A General Non-Probabilistic Theory of Inductive Reasoning
Authors: Wolfgang Spohn
Date: 2013-03-27 00:00:00

Probability theory, epistemically interpreted, provides an excellent, if not
the best available account of inductive reasoning. This is so because there are
general and definite rules for the change of subjective probabilities through
information or experience; induction and belief change are one and same topic,
after all. The most basic of these rules is simply to conditionalize with
respect to the information received; and there are similar and more general
rules. 1 Hence, a fundamental reason for the epistemological success of
probability theory is that there at all exists a well-behaved concept of
conditional probability. Still, people have, and have reasons for, various
concerns over probability theory. One of these is my starting point:
Intuitively, we have the notion of plain belief; we believe propositions2 to be
true (or to be false or neither). Probability theory, however, offers no formal
counterpart to this notion. Believing A is not the same as having probability 1
for A, because probability 1 is incorrigible3; but plain belief is clearly
corrigible. And believing A is not the same as giving A a probability larger
than some 1 - c, because believing A and believing B is usually taken to be
equivalent to believing A & B.4 Thus, it seems that the formal representation
of plain belief has to take a non-probabilistic route. Indeed, representing
plain belief seems easy enough: simply represent an epistemic state by the set
of all propositions believed true in it or, since I make the common assumption
that plain belief is deductively closed, by the conjunction of all propositions
believed true in it. But this does not yet provide a theory of induction, i.e.
an answer to the question how epistemic states so represented are changed
tbrough information or experience. There is a convincing partial answer: if the
new information is compatible with the old epistemic state, then the new
epistemic state is simply represented by the conjunction of the new information
and the old beliefs. This answer is partial because it does not cover the quite
common case where the new information is incompatible with the old beliefs. It
is, however, important to complete the answer and to cover this case, too;
otherwise, we would not represent plain belief as conigible. The crucial
problem is that there is no good completion. When epistemic states are
represented simply by the conjunction of all propositions believed true in it,
the answer cannot be completed; and though there is a lot of fruitful work, no
other representation of epistemic states has been proposed, as far as I know,
which provides a complete solution to this problem. In this paper, I want to
suggest such a solution. In [4], I have more fully argued that this is the only
solution, if certain plausible desiderata are to be satisfied. Here, in section
2, I will be content with formally defining and intuitively explaining my
proposal. I will compare my proposal with probability theory in section 3. It
will turn out that the theory I am proposing is structurally homomorphic to
probability theory in important respects and that it is thus equally easily
implementable, but moreover computationally simpler. Section 4 contains a very
brief comparison with various kinds of logics, in particular conditional logic,
with Shackle's functions of potential surprise and related theories, and with
the Dempster - Shafer theory of belief functions.