The Paradox of the Ravens (a.k.a,, The Paradox of Confirmation) is indeed an old chestnut. A great many things have been written and said about this paradox and its implications for the logic of evidential support. The first part of this paper will provide a brief survey of the early history of the paradox. This will include the original formulation of the paradox and the early responses of Hempel, Goodman, and Quine. The second part (...) of the paper will describe attempts to resolve the paradox within a Bayesian framework, and show how to improve upon them. This part begins with a discussion of how probabilistic methods can help to clarify the statement of the paradox itself. And it describes some of the early responses to probabilistic explications. We then inspect the assumptions employed by traditional (canonical) Bayesian approaches to the paradox. These assumptions may appear to be overly strong. So, drawing on weaker assumptions, we formulate a new-and-improved Bayesian confirmation-theoretic resolution of the Paradox of the Ravens. (shrink)

In the Paradox of the Ravens, a set of otherwise intuitive claims about evidence seems to be inconsistent. Most attempts at answering the paradox involve rejecting a member of the set, which seems to require a conflict either with commonsense intuitions or with some of our best confirmation theories. In contrast, I argue that the appearance of an inconsistency is misleading: ‘confirms’ and cognate terms feature a significant ambiguity when applied to universal generalisations. In particular, the claim (...) that some evidence confirms a universal generalisation ordinarily suggests, in part, that the evidence confirms the reliability of predicting that something which satisfies the antecedent will also satisfy the consequent. I distinguish between the familiar relation of confirmation simpliciter and what I shall call ‘predictive confirmation’. I use them to formulate my answer, illustrate it in a very simple probabilistic model, and defend it against objections. I conclude that, once our evidential concepts are sufficiently clarified, there is no sense in which the initial claims are both plausible and inconsistent. (shrink)

Philosophers such as Goodman, Scheffler and Glymour aim to answer the Paradox of the Ravens by distinguishing between confirmation simpliciter and selective confirmation. In the latter concept, the evidence both supports a hypothesis and undermines one of its "rivals". In this article, I argue that while selective confirmation does seem to be an important scientific notion, no attempt to formalise it thus far has managed to solve the Paradox of the Ravens.

The Paradox of the Ravens (a.k.a,, The Paradox of Confirmation) is indeed an old chestnut. A great many things have been written and said about this paradox and its implications for the logic of evidential support. The first part of this paper will provide a brief survey of the early history of the paradox. This will include the original formulation of the paradox and the early responses of Hempel, Goodman, and Quine. The second part (...) of the paper will describe attempts to resolve the paradox within a Bayesian framework, and show how to improve upon them. This part begins with a discussion of how probabilistic methods can help to clarify the statement of the paradox itself. And it describes some of the early responses to probabilistic explications. We then inspect the assumptions employed by traditional (canonical) Bayesian approaches to the paradox. These assumptions may appear to be overly strong. So, drawing on weaker assumptions, we formulate a new-and-improved Bayesian confirmation-theoretic resolution of the Paradox of the Ravens. (shrink)

Hempel's famous Raven Paradox derives from Nicod's criteria for confirmation and the Equivalence Condition, the unintuitive conclusion that things like white roses, green T-shirts and ice cubes confirm the raven hypothesis ‘All ravens are black.’ By a small rearrangement of the Equivalence Condition, I show that we can also derive the conclusion, which sounds even more intuitively intolerable, that observation of black ravens fails to confirm the raven hypothesis. We are left with the contradictory result that black (...) ravens both confirm and do not confirm the raven hypothesis. (shrink)

The canonical Bayesian solution to the ravens paradox faces a problem: it entails that black non-ravens disconfirm the hypothesis that all ravens are black. I provide a new solution that avoids this problem. On my solution, black ravens confirm that all ravens are black, while non-black non-ravens and black non-ravens are neutral. My approach is grounded in certain relations of epistemic dependence, which, in turn, are grounded in the fact that the kind (...) raven is more natural than the kind black. The solution applies to any generalization “All F’s are G” in which F is more natural than G. (shrink)

Think of confirmation in the context of the Ravens Paradox this way. The likelihood ratio measure of incremental confirmation gives us, for an observed Black Raven and for an observed non-Black non-Raven, respectively, the following “full” likelihood ratios.

