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Journal Article

# Three Propositional Calculi of Probability

Herman Dishkant
Studia Logica: An International Journal for Symbolic Logic
Vol. 39, No. 1 (1980), pp. 49-61
Stable URL: http://www.jstor.org/stable/20014970
Page Count: 13
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## Abstract

Attempts are made to transform the basis of elementary probability theory into the logical calculus. We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the Łukasiewicz logic Ł\${}_{\aleph _{0}}\$ (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which one may only add and subtract probabilities. The second calculus MP is a usual modal propositional calculus. It has the modal rules x ⊦ □ x, x ⊃ y ⊦ □ x ⊃ □ y, x ⊦ ⅂ □ ⅂ x, x ⊃ y ⊦ □ (y ⊃ x) ≡ □ (□ y ⊃ □ x), in addition to the rules of classical propositional logic. One may read □x as "x is probable". Imbeddings of NP and of Ł\${}_{\aleph _{0}}\$ into MP are given. The third calculus ŁP is a modal extension of Ł\${}_{\aleph _{0}}\$. It may be obtained by adding the rule □ ((∼ □ x → □ y) → □ y) ⊦ □ x → □ y to the modal logic of quantum mechanics ŁQ [5]. One may read □x in ŁP as "x is observed". An imbedding of NP into ŁP given.

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