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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 (normed) 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)

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Just to put the first sentence into perspective from the historical point of view: Boole 1815-1864, Kolmogorov 1903-1987. –  Goldstern Aug 12 '13 at 20:08
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up vote 11 down vote accepted

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 $\mathcal{O}(\mathbb{R}) \to \mathcal{H}$.

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