Conditional Quantification, or Poor Man's Probability

The purpose of this note is to take a few steps toward a first order logic of a particular type of incomplete information: incomplete knowledge about the value of a variable, where the ‘degree of incompleteness’ is given by some algebraic structure. The specific form of incomplete information we have in mind is best introduced by means of an example from probability theory. Let Ω be a sample space equipped with a second countable Hausdorff topology, and let X : Ω −→ IR be a random variable. Think of X as representing some measurement apparatus. If B is the Borel σ-algebra on Ω, then the fact that singletons are in B represents the assumption that an outcome can be observed with arbitrary precision. In practice, however, it may be impossible to observe an outcome X(ω) with arbitrary precision. For instance, the best we can do may be to locate X(ω) in some element of a partition of IR into intervals of length . Observe that two notions of incomplete information should be distinguished here. Let a probability measure P : B −→ IR be given. Now if A is a Borel subset of IR, we may not know for sure whether measurement of X will yield an outcome in A, but at least P (X ∈ A) is welldefined, so we know the probability to find an outcome in A. This presupposes however that it is possible to determine the outcomes of X exactly; otherwise the statement X ∈ A will not obey the requisite two-valued logic. If on the other hand the outcomes of X

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