Hempel's paradox of the ravens arises from the inconsistency of three prima facie plausible principles of confirmation. This paper uses Carnapian inductive logic to (a) identify which of the principles is false, (b) give insight into why this principle is false, and (c) identify a true principle that is sufficiently similar to the false one that failure to distinguish the two might explain why the false principle is prima facie plausible. This solution to the paradox is compared (...) with a variety of other responses and is shown to differ from all of them. (shrink)

The discussion about the Raven Paradox is ever-renewing: after nearly 70 years, many authors propose from time to time new solutions, and many authors state that these solutions are unsatisfactory. It is worthy to be carefully noted that though most arguments in favor or against the paradox are based on the notion of “probability” and on the application of Bayes’ law, not one of them makes use of the Kolmogorov axiomatic theory of probability and on the subsequent notion (...) of “random variable”. This seems to be due to a preference for the purely logical interpretation of the notion of “probability” which makes it possible to assign a probability value to propositions expressing natural (universal) laws. This paper aims at presenting an attempt to contribute a solution to the paradox by the proposal of a different way to express universal natural laws and by the use of a language which does not allow one to apply probability to universal statements. (shrink)

Simple random sampling resolutions of the raven paradox relevantly diverge from scientific practice. We develop a stratified random sampling model, yielding a better fit and apparently rehabilitating simple random sampling as a legitimate idealization. However, neither accommodates a second concern, the objection from potential bias. We develop a third model that crucially invokes causal considerations, yielding a novel resolution that handles both concerns. This approach resembles Inference to the Best Explanation (IBE) and relates the generalization’s confirmation to confirmation of (...) an associated law. We give it an objective Bayesian formalization and discuss the compatibility of Bayesianism and IBE. (shrink)

"All ravens are black" is logically but not confirmationally equivalent with "all non-black things are non-ravens." But this is impossible, given that logical equivalence guarantees confirmational equivalence. In this paper, this paradox is solved.

In this paper, I provide a new argument in support of a concessive response to the Ravens Paradox. The argument I offer stems from Mark Schroeder's Gricean explanation for why existential judgments about normative reasons for action are unreliable. In short, I argue that Schroeder's work suggests that, in the case of the Ravens Paradox, people are running together the issue of what's assertible about evidence with what's true about evidence. Once these issues are pulled apart, (...) we have reason to think that the negative existential judgment about evidence that drives the Ravens Paradox is mistaken, and thus that there is in fact no paradox here at all. (shrink)

I argue that the standard Bayesian solution to the ravens paradox— generally accepted as the most successful solution to the paradox—is insufficiently general. I give an instance of the paradox which is not solved by the standard Bayesian solution. I defend a new, more general solution, which is compatible with the Bayesian account of confirmation. As a solution to the paradox, I argue that the ravens hypothesis ought not to be held equivalent to its (...) contrapositive; more interestingly, I argue that how we formally represent hypotheses ought to vary with the context of inquiry. This explains why the paradox is compelling, while dealing with standard objections to holding hypotheses inequivalent to their contrapositives. (shrink)

Reflecting the influence of both Bayesian and Popperian methodology, this paper presents an interpersonal view of confirmation which: (i) explains why a non-black non-raven may constitute evidence for and a black non-raven fails to constitute evidence for the hypothesis 'All ravens are black;' (ii) accounts for the common sense belief that black ravens confirm the raven hypothesis more than do non-black non-ravens; and (iii) applies to lawlike sentences understood as subjunctive conditionals as well as to accidental generalizations.

