For instance, if A is the proposition that a coin tossed at time t will fall heads, X is the proposition that the chance of A at time t is x , and E is the available evidence that does not contradict X ; the Principal Principle then says that x equals the actual degree of belief that the coin falls heads, conditionally on the proposition that its chance of falling heads is x . In other words, the chance of A equals the degree to which an agent believes in A .
In fully personalistic approaches coherence is nec and suff for assignments of prior proba. So, the objection: aren’t frequencies just a pedagogical need? Carnap gives an interesting answer (Philosophical Foundations of Probability, §50-51, 41C). 1. we can do without frequencies if inductive logic is accepted. Reason 1: prob1 (subj) can be explicated as estimate of proba2 (obj). So, if proba2 is know, then proba1 just equals this value. Reason 2: even if proba2 is unknown, we can still compute proba1 as estimate of the unknown proba2 from *frequencies* in the sample But the subjectivist can still play a last card: the exchangeability argument. Briefly and informally, De Finetti shows that different agents may start with different prior probabilities, but, as evidence accumulates, their posterior proba will tend to converge, thus giving the illusory impression that objective probability exists.
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Objective Bayesianism and causal modelling in the social sciences Federica Russo Philosophy, Louvain & Kent
Let C be any reasonable initial credence function. Let t be any time. Let x be any real number in the unit interval. Let X be the proposition that the chance, at time t , of A 's holding equals x . Let E be any proposition compatible with X that is admissible at time t . Then, C ( A | XE ) = x .
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The Janus you choose makes the whole difference
Lewis 1986 :
Carnap did well to distinguish two concepts of probability, insisting that both were legitimate and useful and that neither was at fault because it was not the other. I do not think Carnap chose quite the right two concepts, however. In place of his ‘degree of confirmation’ I would put credence or degree of belief; in place of his ‘relative frequency in the long run’ I would put chance or propensity, understood as making sense in the single case. The division of labor between the two concepts will be little changed by these replacements. Credence is well suited to play the role of Carnap's probability1, and chance to play the role of probability2 .
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