The principles
Here are two plausible Principles of Uniformity:
That an observed emerald is green justifies that all emeralds are green.1 That an observed raven is black justifies that all ravens are black.
In an abstract schema, where F and G are properties:
Instance
That an observed F is G justifies that all Fs are G.
In the quest for substantive principles for inferring from the observed to the unobserved, Instance is one of the best candidates for a first step. It should seem obviously true.
(Comparison with previous chapter: Notice that the hypothesis confirmed is that all Fs are G. This is the conjunction of “observed Fs are G” and
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“unobserved Fs are G.” The previous two chapters discussed only the latter.
And indeed it is the unobserved that we are primarily concerned with. But it’s useful to start with a hypothesis about all Fs because that’s what the literature does, and because it makes one of the other principles (Equivalence) easier to understand. Hypotheses about the observed and the unobserved will be separated once more in 8.3. Also, the evidence earlier concerns only a single observed F rather than “all” the observed F, which follows the literature and makes things a bit simpler.)
But Instance quickly leads to paradox. We need two more principles to generate the problem. First:
Equivalence “All Fs are Gs” says the same as “All non-Gs are non-Fs.”
Equivalence is not obvious, but some examples should make it intuitive.
Consider “all humans breathe.” This says the same as “all non-breathers are non-human.” Both sentences are true if and only if there are no nonbreathing humans. (Careful: “All Gs are Fs” is different. The recipe to generate what’s called the contrapositive is to reverse the F and G, and negate both.)
We can also give a cards example. Assume that every card in a special deck is either black or red, and either a club or a heart (we’ve removed the spades and diamonds for simplicity). Then, “all clubs are black” says the same as “all reds are hearts.” Both of these sentences are true as long as there are no red clubs. A picture will help:
So both sentences make the same claim about the distribution of objects:
there is nothing in the bottom left box.
The second principle for generating the ravens paradox is:
Substitution For any H1 and H2, if H1 and H2 say the same thing and E justifies H1, then E justifies H2.
Substitution is intuitive. If two sentences say the same thing then anything that justifies one must justify the other. Think of a Venn diagram—two sentences that say the same thing correspond to two areas that perfectly overlap, that is, there is really only one area, picked out in different ways. We won’t question Substitution.2
“All clubs are black” says anything in the left column is in
the top box “All reds are
hearts” says
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The Paradox The Paradox
Now let’s use these principles to derive the paradox. Getting your head round the paradox is not easy, so we’ll start with a cards example. Assume that everything is either black or red, and either a spade or a heart:
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 “all clubs are black” (by Equivalence )
Conclusion Therefore, that an observed red is a heart justifies that all clubs are black (by 2, 3, and Substitution).
This is a paradoxical conclusion. Why should observing a red heart justify that all clubs are black? Repeating the table from the previous page, why should finding something in the bot tom right box justify that there is nothing in the bottom left box?
We’ll look at several examples of this problem, but the cards example is the most straightforward, so I recommend thinking hard about this case before moving on. Do you think the conclusion is true? If so, why does it seem odd? If you think the conclusion is false, which premise would you reject?
Here’s another example that is relatively easy to get your head around.
Suppose you’re at a ride at the fairground where riders have to be taller than 4 ft. Let’s divide everyone in the world into “riders,” who are on the ride at a particular time, and “spectators,” who are not. And divide everyone into those who are more than 4 ft tall and those who are less than 4 ft tall (pretend no- one is exactly 4 ft):
1 An observed 5 ft rider is an instance of a rider more than 4 ft tall.
2 Therefore, that an observed rider is 5 ft justifies that all riders are more than 4 ft tall. (By Instance )
C
Clluub b HHeeaarrtt
Black Black
Red Red
Black club Black heart
Red club Red heart E
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3 “All riders are more than 4 ft tall” says the same as “everyone less than 4 ft tall is a spectator.” (By Equivalence )
Conclusion Therefore, that an observed rider is 5 ft justifies that everyone less than 4 ft tall is a spectator. (By 2, 3, and Substitution) Again, the conclusion is paradoxical—why should seeing someone on the ride of 5 ft justify that people less than 4 ft are not on the ride? You shouldn’t be able to learn about spectators by observing riders.
At this point, you might be wondering why this chapter is called “the paradox of the ravens.” The reason is that this paradox is usually presented in the following form, where we assume for simplicity that everything is either black or white, and either a sneaker or a raven:
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 “all ravens are black.” (By Equivalence )
Conclusion Therefore, that an observed white thing is a sneaker justifies that all ravens are black. (By 2, 3, and Substitution)
This conclusion is crazy. It says that if we go to the closet and see a white sneaker, it justifies that all ravens are black. Which means we don’t need to go outside and look at birds to find out if ravens are black, we can just look in the closet—ornithology has become an indoor pursuit.
The argument says that finding something in the bottom right box justifies that there is nothing in the bottom left box.
Can we reject a premise of the paradox? In 8.2 we discuss an attempt to deny Equivalence , and argue that the paradox is not avoided. In 8.3 and 8.4, we consider Instance and argue that there are several ways it can be false.
R
Raavveen n SSnneeaakkeerr
Black Black
White White
Black raven Black sneaker
White raven White sneaker E
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8.2 Deny equivalence 8.2 Deny equivalence
Equivalence is a principle of logic. But speakers don’t always interpret the sentences expressing the paradox in such a way that they come out equivalent.3 Recall the explanation of Equivalence:
It was claimed earlier that both sentences make the same claim about the distribution of objects: there is nothing in the bottom left box.
But imagine that all the clubs have been removed from the deck. Many people say that “all clubs are black” is no longer true! But clearly “all reds are hearts” remains true. This means that people are not interpreting these claims as saying the same thing. Therefore when we interpret the hypotheses the way normal speakers do, Equivalence is false; “all clubs are black” does not say the same as “all reds are hearts.”
(By contrast, philosophers interpret the claims in a way that makes Equivalence true, as they interpret “all F are G” as being true if there are no F.
This oddity is a version of the paradox of material implication.)