What is the Best Response to the Paradox of the Ravens?
In this essay, I will first describe the origin and nature of the Paradox of the Ravens. Then I will briefly describe some of the less popular responses to the puzzle, before continuing with a description of the response most widely accepted in the literature. I will, however, side-step the issue of how to define "best" in this context. Instead, I will conclude by documenting why I agree with the general opinion that the Bayesian analysis is sufficiently useful to be regarded as "the" response to the paradox of the ravens.
The "Paradox of the Ravens" was proposed by the German logician Carl Gustav Hempel in the 1940s to illustrate a problem where inductive logic seems to violate intuition(1). Hempel proposed the inductive hypothesis "All ravens are black", and explored how it is that we confirm this hypothesis. In strict logical terms, by the Contrapositive Law of deductive logic, this hypothesis is equivalent to "Everything that is not black is not a raven". Clearly, the observation of a white shoe is evidence in support of this latter hypothesis -- a white shoe is not black and is not a raven. But since the two hypotheses are logically equivalent, the observation of a white shoe must also be accepted as evidence in support of the hypothesis that "All ravens are black". Which appears to be counter-intuitive.
The apparent paradox relies on three premises that are regarded as intuitively plausible, yet are logically inconsistent.
Premise 1 (Nicod's Condition) -- In the absence of other evidence, the observation that some object a is both F and G confirms the generalization that all F are G. It is readily acceptable that observing a black raven confirms the generalization that all ravens are black.
Premise 2 (Equivalence Condition) -- If evidence confirms a proposition, then it also confirms any logically equivalent proposition. The rules of deductive logic demand that the truth conditions of logically equivalent propositions are the same.
Premise 3 -- In the absence of other evidence, a white shoe does not confirm the hypothesis that all ravens are black.
To "resolve" the paradox, obviously one of these three premises must be denied. And to provide a "good" resolution for the paradox (again, without getting into what would define "good" in this context), that denial must rest on some solid rationale.
I.J. Good(2) and Patrick Maher(3) are two philosophers who have explored the potential of denying Nicod's Condition. They each have pointed out that in certain conditions of background knowledge, Nicod's Condition turns out to be false. Consider, for example, someone who is familiar only with birds in the parrot family. Parrots come in a delightful multitude of colours. When advised that a raven is a bird, such a person might reasonably conclude that ravens, like other birds, would come in a multitude of colours. Seeing a black raven would then not confirm the hypothesis that "all ravens are black", but would instead confirm the hypothesis that "all ravens are multi-coloured". Drawing on the equivalency condition, this would entail that observing a white shoe would also not confirm the hypothesis that all ravens are black. So there is a reasonable case to be made that Nicod's Condition ought to be dropped from the paradoxical argument.
Hempel was aware of this argument, but responded that the paradox must be understood in the context of either no background information, or actual rather than fanciful background information(4). Dropping Nicod's Condition might be justifiable under some imaginable conditions, but is clearly not justifiable given our actual background knowledge of ravens in particular, and birds in general (i.e. that birds in the same species are usually similarly coloured). Similarly, it is clearly not justifiable if all background information is to be rejected when considering the logic of the paradox.
W.V.O. Quine offered another approach to justify dropping Nicod's Condition. Quine argued that only certain "natural kind" predicates obey Nicod's Condition, while artificially contrived predicates do not(5). Hence, Nicod's Condition holds for black ravens because black ravens form a "natural kind". But Nicod's Condition does not hold for the artificial predicate "non-black non-ravens" because it does not delimit a "natural kind". This approach was offered in the context of Nelson Goodman's example of the predicate "grue". But it serves equally well as a resolution to the paradox of the ravens. Hempel's ravens argument appears paradoxical, on this basis, because we automatically apply Nicod's Condition to all predicates, when in fact it is only valid when applied to "natural kinds".
Of course, Quine's response to the paradox only works if one accepts the "natural kinds" approach to linguistic predicates. There are many objections to the concept of "natural kinds", but I will side-step that debate in this essay by simply noting that as a resolution of the ravens paradox, the "natural kinds" justification for dropping Nicod's Condition is not widely accepted.
