Abstract
The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability theory are restated in this framework.
This paper is a shortened version of http://arxiv.org/abs/1307.5173v4.
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Notes
- 1.
Events are closed under \(-\vee -\), which represents alternative occurrence and \(-\wedge -\), which represents simultaneous occurrence, and under negation.
- 2.
Rational numbers and real numbers are instances of values.
- 3.
We will exclude probability functions with negative values, a phenomenon known in non-commutative probability theory, leaving the exploration of that kind of generalization to future work.
- 4.
In some cases the restriction to a single probability function P is impractical and providing a dedicated sort for such functions brings more flexibility and expressive power. This expansion may be achieved in different ways.
- 5.
We assume that in a context of partial functions an identity \(t=r\) is valid if either both sides are undefined or both sides are defined and equal. This convention, however, leaves room for alternative readings of the expressions at hand. In particular the definition given for \(x \lhd y \rhd z\) implies that whenever t is undefined, so is \(t \lhd r \rhd s\). That is not a very plausible feature of the conditional and in the presence of partial operations the conditional operator requires a different definition. These complications are to some extent avoided, or rather made entirely explicit, when working with total functions. The use of the notation \(P^\star (-|-)\) instead of the common notation \(P(-|-)\) is justified by the fact that unavoidably \(P^\star (-|-)\) inherits properties from the equational specification of the functions from which it has been made up. Such properties need not not coincide with what is expected from \(P(-|-)\).
- 6.
- 7.
More generally, \(\textit{BA}+\textit{Md}+\textit{Sign}+\textit{PF}_P\) is sound for the class of
-structures with 
a signed cancellation meadow. - 8.
From this pair of inequalities one can derive the original Bell inequalities from [3]. The key observation of Bell was that quantum mechanics gives rise to the hypothesis that a 4-dimensional event space exists in which a family of joint probabilities for at most two dimensions can be found that violates the inequalities from the theorem.
- 9.
This meeting took place at Science Park Amsterdam, Friday June 7, 2013 under the heading “Frontiers of Forensic Science”, and was organized by Andrea Haker.
-structures with 
a signed cancellation meadow.References
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Bergstra, J.A., Ponse, A. (2017). Probability Functions in the Context of Signed Involutive Meadows (Extended Abstract). In: James, P., Roggenbach, M. (eds) Recent Trends in Algebraic Development Techniques. WADT 2016. Lecture Notes in Computer Science(), vol 10644. Springer, Cham. https://doi.org/10.1007/978-3-319-72044-9_6
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