The aim of the paper is to prove the completeness theorem for biprobability models. This also solves Keisler's Problem 5.4 (see [4]).
Let
be a countable admissible set and ω ∈
. The logic
is similar to the standard probability logic
. The only difference is that two types of probability quantifiers
and
are allowed.
A biprobability model is a structure (
, μ1, μ2) where
is a classical structure without operations and μ1, μ2 are two types of probability measures such that μ1 is absolutely continuous with respect to μ2, i.e. μ1 ≪ μ2.
The quantifiers are interpreted in the natural way, i.e.
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200030784/resource/name/S0022481200030784_eqnU1.gif?pub-status=live)
for i = 1, 2. (The measure
is the restriction of the completion of
to the σ-algebra generated by the measurable rectangles and the diagonal sets ![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151027033958516-0657:S0022481200030784_inline10.gif?pub-status=live)
![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151027033958516-0657:S0022481200030784_inline11.gif?pub-status=live)
Axioms and rules of inference are those of
, as listed in [2] with the axiom B4 from [4], with the remark that both P1 and P2 can play the role of P, together with the following axioms:
Axioms of continuity.
1)
.
2)
.
Axiom of absolute continuity:
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200030784/resource/name/S0022481200030784_eqnU2.gif?pub-status=live)
where
and Φn = {φ ∈ Φ: φ has n free variables}.