It is an old and familiar challenge to normative theories of personal probability that they do not make room
for non-trivial uncertainties about (the non-controversial parts of) logic and mathematics. Savage (1967) gives a frank presentation of the problem, noting that his own (1954) classic theory of rational preference serves as a poster-child for the challenge.
Here is the outline of this presentation:
Based on joint work with J.B.Kadane and M.J.Schervish
First is a review of the challenge.
Second, I comment on two approaches that try to solve the challenge by making surgical adjustments to the canonical theory of coherent personal probability. One approach relaxes the Total Evidence Condition: see Good (1971). The other relaxes the closure conditions on a measure space: see Gaifman (2004). Hacking (1967) incorporates both of these approaches.
Third, I summarize an account of rates of incoherence, explain how to model uncertainties about logical and mathematical questions with rates of incoherence, and outline how to use this approach in order to guide the uncertain agent in the use of, e.g., familiar numerical Monte Carlo methods in order to improve her/his credal state about such questions (2012).
Gaifman, H. (2004) Reasoning with Limited Resources and Assigning Probabilities to Arithmetic
Statements. Synthese 140: 97-119.
Good, I.J. (1971) Twenty-seven Principles of Rationality. In Good Thinking, Minn. U. Press (1983): 15-19.
Hacking, I. (1967) Slightly More Realistic Personal Probability. Phil. Sci. 34: 311-325.
Savage, L.J. (1967) Difficulties in the Theory of Personal Probability. Phil. Sci. 34: 305-310.
Seidenfeld, T., Schervish, M.J., and Kadane, J.B. (2012) What kind of uncertainty is that?
J.Phil. 109: 516-533.