“The sun has risen ten times, so it will rise again” has a counterexample row, however large the number. It gains force only under a prior over hypotheses about how reliable the sun is, and the same ten sunrises give a different prediction under a different prior.
Core ideas
Induction is reasoning from observed cases to unobserved ones. The question is not whether we do it, but what we are assuming when we do.
From “every observed case was F” to “the next case will be F” there is always a row of the truth table on which the premises hold and the conclusion fails. Adding more premises of the same kind adds more cases to the premises and leaves that row in place.
To let the past bear on the future you must say, before looking, how likely the various regularities are. That is a prior over hypotheses, and the data then reweight it.
The question is not whether induction has worked. It is whether anything non-circular shows that it will go on working, and the answer Hume gave is that nothing does.
The material
First the argument, then the row that defeats it, then the extra premise that would repair it.
Take the plainest inductive argument there is: the sun has risen on each of the last ten days, so it will rise on the eleventh. Write R₁ for “the sun rises on day 1”, and so on. The argument has ten premises and one conclusion, and “Validity by Truth Table” gives the test: look for a row on which every premise is true and the conclusion false.
The row exists. Nothing in the ten premises mentions day 11, so the sentence about day 11 is free to be false, and the argument is invalid. Replace ten by ten thousand and the row is still there: the premises grow, the conclusion does not move into them.
Enumerative induction is the inference from “all observed cases of F have been G” to “the next case of F will be G”. As a form it is invalid, so whatever force it has comes from a premise it does not state.
The missing premise can be written as a probability. Suppose the sun rises each day with some fixed chance, and consider five hypotheses about that chance: 1, 3/4, 1/2, 1/4 and 0. Put an equal prior of 1/5 on each. This is the rule of succession reduced to five points, which keeps every number an exact fraction; it is a discretisation and it says nothing about hypotheses in between.
Ten sunrises in a row have probability chance¹⁰ under each hypothesis, so they favour the high chances heavily. After the data, the hypothesis that the sun always rises carries the posterior the lab prints in its first tile, and the probability of an eleventh sunrise is each hypothesis's chance weighted by its posterior.
The probability that the next case is a success is the sum, over the hypotheses, of the posterior of each times its chance of a success: P(next | data) = Σ P(h | data)·chance(h). It is below 1 whenever some hypothesis with a chance under 1 keeps any posterior.
Give a skeptic a prior of 1/1000 on a chance of 1, 1/10 on each of 3/4, 1/2 and 1/4, and 699/1000 on a chance of 0. The ten sunrises are the same, the likelihoods are the same, and the predictive probability falls from just under 1 to about 0.78. The data did not change; the prior did.
That is the force of Hume's point. His argument can be set out as a valid argument, and its premises are worth stating at full strength:
The lab does not answer this. It shows exactly where the answer would have to go. Reject the third premise and you owe an account of a third source of support, or of why resemblance is necessary after all; reject the fourth and you owe an account of why a circular defence is not vicious; accept the conclusion and you owe an account of why anyone should expect the sun at all. Either way the reader decides, and the arithmetic is the same.
Interactive
prior times likelihood, normalised; the Bayes factor is the weight of the data
Method
Four moves, in this order, for any argument from past cases to a new one.
Write the observed cases as premises and the new case as the conclusion, and set every premise true and the conclusion false. If nothing forbids that, the argument is invalid.
Say what would have to be true of the unobserved for the past to bear on it. For the sun it is that the chance of rising is stable from one day to the next.
List the hypotheses about that stability, with a probability on each before any data. The choice is yours, and it is the part the data cannot check.
Read the posterior and the predictive probability from the lab, then change the prior and read them again. The gap between the two answers is how much the conclusion owes to the premise rather than to the data.
Take the five chances and the uniform prior. Weight each by the probability of the data under it, then normalise.
The first predictive probability is 1 − 611/44346, so the uniform prior is certain enough to bet on and never certain. The second is about 0.78. The ten sunrises were identical; the skeptic's prior started with almost nothing on the hypotheses that ten sunrises favour, and ten sunrises were not enough to undo that.
Given an inductive argument, you should be able to write the row that makes it invalid, state the premise a prior would supply, and compute the predictive probability under two priors from the lab, saying which part of the difference is the prior's.
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