Modal operators with probabilistic interpretations, I

Studia Logica 46 (4):383 - 393 (1987)
<span class='Hi'></span> We present a class of normal modal calculi PFD,<span class='Hi'></span> whose syntax is endowed with operators M r <span class='Hi'></span>(and their dual ones,<span class='Hi'></span> L r)<span class='Hi'></span>, one for each r <span class='Hi'></span>[0,1]<span class='Hi'></span>: if a is sentence,<span class='Hi'></span> 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.<span class='Hi'></span> Every such a model is a kripkean model,<span class='Hi'></span> enriched by a family of regular <span class='Hi'></span>(see below)<span class='Hi'></span> probability evaluations with range in a fixed finite subset F of <span class='Hi'></span>[0,1]<span class='Hi'></span>: there is one such a function for every world w,<span class='Hi'></span> P F(w,<span class='Hi'></span>-)<span class='Hi'></span>, and this allows to evaluate M ra as true in the world w iff p F(w,<span class='Hi'></span> )<span class='Hi'></span> r.For every fixed F as before,<span class='Hi'></span> suitable axioms and rules are displayed,<span class='Hi'></span> 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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