Three prepositional calculi of probability

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_{ℵ 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_{ℵ 0} into MP are given.

The third calculus ŁP is a modal extension of Ł_{ℵ 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 is given.