In practice, we start from an empirical domain of random, chance, uncertain, or indeterminate phenomena of concern to us. These domains are supplied with meanings or, as they are commonly called, interpretations, that determine the structure of their representation through mathematical probability P. These meanings are first classified as either objective or personalist/subjective.Objective meanings of interest to us are those connected to outcomes of indefinitely repeated and unlinked physical random experiments (say, through limiting relative frequencies as introduced by Bernoulli in the 17thC.) or to the formal logical notion ofinductive support provided by an evidence statement to an hypothesis statement (Carnap, Solomonoff, Chaitin). Probability for inductive support will be referred to as epistemic probability. Rochesterís Henry E. Kyburg, Jr., was the leading exponent of epistemic probability. Subjective probability, whose modern father was de Finetti, refers to a formalization of degrees of belief (a propositional attitude) held by an individual concerning the truth or the occurrence of possible outcomes or events.
It is our contention that standard probability is not always the appropriate mathematical structure for probability representing one of the above meanings. We will support this contention by looking at some issues in mathematical modeling, particularly the precision or expressivity of the probability model, and looking more closely at the empirical domains themselves.
We close by briefly pointing to the following alternative mathematical structures for probability: chaotic probability as a set M of standard probabilities that is akin to a statistical hypothesis; probability as a pair of functions called upper and lower probabilities that can often be derived as infima and suprema of the standard probabilities contained in M; probability as a non-quantitative partial ordering defined on the pairs of events in the algebra A; classificatory or modal probability.
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