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**Characterization of the Almost Agreeing Probabilities for the
Kraft-Pratt-Seidenberg Structures**

**Petros Hadjicostas**

### Abstract:

*When ``'' we say that ``event is at least as probable as
event ''. This is a qualitative probability judgment. A qualitative
probability structure, , where is an algebra
on the set and is a binary relation on , satisfies
connectedness, transitivity, nontriviality, nonnegativity, and additivity.
In 1959, Kraft, Pratt, and Seidenberg answered in the negative the question
of whether every finite structure of qualitative probability has a finitely
additive, order-preserving probability representation.
They achieved that by constructing an ordering over a 32-element algebra, the
powerset of a 5-atom set, that does not have a probability representation.
Their ordering satisfies four crucial conditions that prevent the existence of
strictly agreeing quantitative probability representations. In addition, the
three authors showed that there is an almost agreeing probability for their
example. They went even further and constructed an example that does not even
have an almost agreeing probability representation. The purpose of this paper
is to review the proofs of the three authors, and to characterize all almost
agreeing probabilities for all qualitative probability structures on the
powerset of a 5-atom set that satisfy the four crucial conditions
I mentioned above. In doing so, I prove some theorems that facilitate showing
that (with finite) satisfies additivity.
**
*