Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 475–489 | Cite as

A logic for arguing about probabilities in measure teams

  • Tapani Hyttinen
  • Gianluca PaoliniEmail author
  • Jouko Väänänen


We use sets of assignments, a.k.a. teams, and measures on them to define probabilities of first-order formulas in given data. We then axiomatise first-order properties of such probabilities and prove a completeness theorem for our axiomatisation. We use the Hardy–Weinberg Principle of biology and the Bell’s Inequalities of quantum physics as examples.


Probability logic Team semantics Dependence logic 

Mathematics Subject Classification

03B48 03B60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramsky, S., Hardy, L.: Logical Bell Inequalities. Phys. Rev. A 85(062114), 1–11 (2012)Google Scholar
  2. 2.
    Carnap, R.: Logical Foundations of Probability. The University of Chicago Press, Chicago (1950)zbMATHGoogle Scholar
  3. 3.
    Gaifman, Haim: Concerning measures in first order calculi. Israel J. Math. 2, 1–18 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Goldblatt, R.: Lectures on the Hyperreals: An Introduction to Non-Standard Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hardy, G.H.: Mendelian proportions in a mixed population. Science 28(706), 49–50 (1908)CrossRefGoogle Scholar
  6. 6.
    Hurd, A.E., Loeb, P.A.: An Introduction to Non-Standard Real Analysis. Academic Press, London (1985)zbMATHGoogle Scholar
  7. 7.
    Hyttinen, T., Paolini, G., Väänänen, J.: Quantum team logic and Bell’s inequalities. Rev. Symb. Log. 08(04), 722–742 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Meyn, S., Tweedie, R.: Markov Chains and Stochastic Stability. Springer, London (1993)CrossRefzbMATHGoogle Scholar
  9. 9.
    Väänänen, J.: Dependence Logic. Cambridge University Press, London (2007)CrossRefzbMATHGoogle Scholar
  10. 10.
    Weinberg, W.: Über den Nachweis der Vererbung beim Menschen. Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg 64, 368–382 (1908)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Tapani Hyttinen
    • 1
  • Gianluca Paolini
    • 1
    Email author
  • Jouko Väänänen
    • 1
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  2. 2.Institute for Logic, Language and ComputationUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations