Modal operators with probabilistic interpretations, I

Studia Logica 46 (4):383-393 (1987)
Abstract
We present a class of normal modal calculi PFD, whose syntax is endowed with operators M r, one for each r [0,1] : if a is sentence, M r is to he read the probability that a is true is strictly greater than r and to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model. Every such a model is a kripkean model, enriched by a family of regular probability evaluations with range in a fixed finite subset F of [0,1] : there is one such a function for every world w, P F, and this allows to evaluate M ra as true in the world w iff p F r.For every fixed F as before, suitable axioms and rules are displayed, so that the resulting system P FD is complete and compact with respect to the class of all the F-restricted probabilistic kripkean models
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DOI 10.1007/BF00370648
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W. D. Hart (1972). Probability as Degree of Possibility. Notre Dame Journal of Formal Logic 13 (2):286-288.

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