Deductive Probability, Physical Probability and Partial Entailment

A new kind of logical probability is introduced, based on the relationships between logical dependence and probabilistic dependence rather than on the principle of indiffer-ence or axioms of invariance. Given its deductive nature, this kind of probability is called deductive probability. To define it, an extensive investigation is carried out into the idea of logical independence. This investigation leads to the conclusion that only the notion of logical separability, introduced by the author in a previous paper (1990), is suitable to properly characterise deductive probability. If the definition of deductive probability is so modified that reference to logical independence is replaced with reference to physical independence, a new notion of physical probability is obtained. Since the idea of logical independence can be characterised as a degenerate case of physical independence (just as a logical truth is a degenerate case of factual proposition), deductive probability can be viewed as a degenerate case of physical probability. The importance of deductive and physical probability (as here characterised) rely on a theorem by which both deductive and physical probability definitions uniquely determine a single probability function: the function that distributes probability uniformly among the logically possible or, respectively, physically possible elementary cases. So the present account allows for a new justification of the uniform distribution across elementary cases and therefore a new way to interpret the formulas of statistical mechanics. More precisely, the uniform distribution is neither reached via epistemic principles nor it is considered as a physical probabilistic hypothesis about frequencies, propensities or chances. It is rather considered as the only probability function preserving, by means of the probabilistic dependencies that it induces, the physical dependencies that derive from the assumed nonprobabilistic laws. By the notion of deductive probability, a new account of partial entailment is developed. Two meanings of the term ‘partial entailment’ are distinguished. They generalise two distinct aspects of deductive total entailment. It is argued that epistemic inductive probability is adequate as an explicatum of partial entailment with respect to the first meaning while it is at odds with the second one.