Argument
Lottery paradox
Overview
The lottery paradox arises from Henry E. Kyburg Jr. considering a fair 1,000-ticket lottery that has exactly one winning ticket. If that much is known about the execution of the lottery, it is then rational to accept that some ticket will win.
Suppose that an event is considered "very likely" only if the probability of it occurring is greater than 0.99. On those grounds, it is presumed to be rational to accept the proposition that ticket 1 of the lottery will not win. Since the lottery is fair, it is rational to accept that ticket 2 will not win either. Indeed, it is rational to accept for any individual ticket i of the lottery that ticket i will not win. However, accepting that ticket 1 will not win, accepting that ticket 2 will not win, and so on until accepting that ticket 1,000 will not win entails that it is rational to accept that no ticket will win, which entails that it is rational to accept the contradictory proposition that one ticket wins and no ticket wins.
The lottery paradox was designed to demonstrate that three attractive principles governing rational acceptance lead to contradiction:
It is rational to accept a proposition that is very likely true.
It is irrational to accept a proposition that is known to be inconsistent and is jointly inconsistent.
If it is rational to accept a proposition A and it is rational to accept another proposition A', it is rational to accept A and A'.
The paradox remains of continuing interest because it raises several issues at the foundations of knowledge representation and uncertain reasoning: the relationships between fallibility, corrigible belief and logical consequence; the roles that consistency, statistical evidence and probability play in belief fixation; the precise normative force that logical and probabilistic consistency have on rational belief.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Since therefore an entire indifference is essential to chance, no one chance can possibly be superior to another, otherwise than as it is composed of a superior number of equal chances. For if we affirm that one chance can, after any other manner, be superior to another, we must at the same time affirm, that there is something, which gives it the superiority, and determines the event rather to that side than the…
David Hume, A Treatise of Human Nature
Close in meaning to Lottery paradox; the term itself is not named.
According to the view here taken, these are the only fundamental rules for the calculation of chances. An additional one, derived from a different conception of probability, is given in some treatises, which if it be sound might be made the basis of a theory of reasoning. Being, as I believe it is, absolutely absurd, the consideration of it serves to bring us to the true theory; and it is for the sake of this…
Charles S. Peirce, Chance, Love, and Logic
States what Lottery paradox is.
This is a question which we cannot be exempted from answering: but the consideration of it belongs to what is called the Theory of Probability, which will form the subject of a chapter hereafter. It is seen, however, at once, that the conclusion does amount to a practical certainty after a sufficient number of instances, and that the method, therefore, is not radically vitiated by the characteristic imperfection…
John Stuart Mill, A System of Logic
States what Lottery paradox is.
Related ideas
- Philosophical logic
- Logical consequence
- Association for Symbolic Logic
- Thought experiment
- Epistemology
- David Makinson
Place in the tradition
Part of Contextualism.