Israel Scheffler and Nelson Goodman offered an argument for dropping the Equivalence Condition premise(6). Their notion of "Selective Confirmation" draws upon the theory of Karl Popper(7) that scientific hypotheses are never confirmed, only falsified. The concept of "selective confirmation" of a proposition is understood as a falsification of the contrary proposition. The observation of a black raven thus does not confirm the hypothesis that "all ravens are black". Rather, it falsifies the contrary hypothesis that "no ravens are black". In other words, selective confirmation violates the equivalency condition because observing a black raven selectively confirms "all ravens are black" (by falsifying its contrary), but not "all non-black things are non-ravens" (because it does not falsify its contrary). A white shoe (a non-black-non-raven) therefore does not selectively confirm the hypothesis that "all ravens are black" because it is consistent with both "all ravens are black" and "no ravens are black".
The notion of selective confirmation is one example of an approach to the concept of "supporting evidence" that does not coincide with the concept of "increased likelihood". Since by the laws of the probability calculus, logically equivalent propositions must have the same probabilities, any suggested resolution to the ravens paradox that denies the equivalency condition must interpret "supporting evidence" in terms that do not coincide with a notion of probability. There are a number of approaches of this sort that have been offered in the vast literature on the ravens paradox, but none has attracted wide support. In the interests of brevity, therefore, I will not pursue them further.
The solution to the paradox that has achieved the widest acceptance, is the denial of the 3rd premise. This means accepting the apparently paradoxical conclusion that the observation of a white shoe (a non-black non-raven) does indeed support the hypothesis that all ravens are black.
The most widely accepted argument in support of this response to the ravens paradox, is the Bayesian analysis. However, as Chihara(8) has pointed out, there is not one single Bayesian analysis. Rather there is a multitude of slightly different analyses. All of which take an anti-Popperian view of hypothesis confirmation, and all of which adopt a "probability raising" notion of "supporting evidence". Despite the range of alternative analyses of the scenario, there is a common conclusion. Because of the relative (subjectively estimated or historically observed) infrequency of ravens to non-ravens and black things to non-black things, the observation of a white shoe does increase the probability of the hypothesis that "all ravens are black" but only to a very miniscule degree.
Formally, in one version, Bayes' Theorem states --
P(H/e.k) = P(e/H.k) x P(H/k)
In words -- the (posterior) probability of the hypothesis H, given the evidence e and the background knowledge k is equal to the probability of the evidence e given the hypothesis H and the background knowledge k, times the (antecedent) probability of the hypothesis H given the background k, all divided by the probability of the evidence e given the background k.
In the Ravens scenario where the hypothesis H is "All ravens are black", and e is "a white shoe", the P(e/H.k) is almost equal to P(e/k), so P(H/e.k) is almost equal to P(H/k). P(e/k) is only minimally less than P(e/H.k) because the hypothesis that "all ravens are black" minimally reduces the vast number of things that are non-black and non-raven, when compared to the situation without that hypothesis. The probability of seeing a white shoe, given all that we know about the world, is only minimally less than the probability of seeing a white shoe, given all that we know about the world combined with the hypothesis that all of one collection of things (the ravens) will be black.
Premise 3 above gains its appearance of plausibility because the degree to which a white shoe confirms that "all ravens are black" is so very close to zero that for all practical purpose it can be (and hence is) treated as zero.
Despite the wide acceptance of the Bayesian response to the Ravens paradox, the Bayesian approach does suffer from a couple of difficulties. One is major and serious, and one is minor and only sometimes serious. The major and serious difficulty with any Bayesian analysis is that it depends upon probabilities -- and we do not have a very good understanding of just what constitutes a probability (or likelihood, or tendency). In situations where we can approximate the probabilities involved using the history of previously observed frequencies, this may not pose much of a road-block. But in cases where we have to provide some estimation of the antecedent likelihood of some hypothesis, given only our current background information, this becomes a serious difficulty. The philosophical discussion of the meaning of such subjectively estimated probabilities is voluminous. I do not intend to get into that morass here, except to note this issue remains a problem for the Bayesian response to the ravens paradox.
The minor and only sometimes serious difficulty is that any Bayesian analysis presupposes that the probability of the observation (given background information) is independent of the hypothesis (or theory) being examined(9). In other words, the Bayesian analysis presupposes that P(e|H.k) ~= P(e.k). In a case like the ravens hypothesis we can be reasonably sure from background information that the presupposition holds good. But this is not always the case. The presupposition may be particularly problematic, for example, in quantum physics -- where it is recognized that any observation of the evidence alters the situation. To apply a Bayesian analysis to any hypothesis in general, therefore, demands that we first gain some reasonable assurance that the presupposition holds true and that the formulation of the hypothesis does not impact the probability of the evidence. This issue gains in importance when one considers the issue of the theory-ladenness of observation, and the extent to which the theory we adopt can change how we view the world, and the probabilities we attach to observing the evidence we observe. But that will remain the topic of another essay.
One of the lessons that we can learn from the Bayesian response to the paradox of the ravens, is that whether or not observations confirm hypotheses is never independent of background information. Probabilities are the core of any Bayesian analysis of scenarios like that of the white shoe and the ravens hypothesis. And probabilities, whether conceived as subjective estimations, or as observed historical frequencies, are "all things considered" evaluations. As for that matter, is the assurance that the necessary presupposition of hypothesis independence holds good.
I choose the Bayesian analysis of the Paradox of the Ravens as the "best" response because it does not attack the intuitively obvious premise that observing a black raven confirms (admittedly to some small degree) the hypothesis that all ravens are black (Nicod's Condition). Without that condition, we could not do the kind of inductive pattern recognition that we are so good at. And because it does not challenge the defined rules of Deductive Logic and the Probability Calculus that logically equivalent propositions have equivalent truth conditions (the Equivalency Condition). And because the Bayesian analysis provides a ready explanation of why the Premise 3 appears so intuitively reasonable while remaining strictly false. Despite the challenge presented by a lack of a complete philosophical understanding of the probabilities involved, and the potential challenge presented by the theory-ladenness of observation, the Bayesian approach to related problems of experimental confirmation of scientific hypotheses has proved remarkably (albeit pragmatically) resourceful.
Notes & References
(1) Wikipedia contributors. "Raven paradox" in Wikipedia, The Free Encyclopedia. URL=<http://en.wikipedia.org/w/index.php?title=Raven_paradox&oldid=455568165>.
(2) Good, I.J.; "The White Shoe is a Red Herring" in The British Journal for the Philosophy of Science, Vol. 17, No. 4 (Feb 1967), p322.
(3) Maher, Patrick; "Inductive Logic and the Ravens Paradox" in Philosophy of Science, Vol 66, No 1 (Mar 1999), pp 50-70.
(4) Hempel, C.G.; "The White Shoe - No Red Herring" in The British Journal for the Philosophy of Science, Vol. 18, No. 3 (1967), p. 239
(5) Quine, W.V.O.; "Natural Kinds" in Ontological Relativity and other Essays. Columbia University Press, New York, New York, 1969. p114
(6) Scheffler, Israel & Goodman, Nelson; "Selective Confirmation and the Ravens: A Reply to Foster" in The Journal of Philosophy, Vol. 69, No. 3 (Feb. 10, 1972), pp. 78-83
(7) Popper, Karl; The Logic of Scientific Discovery, RC Series Bundle, Routledge Classics, Routledge, New York New York, 2002 (1934), ISBN 0-415-27844-9.
(8) Chihara, Charles, S.; "Some Problems for Bayesian Confirmation Theory" in The British Journal for the Philosophy of Science, Vol. 38, No. 4 (Dec 1987), pp 551-560.
(9) Vranas, Peter B.M.; "Hempel's Raven Paradox: A Lacuna in the Standard Bayesian Solution" in The British Journal for the Philosophy of Science, Vol 55, No 3 (2004), pp 545-560.
This question bears some similarity to Moore’s class of paradoxes of the form “P, but I do not believe that P”. The similarity relates to the fact that we have an intuitive reaction against both claims, though they may not in fact be logically contradictory. To be contradictory, a claim must violate the law of non-contradiction and assert P & ¬ P. Moore’s paradox does not actually do this, but it carries an impression that it does, because we assume that someone asserting P must believe it.
Similarly, the assertion made here contains no logical contradiction. There are no logical reasons why any observation P should not confirm any claim Q. However, we intuitively feel that any observation purporting to confirm P must have ‘something to do with’ P. Surely nothing can be learned about ravens by observing herring. This essay will argue that here intuition is wrong and logic is right in this case.
The problem originates with the logical equivalence of hypotheses, as illustrated by the pair below.
