Arch. Math. Logic (2017) 56:475–489
A logic for arguing about probabilities in measure teams
· Gianluca Paolini
Received: 20 September 2015 / Accepted: 7 April 2017 / Published online: 13 April 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract 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 prop-
erties 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 quan-
tum physics as examples.
Keywords Probability logic · Team semantics · Dependence logic
Mathematics Subject Classiﬁcation 03B48 · 03B60
The logic of propositions with assigned probabilities is usually associated with nond-
eductive methods such as inductive reasoning . The concept of probability in such
an approach is the degree of conﬁrmation or belief. Instead, in this paper we assign
probabilities to propositions using the frequency interpretation and study properties of
such probabilities. Thus, while probability logic usually focuses on the question how
to assign probabilities to composite formulas, we focus on the symmetric question how
The research of the second author was supported by the Finnish Academy of Science and Letters (Vilho,
Yrjö and Kalle Väisälä foundation).
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam,