Probabilistic and Non-Monotonic Inference
  (l) I have enough evidence to render the sentence S probable. (la) So,
relative to what I know, it is rational of me to believe S. (2) Now that I have
more evidence, S may no longer be probable. (2a) So now, relative to what I
know, it is not rational of me to believe S. These seem a perfectly ordinary,
common sense, pair of situations. Generally and vaguely, I take them to embody
what I shall call probabilistic inference. This form of inference is clearly
non-monotonic. Relatively few people have taken this form of inference, based
on high probability, to serve as a foundation for non-monotonic logic or for a
logical or defeasible inference. There are exceptions: Jane Nutter [16] thinks
that sometimes probability has something to do with non-monotonic reasoning.
Judea Pearl [ 17] has recently been exploring the possibility. There are any
number of people whom one might call probability enthusiasts who feel that
probability provides all the answers by itself, with no need of help from
logic. Cheeseman [1], Henrion [5] and others think it useful to look at a
distribution of probabilities over a whole algebra of statements, to update
that distribution in the light of new evidence, and to use the latest updated
distribution of probability over the algebra as a basis for planning and
decision making. A slightly weaker form of this approach is captured by Nilsson
[15], where one assumes certain probabilities for certain statements, and
infers the probabilities, or constraints on the probabilities of other
statement. None of this corresponds to what I call probabilistic inference. All
of the inference that is taking place, either in Bayesian updating, or in
probabilistic logic, is strictly deductive. Deductive inference, particularly
that concerned with the distribution of classical probabilities or chances, is
of great importance. But this is not to say that there is no important role for
what earlier logicians have called "ampliative" or "inductive" or "scientific"
inference, in which the conclusion goes beyond the premises, asserts more than
do the premises. This depends on what David Israel [6] has called "real rules
of inference". It is characteristic of any such logic or inference procedure
that it can go wrong: that statements accepted at one point may be rejected at
a later point. Research underlying the results reported here has been partially
supported by the Signals Warfare Center of the United States Army.