S1: All ravens are black
S2: All non-black objects are not ravens
It seems at first that the two claims i) and ii) are different because they have different subjects. But in fact they say the same thing. They are both made true by the blackness of all ravens, despite the appearance that i) is about ravens while ii) is about anything that is not black.
We now introduce what appears to be a basic rule of confirmation, the Equivalence Condition:
EC: Anything that confirms a hypothesis also confirms any logically equivalent hypothesis.
This simply means that hypotheses come together with any further claims they entail. If I have evidence, perhaps visual, that Pierre is in the café, then this is equally good evidence for the claim that he is not in the park, because the truth of the first claim entails the falsity of the latter and many other similar assertions. It would be strange if I could be certain that Pierre was in the café but unsure about whether he was also in the park.
Note that the strangeness is not related to the certainty here. If someone has secretly selected a ball from a box containing three white balls and one black ball, I know that there is a 75% chance that they have a white ball and equally a 75% chance that they do not have a black ball, because if it is white then it is (perhaps logically but certainly in some fashion) entailed that it is not black.
This leads to the paradox. For an observation of a red herring confirms b). So it must also confirm a). And how can that be?
2.1. Origins of the Paradox
Hempel introduces Nicod’s Criterion, as below.
NC: A hypothesis can only be confirmed or dis-confirmed by one of its instances.
It can be seen that despite the fact that this criterion was widely accepted, it already has the appearance of being in conflict with A above. Hempel considers hypotheses i) and ii) in the light of a universe of four objects, as specified below:
a) a black raven;
b) a non-black raven;
c) a black non-raven;
d) a non-black non-raven.
By NC, these objects would have the following effects in relation to S1 and S2.
Object S1 S2
a Confirms Neutral
b Disconfirms Disconfirms
c Neutral Neutral
d Neutral Confirms
The different effects of a) and d) mean that Hempel is able to bring the apparently fatal objection that NC “makes confirmation depend not only on the content of the hypothesis, but also on its formulation”. It is therefore clear that in choosing between NC and EC, EC is to be preferred. This then leads directly to the paradox that a red herring confirms both S1 and S2.
Hempel has a further approach to argue that the intuitive conflict is purely a result of psychological factors, by using the well-known oddities in the behavior of the logical-IF statement. The relevant truth table allows for a material conditional to be false in only one circumstance: where the antecedent is true and the consequent is false. This has the unusual result that the conditional is true if the antecedent is false even if the consequent is also false.
Thus, the proposition ‘all mermaids are green’ is true. The logical explanation of this is that we could only falsify it by observing a non-green mermaid, and since we cannot observe any mermaids at all, this cannot be done. Hempel notes the Russellian point that we are probably subconsciously attaching existential import to the proposition and expanding it to ‘there is something which is a mermaid and it is green’. The first conjunct is false and so the proposition is false on that expansion. Similarly, we find it strange to say truly of someone that all their daughters are clever when that person has no daughters.
It is known that all sodium salts burn yellow. This may be phrased conditionally as if x is sodium then x burns yellow. Note that we do no expect this hypothesis to be confirmed by an observation of ice, which does not contain sodium, not burning yellow. Yet this is exactly the same fallacy. The proposition is still true and is confirmed by the ice observation because the antecedent is false.
Hempel seeks to illustrate this further by considering the order in which the observations are made or what background knowledge we are using. If an unknown substance is burned, we would interpret the results differently. If the unknown substance does not burn yellow, we would conclude that it did not contain sodium salts. If on the other hand we know already that it is ice, we would be tempted to conclude that its failure to burn yellow tells us nothing about the sodium salt hypothesis. But we need to note that this is still consistent with the hypothesis and thus confirms it.
If the flame has burned yellow, we could subsequently have determined otherwise that the material contained sodium salt and confirmed the hypothesis. If it did not, we could have determined otherwise that the material was ice and this confirmed the other formulation of the hypothesis – that anything that does not burn yellow is not a sodium salt. Hempel claims that this makes the paradoxes disappear but it can be noted that the strong temptation remains ignore the reasoning.
2.2. A Definition of Confirmation
Hempel notes that there had been two candidates for definitions of confirmation, NC and also predictive success. The latter would be primarily associated with A J Ayer and could take the plausible form that a hypothesis is confirmed to the extent that it is able to predict observations successfully.
