Comment author:AnthonyC
29 March 2011 06:22:18PM
0 points
[-]

Consider the set of all possible hypotheses. This is a countable set, assuming I express hypotheses in natural language. It is potentially infinite as well, though in practice a finite mind cannot accomodate infintely-long hypotheses.
To each hypothesis, I can try to assign a probability, on the basis of available evidence.
These probabilities will be between zero and one.
What is the probability that a rational mind will assign at least one hypothesis the status of absolute certainty?
Either this is one (there is definitely such a hypothesis), or zero (there is definitely not such a hypothesis, which cannot be, because the hypothesis "there is definitely not such a hypothesis" is then a counterexample), or somewhere in between (there may be, somewhere, a hypothesis that a rational mind would regard as being absolutely certain).
So I cannot accept your hypothesis that there does not exist, anywhere, ever, a hypothesis that I should regard as being absolutely certain.

Comment author:jimrandomh
29 March 2011 07:01:57PM
1 point
[-]

Self-referential hypotheses do not always map to truth values, and "a rational mind will assign at least one hypothesis the status of absolute certainty" is self-referential. The contradiction you've encountered arises from using a statement isomorphic to "this statement is false" and requiring it to have a truth value, not to a problem with excluding 0 and 1 as probabilities.

## Comments (128)

OldConsider the set of all possible hypotheses. This is a countable set, assuming I express hypotheses in natural language. It is potentially infinite as well, though in practice a finite mind cannot accomodate infintely-long hypotheses. To each hypothesis, I can try to assign a probability, on the basis of available evidence. These probabilities will be between zero and one. What is the probability that a rational mind will assign at least one hypothesis the status of absolute certainty? Either this is one (there is definitely such a hypothesis), or zero (there is definitely not such a hypothesis, which cannot be, because the hypothesis "there is definitely not such a hypothesis" is then a counterexample), or somewhere in between (there may be, somewhere, a hypothesis that a rational mind would regard as being absolutely certain). So I cannot accept your hypothesis that there does not exist, anywhere, ever, a hypothesis that I should regard as being absolutely certain.

Self-referential hypotheses do not always map to truth values, and "a rational mind will assign at least one hypothesis the status of absolute certainty" is self-referential. The contradiction you've encountered arises from using a statement isomorphic to "this statement is false" and requiring it to have a truth value, not to a problem with excluding 0 and 1 as probabilities.