Geodesic
A Logic With Higher Order Conditional Probabilities
Publications de l'Institut Mathématique, _N_S_82 (2007) no. 96, p. 141 .

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We investigate probability logic with the conditional probability operators. This logic, denoted $LCP$, allows making statements such as: $P_{\geq s}\alpha$, $CP_{\geq s}(\alpha\mid\beta)$, $CP_{\leq 0}(\alpha\mid\beta)$ with the intended meaning ``the probability of $\alpha$ is at least $s$", ``the conditional probability of $\alpha$ given $\beta$ is at least $s$", ``the conditional probability of $\alpha$ given $\beta$ at most $0$". A possible-world approach is proposed to give semantics to such formulas. Every world of a given set of worlds is equipped with a probability space and conditional probability is derived in the usual way: $P(\alpha\mid\beta)=\frac{P(\alpha\wedge\beta)}{P(\beta)}$, $P(\beta)>0$, by the (unconditional) probability measure that is defined on an algebra of subsets of possible worlds. Infinitary axiomatic system for our logic which is sound and complete with respect to the mentioned class of models is given. Decidability of the presented logic is proved.
DOI : 10.2298/PIM0796141O
Classification : 68T37 68T27, 03B48, 03B70 
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     title = {A {Logic} {With} {Higher} {Order} {Conditional} {Probabilities}},
     journal = {Publications de l'Institut Math\'ematique},
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Zoran Ognjanović; Nebojša Ikodinović. A Logic With Higher Order Conditional Probabilities. Publications de l'Institut Mathématique, _N_S_82 (2007) no. 96, p. 141 . doi : 10.2298/PIM0796141O. http://geodesic.mathdoc.fr/articles/10.2298/PIM0796141O/

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