Justification does not aggregate: de-generalizing the lottery paradox

Abstract Preface and Lottery paradoxes have shown that competently deducing a novel conclusion from multiple justified premises does not necessarily result in a justified belief. However, those paradoxes do not arise if only a single premise is used. Yet, in recent years, justification closure for single premise deduction has also been challenged. Schechter has presented a case that, according to him, thwarts any attempt to formulate a general principle that links competent single premise deduction to justification. This paper develops a justification closure principle for single premise deduction that accommodates Schechter’s case, thus showing that his pessimism regarding the possibility of such principles is mistaken. I also argue that an alternative reply to Schechter’s case fails and that safeguarding justification closure constitutes progress for establishing knowledge closure for single premise deduction.

Abstract Ranking Theory has been developed as an alternative to subjec- tive probability for modelling partial belief. It goes further than probability in that it jointly models both full and partial belief. One selling feature is that the Lottery Paradox does not arise in Ranking Theory. However, all is not well when it comes to modelling lottery beliefs with ranks. In this paper, I show how lotteries create problems for Ranking Theory.

In his Tractatus, Wittgenstein proposed a method for calculating probability using truth tables, which served as inspiration for Carnap and Ramsey's work on probability. Despite this, Wittgenstein's idea was not widely considered in the literature. This method involves comparing two propositions, where the first is considered only in true instances, while the other is analyzed only when the first is true. This approach is not dissimilar from Makinson's supraclassical logic, despite the use of different methods. The aim of this work is to shed light on Wittgenstein's method, exploring its foundational aspects and demonstrating the relationship between Wittgenstein's probability and Makinson's supraclassical logic. By doing so, we argue that Wittgenstein anticipated some modern developments in logic, proposing one of the earliest systems capable of incorporating beliefs within a formal calculus. In the final section, we will discuss how Wittgenstein's approach resolves (or, better, dissolves) the Lottery Paradox, showing that within this framework, the paradox ceases to exist.

Abstract This paper generalizes the preface paradox beyond the conjunctive aggregation of beliefs and constructs an analogous paradox for deontic reasoning. The analysis of the deontic case suggests a systematic restriction of intuitive rules for reasoning with obligations. This proposal can be transferred to the epistemic case: It avoids the preface and the lottery paradox and saves one of the two directions of the Lockean Thesis (i.e. high credence is sufficient, but not necessary for rational belief). The resulting account compares favorably to competing proposals; in particular, we can formulate the rules of correct doxastic reasoning without reference to probabilistic features of the involved propositions.

This paper fills a gap in the existing metaphilosophical research on paradoxes by focusing on the role of scenarios. Typical philosophical paradoxes contain a scenario description whose contribution to paradoxes remains unexplored. I argue that scenarios are examples or instantiations of the abstract schema of paradoxes. As such, scenarios contribute to paradoxes on two levels. First, they make the argument more concrete, thus enhancing the dialectical force of paradoxes and facilitating their understanding, especially for non-experts. This function is external to the paradox itself, but has important practical implications for the use of paradoxes and their effect on philosophical debates, and it contributes to explaining why philosophical paradoxes are usually introduced by a scenario. Second, and more crucially, scenarios are essential to the epistemic dimension of paradoxes. By definition, paradoxes have two necessary components: the argumentative structure and the plausibility/implausibility of the premises/conclusion. By providing examples of the abstract schema, scenarios contribute to making the premises plausible. In particular, scenarios are the source of plausibility and justification for those premises that contain an empirically grounded assertion of existence. Examples of such paradoxes are the Sorites paradox, the Lottery paradox, and the Grue paradox. Contrary to the dialectical role, the epistemic function of scenarios is indispensable, as it connects paradoxes to the real world and underscores their significance in specific debates.

Most of the literature surrounding virtue reliabilism revolves around issues pertaining to the analysis of knowledge. With the exception of the lottery paradox, virtue reliabilists have paid relatively little attention to classic epistemological paradoxes, such as Moore’s paradox. This is a significant omission given how central role such paradoxes have in epistemic theorizing. In this essay I take a step towards remedying this shortcoming by providing a solution to Moore’s paradox. The solution that I offer stems directly from the core of virtue reliabilism.

