Objection to denying equivalence
But once we make the intended interpretation explicit to ensure the truth of Equivalence, the paradox does not go away. We can replace “all clubs are black” with “there are no red clubs.” Everyone agrees that the latter is true even if the clubs have been removed. Now the paradoxical argument is that a red heart justifies that there are no red clubs:
1 An observed red heart is an instance of a red heart.
2 Therefore, that an observed red is a heart justifies that all reds are hearts.
(by Instance )
3
“All reds are hearts” says the same as “there are no red clubs” (byEquivalence ) Conclusion Therefore, that an observed red is a heart justifies that there are no red clubs. (by 2, 3, and Substitution)
Black club Black heart
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Paradox retained. Why should a red heart justify that there are no red clubs?
Applying this to the ravens, we replace “all ravens are black” with “there are no white ravens”:
1 An observed white sneaker is an instance of a white sneaker.
2 Therefore, that an observed white thing is a sneaker justifies that all white things are sneakers. (by Instance )
3 “All white things are sneakers” says the same as “there are no white ravens.” (By Equivalence )
Conclusion Therefore, that an observed white thing is a sneaker justifies that there are no white ravens. (By 2, 3, and Substitution) Still crazy. Does a white sneaker really justify that there are no white ravens?
8.3 Instance-observed 8.3 Instance-observed
The real weak point of the paradox is Instance . To investigate it, we need to make a distinction. Instance , which says that an observed FG justifies that all F are G, can be broken up into two parts:4
Instance-observed An observed F is G justifies
All observed Fs are G.
Instance- unobserved An observed F is G justifies
All unobserved Fs are G.
Note: The object described in our evidence always counts as observed; the unobserved objects are those that remain unseen, and which we can only make inductive inferences about. This is slightly awkward use of language, but we’ll just stipulate that that’s how “observed” is to be understood.
We’ll discuss Instance-observed here in 8.3 and Instance-unobserved in 8.4. The same pattern emerges in both sections. First, Instance-observed and Instance-unobserved both need to be restricted. Second, in cases where they hold and the conclusion that there are no white ravens is justified, we can explain why people find this paradoxical.
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8.3.1 Deny Instance-observed 8.3.1 Deny Instance-observed
It might seem that Instance-observed is trivially correct.
Instance-observed An observed F is G justifies
All observed Fs are G.
Applying this to the ravens case, imagine going to your closet and observing a white sneaker (assume that this is all the evidence you have). The relevant inference looks undeniable:
Instance-observed, White Sneakers An observed white thing is a sneaker justifies
All observed white things are sneakers.
The relation between hypothesis and evidence is deductive; the hypothesis (all observed Fs are G) entails the evidence (an observed F is G): P(E|H) = 1.
One might think it follows that the evidence justifies the hypothesis, that is, P(E|H) = 1 > P(E).
But it doesn’t. The reason is that the evidence was certain to be observed, P(E) = 1, and evidence that was certain to be observed can never justify anything.
Let’s go through these two points. First, evidence that is certain to be observed can never justify anything. This is fairly intuitive. If you are certain a piece of evidence is about to be discovered, there is a sense in which you already have the evidence. The evidence doesn’t tell you anything new, so you can’t learn anything from it. Formally, if P(E) = 1, then P(E|H) = P(E), so E does not justify H.5
Second, the evidence was certain to be observed. Here we have to be careful. It was not certain that you would observe a white sneaker. You might have observed a black sneaker. But as the relation between hypothesis and evidence is deductive, the possible evidence can be divided into that which will refute the hypothesis (a white raven) and that which will establish it (anything that’s not a white raven). In the ravens paradox, your evidence is the latter. You were certain that you would obtain this evidence because you imagined looking in the closet, and you can be certain that there are no white ravens in the closet. So you were certain the evidence you have would be observed.
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These two points together show that the observation of the white sneaker fails to justify that all observed white things are sneakers. To repeat, the evidence was certain to be observed, and evidence that is certain to be observed can never justify anything.
For contrast, imagine searching in a forest. You catch sight of something white in a tree. You rush toward it hoping to discover a white raven. But as you get to the tree, you see that it is just a sneaker sitting on a branch. “Hmm,”
you think, “still no observed white ravens.” In this case, it’s plausible that the white sneaker justifies that there are no observed white ravens. In this case, the evidence was not certain to be observed.6
Here’s the diagram again:
The hypothesis says there is nothing observed in the bottom left box. If you were certain all along that you were not going to observe something in the bottom left box, the hypothesis fails to be justified by observing something in the bottom right box. Thus Instance-observed is incorrect, and this removes the threat of indoor ornithology.
8.3.2 Instance-observed and the Wason selection task 8.3.2 Instance-observed and the Wason selection task
Now we come to a different response to the paradox—accept the conclusion, and argue that it only seems paradoxical. For even when Instance-observed is correct, and the resulting inference that there are no observed white ravens is correct, people tend to be surprised that it is. There is a famous psychological experiment, the Wason selection task, which explains why:7
Each card has a number on one side and a colour on the other. Which cards do you need to turn over to justify that every observed card with an even number on one side is black on the other side i.e. there are no observed white evens?8
R
Raavveen n SSnneeaakkeerr
Black Black
White White
Black raven Black sneaker
White sneaker White raven E
3 8
The Wason selection task.
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Most subjects suggest only the eight and the black. But the white card has to be turned over . We have to check if the white card is even, as this would be a counterexample. An observed even white refutes the hypothesis that every observed even is black; an observed odd white justifies that every observed even is black, that is, no observed white evens.9 Most people find this very odd when they first see it because they look for confirming instances and forget to look for counterexamples.We can put the ravens paradox in this form using cards that have a type of object on one side and a color on the other. (Ravens are analogous to evens.) To check whether any observed ravens are white, we have to turn over the white card to check if it says “raven” on the other side. This would be a counter-example to the hypothesis that there are no observed white ravens.
If instead it says “sneaker,” thenthis observation of a white sneaker justifies that there are no observed white ravens .
This promises to give a resolution to the ravens paradox: Instance- observed is correct, and the white sneaker really does justify that there are no observed white ravens; and we find this puzzling because we forget to look for counterexamples.