Confirmation: The Raven Paradox

Examining a paradoxical yet true claim about confirmation provided by observations

1. Introduction

‘Both A Black Raven And A Red Herring Confirm The Claim That All Ravens Are Black.’

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. Hempel

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. Mackie

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

See Also:

Spinoza’s Style Of Argument In Ethics I

Science Is Not A Religion

The Opposition Of Value Systems

Leibniz’s Arguments For Monads: A Summary

By Tim Short

I am a former investment banking and securitisation specialist, having spent nearly a decade on the trading floor of several international investment banks. Throughout my career, I worked closely with syndicate/traders in order to establish the types of paper which would trade well and gained significant and broad experience in financial markets.
Many people have trading experience similar to the above. What marks me out is what I did next. I decided to pursue my interest in philosophy at Doctoral level, specialising in the psychology of how we predict and explain the behaviour of others, and in particular, the errors or biases we are prone to in that process. I have used my experience to write The Psychology of Successful Trading. In this book, I combine the above experience and knowledge to show how biases can lead to inaccurate predictions of the behaviour of other market participants, and how remedying those biases can lead to better predictions and major profits. Learn more on the About Me page.

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