Is there a probability theory developed in intuitionistic logic?

Question

Since Boole it is known that probability theory is closely related to logic.

According to the axioms of Kolmogorov, probability theory is formulated with a (normalized) probability measure $\mbox{Pr}\colon \Sigma \to [0,1]$ on a Boolean $\sigma$-algebra $\Sigma$ (of events).

Realizing these data by a set $X$ (sample space of elementary events) and a corresponding $\sigma$-algebra $\Sigma(X)\subseteq P(X)$ of subsets of $X$, one obtains a probability space $(X,\Sigma(X),\mbox{Pr})$.

The $\sigma$-homomorphisms $f \colon {\cal B}({\mathbb R})\to \Sigma$ (real $\Sigma$-valued measures) are defined on the Borel $\sigma$-algebra ${\cal B}({\mathbb R})$ of the real Borel sets. They can be realized by real-valued measurable functions $F\colon X\to {\mathbb R}$ (random variables).

I wonder how this theory extends from the classical to the intutionistic logic i.e. from the Boolean to the Heyting ($\sigma$-)algebras and what the major differences between the two theories are.

Where can I find precise descriptions of the following topics:

1. Definition and properties of probability measures on a Heyting algebra ${\cal H}$.

2. Definition and properties of real ${\cal H}$-valued measures $f \colon {\cal B}({\mathbb R})\to {\cal H}$.

(Already the discrete case would be of interest.)

(BTW: Boole 1815–1864; Heyting 1898–1980; Kolmogorov 1903–1987)

Answer 1

The title of the question is a bit of a misnomer, or at least has the potential to cause confision. "Intutionistic probability theory" means to me "theory of probability developed in intuitionistic logic". But you seem to be asking whether we can replace $\sigma$-algerbas (which are Boolean algebras) with Hetying algebras.

Regarding your first question, the relevant notion is that of a valuation. This is like a real-valued measure, but it is defined only on the open sets instead of Borel sets. The open sets form a complete Heyting algebra, so this should provide you with a starting point.

Regarding your second question about Heyting-algebra-valued measures on Borel sets, I would just like to observe that a homomorphism $f : \mathcal{B}(\mathbb{R}) \to \mathcal{H}$ preserves complements (because it preserves $0$, $1$, $\land$ and $\lor$), which means that it factors through the regularization of $\mathcal{H}$ (the Boolean algebra consisting of those elements of $\mathcal{H}$ that are closed under double complement). So you might as well replace $\mathcal{H}$ with its Boolean algebra of regular elements, or else consider instead $\mathcal{H}$-valued valuations $\mathcal{O}(\mathbb{R}) \to \mathcal{H}$.

Answer 2

As noted by Andrej Bauer it is good to make a distinction between developing probability theory using intuitionistic logic, and making a new framework for probability compatible with intuitionistic logic.

To clarify, the first program concerns doing measure theory using constructive mathematics. Here probability is viewed as a field of mathematics about bounded measures. I think the earliest substantive research in this direction is Bishop and Cheng's paper "constructive measure theory" (1972) and Chan's "Notes on constructive probability theory" (1974).

The second program is more philosophical and views probability as an extension of logic. To make probability compatible with intuitionistic logic a deviation from Kolmogorov's axiomatization is required. The most direct way is to define probability functions as functions on a Heyting algebra. Proposals in this direction can be found in

Leblanc "Probabilistic semantics for first-order logic" (1979)

Morgan & Leblanc "Probability Theory, Intuitionism, Semantics, and the Dutch Book Argument" (1983)

van Fraassen "Probabilistic Semantics Objectified" (1981)

A more recent approach that uses Kripke semantics for intuitionistic logic is

Weatherson "From Classical to Intuitionistic Probability" (2003)

Answer 3

There is a probability theory developed in Łukasiewicz logic. If you understand the french (and if you are still interested): http://www.brunodefinetti.it/Opere/logiquedelaProbabilite.pdf