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Nicod's condition
An instance of “all F are G”, an F that is G, confirms it. A black raven confirms “all ravens are black”.
Lesson · Science, Induction and Causation · Confirmation
The Raven Paradox
Nicod's condition and the equivalence condition together entail that a white shoe confirms “all ravens are black”. Sampling from the non-black objects shows how little: the Bayes factor is 901/900, against 10/9 for a black raven sampled from the ravens. Both confirm; the asymmetry is in the sampling.
Nicod: a black raven confirms the law
equivalent hypotheses are confirmed together
white shoe: BF 901/900. black raven: 10/9
confirmation comes in degrees, not all-or-none
Two conditions, one paradox
Core ideas
What this lesson establishes
Hempel's paradox is a valid argument from two plausible conditions to an implausible conclusion. The way out is to ask how much.
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The equivalence condition
Whatever confirms a hypothesis confirms every hypothesis logically equivalent to it. “All ravens are black” and “every non-black thing is a non-raven” say the same thing.
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Degree, not kind
Both conditions can stand if confirmation is a matter of degree and the degrees differ enormously with how the evidence was gathered.
The material
A white shoe, and the sampling behind it
First the argument, then the question it leaves, then the arithmetic that answers it.
Write R(x) for “x is a raven” and B(x) for “x is black”. The hypothesis is R(x) → B(x) for every x. Its contrapositive, which “Equivalence, De Morgan and Contraposition” shows to be equivalent, is ¬B(x) → ¬R(x) for every x.
R B | R → B ¬B → ¬R
--------------------------
T T | T T
T F | F F
F T | T T
F F | T T
The columns agree on every row, so the two hypotheses are equivalent. Now run the argument. By Nicod, a non-black non-raven, a white shoe, is an instance of the second hypothesis, so it confirms it. By equivalence it confirms the first. So a white shoe confirms that all ravens are black, and one need not leave the room.
Definition · The two conditions
Nicod's condition: an object that is F and G confirms “all F are G”. Equivalence condition: evidence that confirms one hypothesis confirms every hypothesis logically equivalent to it.
Each condition is hard to reject. What can be rejected is the assumption that the conclusion is absurd, once confirmation is measured. Confirmation depends on a likelihood, and the likelihood depends on how the object was found.
Take a world with ten ravens. If the hypothesis H is true, all ten are black and the non-black things number 900, none of them a raven. If H is false, exactly one raven is not black, so there are 901 non-black things and one of them is a raven. Give H a prior of 1/2.
First case: pick a non-black thing at random and find it is not a raven. Under H that had probability 1. Under not-H it had probability 900/901, since one of the 901 would have been a raven. The Bayes factor is 1 / (900/901) = 901/900.
Second case: pick a raven at random and find it is black. Under H that had probability 1. Under not-H it had probability 9/10. The factor is 10/9.
Theorem · Both confirm, unequally
A white shoe sampled from the non-black things has Bayes factor 901/900; a black raven sampled from the ravens has 10/9. The first is above 1, so the shoe confirms, and it is barely above 1, so it hardly matters.
The asymmetry is therefore not a property of shoes. It comes from the size of the sampled class: there are very many non-black things for a counterexample to hide among, and few ravens. Choose a different class to sample from, such as a thousand ravens, and the numbers move. The lab's limit is the world it describes. The result does not say that colour of shoes is evidence about birds in general, and a reader who samples shoes without a reason to link them to ravens has changed the likelihoods. Good pressed this further: the background can change the sign. If the hypothesis being true would mean there are few ravens and its being false would mean there are a great many, then a black raven drawn from all the birds is more probable when the hypothesis is false, and its factor falls below 1. Rewrite both rows of the lab for a draw from all the birds, with a black raven rarer under the hypothesis than under its rival, and the tile that said confirms will say disconfirms.
Interactive
Exact Bayesian updating
prior times likelihood, normalised; the Bayes factor is the weight of the data
Method
Answering a confirmation paradox
Four steps that apply to any case where evidence looks irrelevant yet follows from two plain principles.
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Write the argument as conditions
State each principle used and check each on its own. Here that is Nicod's condition and the equivalence condition.
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Check the equivalence by table
Build the truth table for the two forms and confirm that they agree on every row.
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Fix how the evidence was gathered
Say what was sampled from what. The likelihoods depend on it, and they are not defined without it.
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Compute the factor for each
Take the ratio of the two likelihoods for each case. Compare them as numbers, and do not ask only whether they exceed 1.
Shoes, ravens and degrees
Score 0 / 0Choose the answer you can justify.
A white shoe and a black raven
Ten ravens; under the hypothesis all black, under its negation one is not. Prior 1/2.
white shoe (sampled from non-black things):
P(non-raven | H) = 900/900 = 1
P(non-raven | not-H) = 900/901
Bayes factor = 901/900
black raven (sampled from ravens):
P(black | H) = 1; P(black | not-H) = 9/10
Bayes factor = 10/9
Both factors exceed 1, so both confirm. The shoe's exceeds 1 by 1/900 and the raven's by 1/9, a hundred times as much. The paradox dissolves because “confirms” was being heard as “confirms appreciably”.
Common mistakes
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Treating confirmation as all-or-nothingA white shoe confirms with factor 901/900 and a black raven with 10/9. Both exceed 1, and the first exceeds it by a hundredth as much. Asking only whether evidence confirms loses the one number that tells the cases apart.
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Thinking the paradox shows that indoor ornithology worksThe 901/900 is the result of sampling from the non-black things at random. A shoe picked for its colour, with no route to a raven, is not that sample, and the lab does not describe it.
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Rejecting the equivalence condition to escapeThe truth table shows the two forms agree on every row. A condition that gives different verdicts to hypotheses that agree on every row is not a theory of evidence for the hypothesis, but of its wording.
Check your understanding
✓
Finish when you can compute the factor for each way of sampling.
Given a hypothesis and a description of how an object was drawn, you should be able to compute the Bayes factor for the object found, compare it with the factor for a different draw, and say why the two differ.
Note: The numbers 10, 900 and 901 are the world the lesson stipulates. A world with more ravens or fewer non-black things changes both factors, and the lab will show it when you edit the likelihoods.