Lp : A Logic for Statistical Information
  This extended abstract presents a logic, called Lp, that is capable of
representing and reasoning with a wide variety of both qualitative and
quantitative statistical information. The advantage of this logical formalism
is that it offers a declarative representation of statistical knowledge;
knowledge represented in this manner can be used for a variety of reasoning
tasks. The logic differs from previous work in probability logics in that it
uses a probability distribution over the domain of discourse, whereas most
previous work (e.g., Nilsson [2], Scott et al. [3], Gaifinan [4], Fagin et al.
[5]) has investigated the attachment of probabilities to the sentences of the
logic (also, see Halpern [6] and Bacchus [7] for further discussion of the
differences). The logic Lp possesses some further important features. First, Lp
is a superset of first order logic, hence it can represent ordinary logical
assertions. This means that Lp provides a mechanism for integrating statistical
information and reasoning about uncertainty into systems based solely on logic.
Second, Lp possesses transparent semantics, based on sets and probabilities of
those sets. Hence, knowledge represented in Lp can be understood in terms of
the simple primative concepts of sets and probabilities. And finally, the there
is a sound proof theory that has wide coverage (the proof theory is complete
for certain classes of models). The proof theory captures a sufficient range of
valid inferences to subsume most previous probabilistic uncertainty reasoning
systems. For example, the linear constraints like those generated by Nilsson's
probabilistic entailment [2] can be generated by the proof theory, and the
Bayesian inference underlying belief nets [8] can be performed. In addition,
the proof theory integrates quantitative and qualitative reasoning as well as
statistical and logical reasoning. In the next section we briefly examine
previous work in probability logics, comparing it to Lp. Then we present some
of the varieties of statistical information that Lp is capable of expressing.
After this we present, briefly, the syntax, semantics, and proof theory of the
logic. We conclude with a few examples of knowledge representation and
reasoning in Lp, pointing out the advantages of the declarative representation
offered by Lp. We close with a brief discussion of probabilities as degrees of
belief, indicating how such probabilities can be generated from statistical
knowledge encoded in Lp. The reader who is interested in a more complete
treatment should consult Bacchus [7].