He replaces EC with a more fundamental Consequence Condition:
CC: If observation confirms a class of propositions K then it also confirms all logical consequences of K.
This is argued for convincingly by noting that in fact the original hypothesis already includes all further statements entailed by the ones specified and thus an observation can only confirm or dis-confirm them as a group – a remark very suggestive of Quine’s later holism. CC has EC as a consequence and so Hempel is able to drop EC as a separate independent criterion, while of course retaining its import within the CC umbrella. CC has the further desirable result of excluding NC, which factor constitutes a further argument for CC.
Hempel now excludes Predictive Success (“PS”) as an element for confirmation by noting that PS, while allowed by EC, is not allowed by CC. If a hypothesis H allows successful prediction of B2 future findings from B1 existing findings, then any equivalent hypothesis to H will also allow this. A weaker hypothesis than H – i.e. one with less predictive power – would not do this.
Any hypothesis H2 that is stronger than H, so that H2 entails H, can also be used to make this prediction. This relates to the famous Under-determination of Theory by Evidence (“UTE”) problem, whereby an apple falling to the ground is equally good evidence for a) gravity and b) gravity plus the moon is made of cheese. So PS would mean that any stronger hypothesis is confirmed by observations while CC says that only weaker ones are. Since CC produces the independently argued for EC and avoids the UTE problem, CC is shown to be a better candidate for a definition of confirmation.
3.1. Intuitive Explanation of Hempel’s Result
Mackie notes Hempel’s suggestion that there are psychological factors in play. He considers an alternative proposal that looks at the question from a probability angle.
The key element of this type of solution involves postulating a potential equivocation on ‘confirmation’. In ordinary language, the term refers to knowledge or strong enough levels of certainty. But here it becomes a term of art and should perhaps be understood as standing in for ‘tends to confirm’ or indeed ‘supports the hypothesis’. As is well known from the work of Popper and the general philosophy of induction, the current scientific paradigm says that observations can never prove a hypothesis. There always remains the possibility of finding a white raven. This is the case even though a single dis-confirmation can provide a falsification.
So perhaps the problem is that while it is true that a red herring confirms in this sense that all ravens are black, the degree of confirmation is less. However, it still remains to be explained how this can be. To see this, note that there are in principle two ways of confirming that all ravens are black. The first way is to observe all of the ravens.
The second way still exists: we could look at everything that is not black and see how many ravens are in that category. If the category of all non-black objects does not include any ravens, we have also shown S1. Since the second way is immensely impracticable, we would for all normal purposes ignore it: this is the source of our intuition against the red herring confirming S1.
In the world as it is, there are many fewer ravens than non-ravens. I have devised an example to make the above argument more intuitive; begin by considering a universe of discourse where these normal circumstances are reversed. It could be an aviary, since it contains many ravens. Imagine that it also contains a large number of non-descript items, all of which are black. These are hard to see and harder to describe. In addition, there are a white piece of chalk and a green leaf. These are the only non-black items. Under these circumstances, we would indeed conclude that all ravens are black by noting that everything that is not black is not a raven.
3.2. Further Cases Beyond Those Considered by Hempel
Mackie allows that Hempel has successfully argued for a consistent definition of confirmation within a limited set of universes of discourse. He then considers an extension he attributes to Watkins termed the Alternative Outcome Principle (“AOP”). This holds that if there are two potential outcomes of an observation, then they cannot both confirm the hypothesis under test. This seems plausible, because it would mean that the experiment or observation had been poorly designed. We must here as ever be on guard against equivocation on ‘confirmation’.
In practice, AOP has strange consequences. “If we inspect an object already known to be a raven and it turns out to be black, this confirms h, for the procedure might have turned out the other way and falsified h; but if we inspect an object already known to be black and it turns out to be a raven, this does not confirm h” and vice versa. The essential reason for this is that under the AOP, not both outcomes can confirm S1 even if they are both consistent with it, as they are – and in fact one outcome must deny the other or reduce its plausibility.
If the raven had not turned out to be black, then S1 would have been falsified. But if the black object had turned out not to be a raven, S1 would not have been falsified on this view – a crucial difference with Hempel – and so the reverse would not have been a confirmation of S1. The view is Popperian in that a test of a hypothesis can only be an attempt to falsify it.
