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.

If you need an accessible version of this item please contact JSTOR User Support

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
  • Get Access
  • Download ($43.95)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Three Propositional Calculi of Probability
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