Hempel‘s paradox of the ravens, and his take on it, are meant to be understood as being restricted to situations where we have no additional background information. According to him, in the absence of any such information, observations of FGs confirm the hypothesis that all Fs are G. In this paper I argue against this principle by way of considering two other paradoxes of confirmation, Goodman‘s 'grue‘ paradox and the 'tacking‘ (or 'irrelevant conjunct‘) paradox. What these (...) paradoxes reveal, I argue, is that a presumption of causal realism is required to ground any confirmation; but once we grant causal realism, we have no reason to accept the central principles giving rise to the paradoxes. (shrink)

In this paper, I critically analyse two strands of Bayesian solution to the paradox: the standard Bayesian solution and the attempts to refute Nicod’s criterion. I argue that the standard Bayesian solution evades the exact challenge of the paradox. I hold that though the NC or instance confirmation is imprecisely formulated, it cannot be ruled out as an invalid form of confirmation. I formulate three conditions of instance confirmation which sufficiently captures our intuitive notion of instance confirmation. Finally (...) on the basis of the conditions of instance confirmation, I show that paradoxical contrapositive instances like white shoe are not contrapositive instances of the raven hypothesis. (shrink)

This chapter advances a contextual view of evidence, through a reconsideration of Hempel's paradox of confirmation. The initial view regarding Hempel's paradox is that a non-black non-raven does confirm ‘All ravens are black’, but only in certain contexts. The chapter begins by reformulating the paradox as a puzzle about how the same entity can have variable evidential values for a given proposition. It then offers a three-stage solution to the reformulated paradox. The situation makes better (...) sense when we reach a deeper propositional understanding of evidence, recognising that each entity can be represented in multiple observational propositions. Some anti-contextualist intuitions can be defused by distinguishing two different senses of the word ‘evidence’, one applying to objects or events and the other applying to propositions; only the latter is relevant to inference. A fuller understanding comes from analysing the constitution and use of evidence in terms of epistemic action. These reflections on the ravens paradox suggest a general philosophical framework more suitable for understanding the function of evidence in scientific and everyday practices. (shrink)

It is shown that the paradoxes of confirmation are closely linked to the paradoxes of material implication and that they can be avoided by formulating natural laws in terms of a genuine if-Connective rather than the material conditional. However, Natural laws so expressed are not confirmed by simple conjunctions. The question then is whether the common assumption that simple conjunctions do confirm universal generalizations is correct. The answer given is that it is not. In particular, A confirming proposition of the (...) form 'this is a black raven' is not equivalent to 'this is a raven and this is black'. (shrink)

In From Instrumentalism to Constructive Realism Theo Kuipers presents a theory of qualitative confirmation that is supposed to not assume the existence of quantitative probabilities. He claims that this theory is able to resolve some paradoxes in confirmation theory, including the ravens paradox. This paper shows that there are flaws in Kuipers' qualitative confirmation theory and in his application of it to the ravens paradox.

To solve the highly counterintuitive paradox of confirmation represented by the statement, “A pair of red shoes confirms that all ravens are black,” Hempel employed a strategy that retained the equivalence condition but abandoned Nicod’s irrelevance condition. However, his use of the equivalence condition is fairly ad hoc, raising doubts about its applicability to this problem. Furthermore, applying the irrelevance condition from Nicod’s criterion does not necessarily lead to paradoxes, nor does discarding it prevent the emergence of paradoxes. (...) Hempel’s approach fails to adequately resolve the paradox. (shrink)

Bayesians regard their solution to the paradox of confirmation as grounds for preferring their theory of confirmation to Hempel’s. They point out that, unlike Hempel, they can at least say that a black raven confirms “All ravens are black” more than a white shoe. However, I argue that this alleged advantage is cancelled out by the fact that Bayesians are equally committed to the view that a white shoe confirms “All non-black things are non-ravens” less than a (...) black raven. In light of this, I reexamine the dialectic between Hempel and the Bayesians. (shrink)

1. Confirmation Theory One, which we shall construct, when applied to the Raven Hypothesis yields the following results: Any black raven confirms the Raven Hypothesis.Any black non-raven confirms the Raven Hypothesis.Any non-black raven disconfirms the Raven Hypothesis.Any non-black non-raven is neutral to the Raven Hypothesis.Theory One consists of two parts: six basic concepts from confirmation theory proper, and the underlying logic.

