Cylindric Probability Algebras

Sometimes it is not enough to prove that a formula φ(

x

) is satisfied or not by an element

a

∈

A

in the model

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$$ \mathfrak{A} $$

we are interested in. It may happen that we are looking for the quantity of those

a

∈

A

for which φ[

a

] is true. Of course, the mathematical discipline for these considerations is probability theory. The logic suitable for this kind of reasoning was introduced by H. J. Keisler in 1976. This logic has formulas similar to those of

L

A

⊆

L

ω1ω

(

A

is a countable admissible set), except that the quantifiers (

Px

≥

r

) (

r

∈

A

∪ [0, 1] is a real number) are used instead of the usual quantifiers (∀

x

) and (∃

x

). A model for this logic is a pair

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$$ \left\langle {\mathfrak{A},\mu } \right\rangle $$

, where

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$$ \mathfrak{A} $$

is a classical structure and μ is a probability measure defined in such a way that definable subsets of the universe

A

are measurable. The quantifiers are interpreted in the natural way, i.e.

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$$ \left\langle {\mathfrak{A},\mu } \right\rangle | = \left( {Px \geqslant r} \right)\phi \left( x \right)\quad \quad iff\quad \quad \mu \left\{ {a \in A:\left\langle {\mathfrak{A},\mu } \right\rangle | = \phi \left[ a \right]} \right\} \geqslant r. $$