder is not justified in believing any of the tickets are losers. My solution avoids the paradoxical result of the standard solution. The solution I defend has been hastily rejected by other philosophers because it appears to lead to skepticism. I defend my solution from the threat of skepticism and give two arguments in favor of my conclusion that the ticket holder in the original lottery case is not justified in believing that his ticket will lose.     

A Pragmatic Dissolution of Harman's Paradox

There is widespread agreement that we cannot know of a lottery ticket we own that it is a loser prior to the drawing of the lottery. At the same time we appear to have knowledge of events that will occur only if our ticket is a loser. Supposing any plausible closure principle for knowledge, the foregoing seems to yield a paradox. Appealing to some broadly Gricean insights, the present paper argues that this paradox is apparent only.     

Self-Knowledge, Rationality and Moore's Paradox

I offer a model of self-knowledge that provides a solution to Moore's paradox. First, I distinguish two versions of the paradox and I discuss two approaches to it, neither of which solves both versions of the paradox. Next, I propose a model of self-knowledge according to which, when I have a certain belief, I form the higher-order belief that I have it on the basis of the very evidence that grounds my first-order belief. Then, I argue that the model in question can account for both versions of Moore's paradox. Moore's paradox, I conclude, tells us something about our conceptions of rationality and self-knowledge. For it teaches us that we take it to be constitutive of being rational that one can have privileged access to one's own mind and it reveals that having privileged access to one's own mind is a matter of forming first-order beliefs and corresponding second-order beliefs on the same basis.     

A geo-logical solution to the lottery paradox, with applications to conditional logic

We defend a set of acceptance rules that avoids the lottery paradox, that is closed under classical entailment, and that accepts uncertain propositions without ad hoc restrictions. We show that the rules we recommend provide a semantics that validates exactly Adams?? conditional logic and are exactly the rules that preserve a natural, logical structure over probabilistic credal states that we call probalogic. To motivate probalogic, we first expand classical logic to geo-logic, which fills the entire unit cube, and then we project the upper surfaces of the geo-logical cube onto the plane of probabilistic credal states by means of standard, linear perspective, which may be interpreted as an extension of the classical principle of indifference. Finally, we apply the geometrical/logical methods developed in the paper to prove a series of trivialization theorems against question-invariance as a constraint on acceptance rules and against rational monotonicity as an axiom of conditional logic in situations of uncertainty.     

The Paradox of Belief Instability and a Revision Theory of Belief

The epistemic paradox of 'belief instability' has recently received notable attention from many philosophers. In this paper I offer a new proposal, which I call a 'revision theory of belief'. This theory is in many respects an application of Gupta's and Belnap's revision theory of truth. They argue that the Liar paradox arises because our notion of truth is circular. I offer a similar proposal for handling the paradox of belief instability. In particular, I argue that our notion involved in the paradox of belief instability is circular, and this circularity of belief is the source of the paradox.     

Methodological and theoretical improvements in the study of superstitious beliefs and behaviour

Via four studies (N = 901), we developed an improved Belief in Superstition Scale (BSS) composed of three distinct components (belief in bad luck, belief in good luck, and the belief that luck can be changed), whose structure was supported through exploratory (Study 1) and confirmatory (Studies 2 and 3) factor analyses using divergent samples. We found that among theoretical predictors, higher ‘chance’ locus of control (i.e., the belief that chance/fate controls one's life) best predicted all three BSS subscales (Studies 2–3). In Study 3, we found that BSS subscale scores were reliable, but largely invariant across age and education with a non‐general psychology sample. In Study 4, the BSS subscales best predicted participants’ superstitious attitudes and behaviour in a new lottery drawing paradigm among other commonly used superstition scales. Taken together, our results indicate that the BSS is a valuable addition to the burgeoning research on superstitious attitudes and behaviour.     

consulting the oracle

Abstract

Is it possible that even logic is not logical? Is it possible that our current belief in a rational and material world is no less a subjective point of view than any other mystical faith? Kurt Gödel in mathematics and Carl Jung in psychology independently established the limits of rationalism half a century ago. Yet the profound implications of their work for our own currently changing world view are not well understood. In this essay, Robertson introduces us to some of the basic themes that underlie the life works of both Gödel and Jung. These themes lead us to a new and startling comprehension of the paradox of self-reference and the limits of rationalism that are inherent in the way we construct our conceptions of reality.     

彩民命运控制对问题购彩的影响:购彩预期和购彩意向的链式中介作用

为考察命运控制对彩民问题购彩的影响,以及购彩预期和购彩意向在其中的链式中介作用,本研究采用问卷法对2538名彩民进行调查。结果发现:(1)彩民问题购彩与命运控制、购彩预期和购彩意向均存在正相关;(2)命运控制不仅能直接预测问题购彩,还能通过两条路径的间接作用影响彩民的问题购彩:购彩意向的中介作用;购彩预期和购彩意向的链式中介作用。本研究构建的中介效应模型在一定程度上揭示了命运控制影响彩民问题购彩的内在机制。