Abstract W. E. Johnson argued that by taking into account both the epistemic and constitutive conditions for using arguments in inferences one could dissolve the paradoxes of material implication. This essay argues that the same sort of consideration can be used to dissolve the paradox of ravens in confirmation theory. It is argued in particular, and in agreement with certain points raised by the Popperians, that those instances of a generalization which are verifying but apparently not confirming cannot (...) raise the posterior of that generalization. (shrink)

Hempel's paradox of the ravens has to do with the question of what constitutes confirmation from a logical point of view; Wason 's selection task has been used extensively to investigate how people go about attempting to confirm or disconfirm conditional claims. This paper presents an argument that the paradox is resolved, and that people's typical performance in the selection task can be explained, by consideration of what constitutes an effective strategy for seeking evidence of the tenability (...) of universal or conditional claims in everyday life. (shrink)

The paper provides a formal proof that efficient estimates of parameters, which vary as as little as possible when measurements are repeated, may be expected to provide more accurate predictions. The definition of predictive accuracy is motivated by the work of Akaike (1973). Surprisingly, the same explanation provides a novel solution for a well known problem for standard theories of scientific confirmation — the Ravens Paradox. This is significant in light of the fact that standard Bayesian analyses of (...) the paradox fail to account for the predictive utility of universal laws like All ravens are black. (shrink)

Historically, Nelson Goodman’s paradox involving the predicates ‘grue’ and ‘bleen’ has been taken to furnish a serious blow to Carl Hempel’s theory of confirmation in particular and to purely formal theories of confirmation in general. In this paper, I argue that Goodman’s paradox is no more serious of a threat to Hempel’s theory of confirmation than is Hempel’s own paradox of the ravens. I proceed by developing a suggestion from R. D. Rosenkrantz into an argument for (...) the conclusion that these paradoxes are, in fact, equivalent. My argument, if successful, is of both historical and philosophical interest. Goodman himself maintained that Hempel’s theory of confirmation was capable of handling the paradox of the ravens. And Hempel eventually conceded that Goodman’s paradox showed that there could be no adequate, purely syntactical theory of confirmation. The conclusion of my argument entails, by contrast, that Hempel’s theory of confirmation is incapable of handling Goodman’s paradox if and only if it is incapable of handling the paradox of the ravens. It also entails that for any adequate solution to one of these paradoxes, there is a corresponding and equally adequate solution to the other. (shrink)

(This is for the Cambridge Handbook of Analytic Philosophy, edited by Marcus Rossberg) In this handbook entry, I survey the different ways in which formal mathematical methods have been applied to philosophical questions throughout the history of analytic philosophy. I consider: formalization in symbolic logic, with examples such as Aquinas’ third way and Anselm’s ontological argument; Bayesian confirmation theory, with examples such as the fine-tuning argument for God and the paradox of the ravens; foundations of mathematics, with examples (...) such as Hilbert’s programme and Gödel’s incompleteness theorems; social choice theory, with examples such as Condorcet’s paradox and Arrow’s theorem; ‘how possibly’ results, with examples such as Condorcet’s jury theorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology. (shrink)

According to Hempel's paradox, evidence (E) that an object is a nonblack nonraven confirms the hypothesis (H) that every raven is black. According to the standard Bayesian solution, E does confirm H but only to a minute degree. This solution relies on the almost never explicitly defended assumption that the probability of H should not be affected by evidence that an object is nonblack. I argue that this assumption is implausible, and I propose a way out for Bayesians. Introduction (...) Hempel's paradox, the standard Bayesian solution, and the disputed assumption Attempts to defend the disputed assumption Attempts to refute the disputed assumption A way out for Bayesians Conclusion. (shrink)

