6 ideas
A probability is not a property of an event or a frequency in the world. It is a logical relation between a conclusion and a body of evidence: the degree of belief it is rational to hold in proposition a, given premises h. Change the evidence and the probability changes, because it is a different relation, even though the event has not changed.
The weight of an argument measures how much relevant evidence supports a probability judgment. Probability measures which way that evidence points. New evidence always increases weight, but it can raise, lower, or leave the probability unchanged, so a 50% estimate resting on one observation differs from a 50% estimate resting on thousands. Decisions should attend to both quantities, not only the probability.
Probabilities are only partially ordered. Some can be ranked against each other, some can be given numbers, and many can be neither measured nor compared. Forcing a single number onto every uncertain judgment invents a precision the evidence does not contain.
Assigning equal probabilities to alternatives only because we know no reason to prefer one leads to contradictions. The contradictions come from the fact that the same situation can be split into alternatives in different ways. Indifference is legitimate only when the alternatives are indivisible and the evidence is symmetric with respect to each of them. Equal ignorance is not evidence of equal likelihood.
Choosing the action with the highest probability times value ignores two things. It ignores the weight of the evidence behind the probabilities, and it ignores the spread of possible outcomes. A reasonable agent may prefer a lower expectation that rests on firmer knowledge, or that carries less risk of ruin. Expected value is therefore an incomplete guide to conduct under uncertainty.
Repeated confirming instances can raise the probability of a generalization, but Keynes argues that a stronger justification, one where confirmations can make a generalization approach certainty, needs a further premise. That premise is a positive prior probability that the world contains only a limited number of independent generating qualities. On this view, strong confidence in induction draws on an assumption about how the world is structured, not on counting instances alone.