Abstract
We present a class of normal modal calculi PFD, whose syntax is endowed with operators M r (and their dual ones, L r), one for each r ε [0,1]: if a is α sentence, M rα 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 (see below) 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(w,-), and this allows to evaluate M ra as true in the world w iff p F(w, α) 〉 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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Fattorosi-Barnaba, M., Amati, G. Modal operators with probabilistic interpretations, I. Stud Logica 46, 383–393 (1987). https://doi.org/10.1007/BF00370648
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DOI: https://doi.org/10.1007/BF00370648