Conditional Quantification, or Poor Man's Probability

The purpose of this note is to take a few steps toward a first order logic of incomplete information, based on the idea that such a logic may be obtained by importing an essential ingredient of probability theory, conditional expectation, into the logic. The reason that we focus on conditional expectation is that it is often used to model a kind of incomplete knowledge that is of interest in itself: incomplete knowledge about the value of a variable, where the ‘degree of incompleteness’ is given by some algebraic structure. Conditional expectation often comes into play when what can be described as a change of granularity is involved. Formal details will be given below, but a typical example is this. 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 2. Let B′ be the σ-algebra generated by those intervals. Then G = X−1B′ is strictly contained in B. The conditional expectation E(X|G) is in a sense X viewed from the perspective of G; the values of X are averaged over an element ∗I am indebted to Samson Abramsky, Alan Bundy, Jaap van der Does, Rosalie Iemhoff, Alekos Kechris, Daniele Turi and Marco Vervoort for helpful comments. Support by the Netherlands Organisation for Scientific research (NWO) under grant PGS 22 262 is gratefully acknowledged. This paper was begun in Amsterdam and finished at HCRC in Edinburgh. I thank Keith Stenning for his friendship and hospitality, and the EPSRC for financial support.