What is required for something to be evidence for a hypothesis? In this fascinating, elegantly written work, distinguished philosopher of science Peter Achinstein explores this question, rejecting typical philosophical and statistical theories of evidence. He claims these theories are much too weak to give scientists what they want--a good reason to believe--and, in some cases, they furnish concepts that mistakenly make all evidential claims a priori. Achinstein introduces four concepts of evidence, defines three of them by reference to "potential" evidence, (...) and characterizes the latter using a novel epistemic interpretation of probability. The resulting theory is then applied to philosophical and historical issues. Solutions are provided to the "grue," "ravens," "lottery," and "old-evidence" paradoxes, and to a series of questions. These include whether explanations or predictions furnish more evidential weight, whether individual hypotheses or entire theoretical systems can receive evidential support, what counts as a scientific discovery, and what sort of evidence is required for it. The historical questions include whether Jean Perrin had non-circular evidence for the existence of molecules, what type of evidence J. J. Thomson offered for the existence of the electron, and whether, as is usually supposed, he really discovered the electron. Achinstein proposes answers in terms of the concepts of evidence introduced. As the premier book in the fabulous new series Oxford Studies in Philosophy of Science, this volume is essential for philosophers of science and historians of science, as well as for statisticians, scientists with philosophical interests, and anyone curious about scientific reasoning. (shrink)

Hempel never met Ramsey, but he knew his work. In his 1958 The Theoretician’s Dilemma: a study in the logic of theory construction, Hempel introduces the term Ramsey sentence, referring to Ramsey’s attempt in Theories to get rid of theoretical terms in formal accounts of scientific theories. In this paper, I draw the attention to another connection between Ramsey’s and Hempel’s works. Hempel’s Deductive-Nomological (DN) account of scientific explanation resembles very closely Ramsey’s account of a certain type of conditional sentences. (...) In the first part of the paper, by introducing a fictional story, I highlight the similarities and differences between the two. In the last part of the paper, I claim that the most relevant difference between Ramsey and Hempel can be used to offer original solutions to Hempel’s Raven Paradox. Two possibilities are presented, arguing that the second, which requires a reconsideration of the formalisation of laws, is the most promising. (shrink)

In this essay I present a new version of the Paradox of the Knower and show that this new paradox vitiates a certain argument against epistemic closure. I then prove a theorem that relates the new paradox to epistemological scepticism. I conclude by assessing the use of the Knower in arguments against syntactical treatments of knowledge.

What is the ethical impact of artificial intelligence assistants on human lives, and specifically how much do they threaten our individual autonomy? Recently, as part of forming an ethical framework for thinking about the impact of AI assistants on our lives, John Danaher claims that if the external automaticity generated by the use of AI assistants threatens our autonomy and is therefore ethically problematic, then the internal automaticity we already live with should be viewed in the same way. He takes (...) advantage of this paradox of internal automaticity to downplay the threats of external automaticity to our autonomy. We respond in this paper by challenging the legitimacy of the paradox. While Danaher assumes that internal and external automaticity are roughly equivalent, we argue that there are reasons why we should accept a large degree of internal automaticity, that it is actually essential to our sense of autonomy, and as such it is ethically good; however, the same does not go for external automaticity. Therefore, the similarity between the two is not as powerful as the paradox presumes. In conclusion, we make practical recommendations for how to better manage the integration of AI assistants into society. (shrink)

Projectible conditions are (roughly) those whose universal generalizations are con firmed by their positive instances. This paper proposes certain modifications to the above definition in order to capture the pre-analytic notion it is supposed to explicate. Then we investigate what logical operations, when performed on projectible conditionals, yield new projectible conditionals. A number of surprising theorems are proven, and these theorems indicate that few conditionals having complex antecedents and consequents are projectible. It is also shown that projectibility is not closed (...) under contraposition, and this is proposed as a solution to the Paradox of the Ravens. Finally, a general conjecture is made concerning just what conditionals are projectible, and it follows from this conjecture that although most conditionals are not projectible, they can still be dealt with inductively, but in ways more complex than recognized by classical confirmation theory. (shrink)