"Lotteries, Knowledge, and Rational Belief: Essays on the Lottery Paradox." Australasian Journal of Philosophy, ahead-of-print(ahead-of-print), pp. 1–2

Abstract Ever since Fred Dretske (1970) questioned closure, a denial of this principle has been among the standard options for a resolution of epistemological paradoxes such as the skeptical paradox (Cohen 1988) and the lottery paradox (Harman 1973). In this article, the author shall argue that all possible solutions of the latter paradox can only be defended if Multi-Premise Closure is rejected. These possible solutions are contextualism and both simple and sensitive moderate invariantism. It will be shown that skepticism and the denial of Single-Premise Closure are not possible solutions of the lottery paradox. The upshot of the discussion here will be that while Single-Premise Closure is beyond reasonable doubt, resolving the lottery paradox forces one to abandon Multi-Premise Closure.

Abstract To resolve the lottery paradox, the “no-justification account” proposes that one is not justified in believing that one's lottery ticket is a loser. The no-justification account commits to what I call “the Harman-style skepticism”. In reply, proponents of the no-justification account typically downplay the Harman-style skepticism. In this paper, I argue that the no-justification reply to the Harman-style skepticism is untenable. Moreover, I argue that the no-justification account is epistemically ad hoc. My arguments are based on a rather surprising finding that the no-justification account implies that people living in Taiwan typically suffer from the Harman-style skepticism.

I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call “the Cut-off Point Problem” and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburg’s puzzle.Very briefly, I show that, in the Lottery Paradox, the premises “ticket n°1 will lose”, “ticket n°2 will lose”… “ticket n°1000 will lose” are equivalent to soritical premises of the form “~(the winning ticket is in {…, (tn)}) ⊃ ~(the winning ticket is in {…, tn, (tn + 1)})” (where “⊃” is the material conditional, “~” is the negation symbol, “tn” and “tn + 1” are “ticket n°n” and “ticket n°n + 1” respectively, and “{}” identify the elements of the lottery tickets’ set. The brackets in “(tn)” and “(tn + 1)” are meant to point out that in the antecedent of the conditional we do not always have a “tn” (and, as a result, a “tn + 1” in the consequent): consider the conditional “~(the winning ticket is in {}) ⊃ ~(the winning ticket is in {t1})”).As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises.In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions.One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox.

In a fair, infinite lottery, it is possible to conclude that drawing a number divisible by four is strictly less likely than drawing an even number; and, with apparently equal cogency, that drawing a number divisible by four is equally as likely as drawing an even number.

Statistical evidence—say, that 95% of your co-workers badmouth each other—can never render resenting your colleague appropriate, in the way that other evidence (say, the testimony of a reliable friend) can. The problem of statistical resentment is to explain why. We put the problem of statistical resentment in several wider contexts: The context of the problem of statistical evidence in legal theory; the epistemological context—with problems like the lottery paradox for knowledge, epistemic impurism and doxastic wrongdoing; and the context of a wider set of examples of responses and attitudes that seem not to be appropriately groundable in statistical evidence. Regrettably, we do not come up with a fully general, fully adequate, fully unified account of all the phenomena discussed. But we give reasons to believe that no such account is forthcoming, and we sketch a somewhat messier account that may be the best that can be had here.

If a reliable friend tells you that a colleague has been badmouthing you, it may be appropriate to resent that colleague. Without reliable evidence, such evidence would not be appropriate, even if in fact the colleague has been badmouthing you. There is room, then, for the evidence law of resentment (and of morality). In this paper, we address one part of this evidence law of morality: Statistical evidence – say, that 95% of your co-workers badmouth each other – can never render resenting your colleague appropriate. The problem of statistical resentment is to explain why. In this paper we put the problem of statistical resentment in several wider contexts: The context of the problem of statistical evidence in legal theory (including the problem that statistical resentment poses for our own past treatment of this topic); the epistemological context – with problems like the lottery paradox for knowledge, epistemic impurism and doxastic wrongdoing; and the context of a wider set of examples of responses and attitudes that seem not to be appropriately groundable in statistical evidence. Regrettably, we do not come up with a fully general, fully adequate, fully unified account of all the phenomena discussed. But we give reasons to believe that no such account is forthcoming, and we sketch a somewhat messier account that may be the best that can be had here.