But this cannot be right, because it means that the order of observation of characteristics is significant. A black raven with its species observed before its color confirms S1 but a black raven with its color observed before its species does not confirm S1. Surely a total observation of a black raven must have the same consequences independent of the order of consideration of its characteristics.
Mackie concludes that the paradox is to be resolved in different ways for different circumstances. In a limited Hempelian universe without background knowledge, we simply deny that observations of red herrings are irrelevant to S1 via a consideration of EC. We are simply wrong to think that this is the case.
Secondly, a numerical approach may be adopted to illustrate the equivocation of ‘confirms’. We are again wrong about the observation of red herrings, but we are wrong because we mistake ‘no confirmation’ for ‘minor confirmation’.
But thirdly, Mackie believes that the best form of confirmation should be on a Popperian basis where the outcome could have falsified the hypothesis. So observations of black ravens confirm S1 only in circumstances where they could have falsified it – we should specifically look out for non-black ravens as opposed to ignore anything not black. Also, “observations of non-black non-ravens confirm [S1] to a worthwhile degree only if they are made in genuine tests of this hypothesis.”
This is an echo of the example in section 3.1 above of the aviary containing mostly black items, many ravens and two non-black items. Counting the white chalk and the green leaf do constitute a good test of S1 in those circumstances. But in the actual world, containing as it does large numbers of ravens and herrings, and vast numbers of other objects of all kinds, there is no real test of S1. A negative outcome would not falsify S1 and so a positive one does not confirm it.
- C Hempel, Studies in the Logic of Confirmation (I.), Mind, New Series, Vol. 54, No. 213 (Jan., 1945), pp. 1-26 and (II.) Mind, New Series, Vol. 54, No. 214 (Apr., 1945), pp. 97-121
- J. L. Mackie, The Paradox of Confirmation, British Journal for the Philosophy of Science, Vol. 13, No. 52 (Feb., 1963), p. 265, (“Mackie”)
Cited as Philosophy, 1960, 10, 319
Author: Tim Short
I went to Imperial College in 1988 for a BSc(hons) in Physics. I then went back to my hometown, Bristol, for a PhD in Particle Physics. This was written in 1992 on the ZEUS experiment which was located at the HERA accelerator in Hamburg (http://discovery.ucl.ac.uk/1354624/). I spent the next four years as a post-doc in Hamburg. I learned German and developed a fondness for the language and people. I spent a couple of years doing technical sales for a US computer company in Ireland. In 1997, I returned to London to become an investment banker, joining the legendary Principal Finance Group at Nomura. After a spell at Paribas, I moved to Credit Suisse First Boston. I specialized in securitization, leading over €9bn of transactions. My interest in philosophy began in 2006, when I read David Chalmers's "The Conscious Mind." My reaction, apart from fascination, was "he has to be wrong, but I can't see why"! I then became an undergraduate in Philosophy at UCL in 2007. In 2010, I was admitted to graduate school, also at UCL. I wrote my Master's on the topic of "Nietzsche on Memory" (http://discovery.ucl.ac.uk/1421265/). Also during this time, I published a popular article on Sherlock Holmes (http://discovery.ucl.ac.uk/1430371/2/194-1429-1-PB.pdf). I then began work on the Simulation Theory account of Theory of Mind. This led to my second PhD on philosophical aspects of that topic; this was awarded by UCL in March 2016 (http://discovery.ucl.ac.uk/1475972/ -- currently embargoed for copyright reasons). The psychological version of this work formed my book "Simulation Theory". My second book, "The Psychology Of Successful Trading: Behavioural Strategies For Profitability" is in production at Taylor and Francis and will be published in December 2017. It will discuss how cognitive biases affect investment decisions and how knowing this can make us better traders by understanding ourselves and other market participants more fully. I am currently drafting my third book, wherein I will return to more purely academic philosophical psychology, on "Theory of Mind in Abnormal Psychology." Education: I have five degrees, two in physics and three in philosophy. Areas of Research / Professional Expertise: Particle physics, Monte Carlo simulation, Nietzsche (especially psychological topics), phenomenology, Theory of Mind, Simulation Theory Personal Interests: I am a bit of an opera fanatic and I often attend wine tastings. I follow current affairs, especially in their economic aspect. I started as a beginner at the London Piano Institute in August 2015 and passed Grade Two in November 2017! View all posts by Tim Short