Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

Three Propositional Calculi of Probability

Herman Dishkant
Studia Logica: An International Journal for Symbolic Logic
Vol. 39, No. 1 (1980), pp. 49-61
Published by: Springer
Stable URL: http://www.jstor.org/stable/20014970
Page Count: 13
  • Download ($43.95)
  • Cite this Item
Preview not available

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.

Page Thumbnails

  • Thumbnail: Page 
[49]
    [49]
  • Thumbnail: Page 
50
    50
  • Thumbnail: Page 
51
    51
  • Thumbnail: Page 
52
    52
  • Thumbnail: Page 
53
    53
  • Thumbnail: Page 
54
    54
  • Thumbnail: Page 
55
    55
  • Thumbnail: Page 
56
    56
  • Thumbnail: Page 
57
    57
  • Thumbnail: Page 
58
    58
  • Thumbnail: Page 
59
    59
  • Thumbnail: Page 
60
    60
  • Thumbnail: Page 
61
    61