The main aim of this paper is to analyse David Lewis’ version of contextualism and his solution to the Gettier problem and the lottery problem through the employment of his Rule of Relevance and Stewart Cohen’s response to these problems. Here I analyse whether Stewart Cohen’s response to David Lewis’ solutions to these problems is on the right track or not. Hence, I try to analyse some concept in David Lewis and Stewart Cohen which has remained unanalysed. Cohen tries to show that when we try to solve some variation of the lottery problem and the Gettier problem by applying Lewis’s Rule of Relevance, then it generates some counterintuitive result. So Cohen gives Lewis some alternatives (which are explained in “The Strategy of Biting the Bullet and the Problem of Interference” and “Biting the Bullet Strategy as a Natural Extension of Contextualist Resolution to the Sceptical Problem and the Pity Poor Bill Variation of the Lottery Problem” sections) to avoid this counterintuitive result; this attempt, however, affects some other presuppositions of the contextual theory of David Lewis. My aim in this paper is to show how without taking these alternatives suggested by Cohen, Lewis can apply his Rules of Relevance to solve the lottery problem and the Gettier problem without any counterintuitive result.

Abstract Several authors have investigated the question of whether canonical logic-based accounts of belief revision, and especially the theory of AGM revision operators, are compatible with the dynamics of Bayesian conditioning. Here we show that Leitgeb’s stability rule for acceptance, which has been offered as a possible solution to the Lottery paradox, allows to bridge AGM revision and Bayesian update: using the stability rule, we prove that AGM revision operators emerge from Bayesian conditioning by an application of the principle of maximum entropy. In situations of information loss, or whenever the agent relies on a qualitative description of her information state—such as a plausibility ranking over hypotheses, or a belief set—the dynamics of AGM belief revision are compatible with Bayesian conditioning; indeed, through the maximum entropy principle, conditioning naturally generates AGM revision operators. This mitigates an impossibility theorem of Lin and Kelly for tracking Bayesian conditioning with AGM revision, and suggests an approach to the compatibility problem that highlights the information loss incurred by acceptance rules in passing from probabilistic to qualitative representations of belief.

Abstract Many theorists hold that outright verdicts based on bare statistical evidence are unwarranted. Bare statistical evidence may support high credence, on these views, but does not support outright belief or legal verdicts of culpability. The vignettes that constitute the lottery paradox and the proof paradox are marshalled to support this claim. Some theorists argue, furthermore, that examples of profiling also indicate that bare statistical evidence is insufficient for warranting outright verdicts.I examine Pritchard's and Buchak's treatments of these three kinds of case. Pritchard argues that his safety condition explains the insufficiency of bare statistical evidence for outright verdicts in each of the three cases, while Buchak argues that her treatment of the distinction between credence and belief explains this. In these discussions the three kinds of cases – lottery, proof paradox, and profiling – are treated alike. The cases are taken to exhibit the same epistemic features. I identity significant overlooked epistemic differences amongst these three cases; these differences cast doubt on Pritchard's explanation of the insufficiency of bare statistical evidence for outright verdicts. Finally, I raise the question of whether we should aim for a unified explanation of the three paradoxes.

Suppose that I hold a ticket in a fair lottery and that I believe that my ticket will lose [L] on the basis of its extremely high probability of losing. What is the appropriate epistemic appraisal of me and my belief that L? Am I justified in believing that L? Do I know that L? While there is disagreement among epistemologists over whether or not I am justified in believing that L, there is widespread agreement that I do not know that L. I defend the two-pronged view that I am justified in believing that my ticket will lose and that I know that it will lose. Along the way, I discuss four different but related versions of the lottery paradox—The Paradox for Rationality, The Paradox for Knowledge, The Paradox for Fallibilism, and The Paradox for Epistemic Closure—and offer a unified resolution of each of them.

Abstract We present a probabilistic justification logic, $\mathsf{PPJ}$, as a framework for uncertain reasoning about rational belief, degrees of belief and justifications. We establish soundness and strong completeness for $\mathsf{PPJ}$ with respect to the class of so-called measurable Kripke-like models and show that the satisfiability problem is decidable. We discuss how $\mathsf{PPJ}$ provides insight into the well-known lottery paradox.

This paper elaborates a new solution to the lottery paradox, according to which the paradox arises only when we lump together two distinct states of being confident that p under one general label of ‘belief that p’. The two-state conjecture is defended on the basis of some recent work on gradable adjectives. The conjecture is supported by independent considerations from the impossibility of constructing the lottery paradox both for risk-tolerating states such as being afraid, hoping or hypothesizing, and for risk-averse, certainty-like states. The new proposal is compared to views within the increasingly popular debate opposing dualists to reductionists with respect to the relation between belief and degrees of belief.