Some Extensions of Probabilistic Logic
  In [12], Nilsson proposed the probabilistic logic in which the truth values
of logical propositions are probability values between 0 and 1. It is
applicable to any logical system for which the consistency of a finite set of
propositions can be established. The probabilistic inference scheme reduces to
the ordinary logical inference when the probabilities of all propositions are
either 0 or 1. This logic has the same limitations of other probabilistic
reasoning systems of the Bayesian approach. For common sense reasoning,
consistency is not a very natural assumption. We have some well known examples:
{Dick is a Quaker, Quakers are pacifists, Republicans are not pacifists, Dick
is a Republican}and {Tweety is a bird, birds can fly, Tweety is a penguin}. In
this paper, we shall propose some extensions of the probabilistic logic. In the
second section, we shall consider the space of all interpretations, consistent
or not. In terms of frames of discernment, the basic probability assignment
(bpa) and belief function can be defined. Dempster's combination rule is
applicable. This extension of probabilistic logic is called the evidential
logic in [ 1]. For each proposition s, its belief function is represented by an
interval [Spt(s), Pls(s)]. When all such intervals collapse to single points,
the evidential logic reduces to probabilistic logic (in the generalized version
of not necessarily consistent interpretations). Certainly, we get Nilsson's
probabilistic logic by further restricting to consistent interpretations. In
the third section, we shall give a probabilistic interpretation of
probabilistic logic in terms of multi-dimensional random variables. This
interpretation brings the probabilistic logic into the framework of probability
theory. Let us consider a finite set S = {sl, s2, ..., Sn) of logical
propositions. Each proposition may have true or false values; and may be
considered as a random variable. We have a probability distribution for each
proposition. The e-dimensional random variable (sl,..., Sn) may take values in
the space of all interpretations of 2n binary vectors. We may compute absolute
(marginal), conditional and joint probability distributions. It turns out that
the permissible probabilistic interpretation vector of Nilsson [12] consists of
the joint probabilities of S. Inconsistent interpretations will not appear, by
setting their joint probabilities to be zeros. By summing appropriate joint
probabilities, we get probabilities of individual propositions or subsets of
propositions. Since the Bayes formula and other techniques are valid for
e-dimensional random variables, the probabilistic logic is actually very close
to the Bayesian inference schemes. In the last section, we shall consider a
relaxation scheme for probabilistic logic. In this system, not only new
evidences will update the belief measures of a collection of propositions, but
also constraint satisfaction among these propositions in the relational network
will revise these measures. This mechanism is similar to human reasoning which
is an evaluative process converging to the most satisfactory result. The main
idea arises from the consistent labeling problem in computer vision. This
method is originally applied to scene analysis of line drawings. Later, it is
applied to matching, constraint satisfaction and multi sensor fusion by several
authors [8], [16] (and see references cited there). Recently, this method is
used in knowledge aggregation by Landy and Hummel [9].