Paradox Lost covers ten of philosophy’s most fascinating paradoxes, in which seemingly compelling reasoning leads to absurd conclusions. The following paradoxes are included: The Liar Paradox, in which a sentence says of itself that it is false. Is the sentence true or false? The Sorites Paradox, in which we imagine removing grains of sand one at a time from a heap of sand. Is there a particular grain whose removal converts the heap to a non-heap? The Puzzle (...) of the Self-Torturer, in which a series of seemingly rational choices has us accepting a life of excruciating pain, in exchange for millions of dollars. Newcomb’s Problem, in which we seemingly maximize our expected profit by taking an unknown sum of money, rather than taking the same sum plus $1000. The Surprise Quiz Paradox, in which a professor finds that it is impossible to give a surprise quiz on any particular day of the week... but also that if this is so, then a surprise quiz can be given on any day. The Two Envelope Paradox, in which we are asked to choose between two indistinguishable envelopes, and it is seemingly shown that each envelope is preferable to the other. The Ravens Paradox, in which observing a purple shoe provides evidence that all ravens are black. The Shooting Room Paradox, in which a deadly game kills 90% of all who play, yet each individual’s survival turns on the flip of a fair coin. Each paradox is clearly described, common mistakes are explored, and a clear, logical solution offered. Paradox Lost will appeal to professional philosophers, students of philosophy, and all who love intellectual puzzles. (shrink)

Philosophers are by now familiar with “the” paradox of supererogation. This paradox arises out of the idea that it can never be permissible to do something morally inferior to another available option, yet acts of supererogation seem to presuppose this. This paradox is not our topic in this paper. We mention it only to set it to one side and explain our subtitle. In this paper we introduce and explore another paradox of supererogation, one which also (...) deserves serious philosophical attention. People who perform paradigmatic acts of supererogation very often claim and believe that their acts were obligatory. Plausibly, this is simply a mistake insofar as the actions really are “above and beyond the call of duty,” as common sense would have it. The fact that moral heroes tend to view their actions in this apparently mistaken way is puzzling in itself, and we might learn something interesting about the moral psychology of such individuals if we could explain this tendency. However, this puzzling aspect of the moral psychology of moral heroes is also the chief ingredient in a deeper puzzle, one perhaps more worthy of the title “paradox.” In this paper we present and try to resolve this paradox. The paradox arises when we combine our initial observation about the moral psychology of moral heroes with three plausible claims about how these cases compare with one in which the agent realizes her act is “above and beyond.” The first of these three additional claims is that the agent who mistakenly claims that the act is obligatory is no less virtuous than someone who performs such an act whilst correctly judging it to be obligatory. The second is that the agent who makes such a mistake would display more moral wisdom if she judged the act to be supererogatory. The third is that there is no other relevant difference between the two agents. These three claims, together with a plausible principle about the way in which the virtues work, give rise to a paradox. We consider several ways in which this paradox might be resolved. We argue that the most plausible resolution is to reject the claim that there is no other relevant difference between the two agents. More specifically, we argue that a relevant difference is that the agent who makes this mistake does so because of the depth of their commitment to certain moral values, and that this is itself an important moral virtue: moral depth. (shrink)

According to Hempel’s raven paradox, the observation of one non-black non-raven confirms the hypothesis that all ravens are black. Bayesians such as Howson and Urbach (Scientific reasoning: the Bayesian approach, 2nd edn. Open Court, Chicago, 1993 ) claim that the raven paradox can be solved by spelling out the concept of confirmation in the sense of the relevance criterion. Siebel (J Gen Philos Sci 35:313–329, 2004 ) disputes the adequacy of this Bayesian solution. He claims that spelling (...) out the concept of confirmation in the relevance sense lets the raven paradox reappear as soon as numerous non-black non-ravens are observed. It is shown in this paper that Siebel’s objection to the Bayesian solution is flawed. Nevertheless, the objection made by Siebel may give us an idea of how Bayesians can successfully handle situations in which we observe more than one non-black non-raven. (shrink)

Wason’s selection task is a paramount experimental problem in the study of human reasoning, often connected with the celebrated ravens paradox in the philosophical literature. Various normative accounts of the selection task rely on a Bayesian approach. Some claim vindication of participants’ rationality. Others don’t, thus following Wason’s original intuition that observed responses are mistaken. In this article we argue that despite claims to the contrary, all these accounts actually speak to the same effect: Wason was right. First, (...) we provide a new accuracy-based analysis of the selection task that includes the existing proposals as special cases. We then show on this basis that none can actually vindicate participants’ rationality. We conclude that all normative renditions considered eventually concur: all in all, Bayesians should follow Wason in the selection task. (shrink)