Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

7.1 Introduction
Predicate Logic is monotonic [3] in the sense that a derived theorem never
contradicts the axioms or already proved theorems [4], from which the former
theorem is derived. Formally, if a theorem T1 is deduced from a set of axioms
A = {a1, a2, a3,...,an}, i.e., A |- T1, then a theorem T2, derived from (A ∪ T1)
(i.e., (A ∪ T1) |- T2), never contradicts T1. In other words T1 and T2 are
consistent. Most of the mathematical theorem provers work on the above
principle of monotonicity. The real world, however, is too complex and there
exist situations, where T2 contradicts T1. This called for the development of
non-monotonic logic.
7.2 Monotonic versus Non-Monotonic Logic
The monotonicity of Predicate Logic can be best described by the following
theorem.
Theorem 7.1: Given two axiomatic theory T and S, which includes a set of
axioms (ground clauses) and a set of first order logical relationships like
Modus Ponens, Modus Tollens, etc. If T is a subset (or proper subset of) S
then Th(T) is also a subset (or proper subset) of Th(S), where Th(T) (or
TH(S)) means the theorems derivable from T (or S) [4].
An interpretation of the above theorem is that adding new axioms to an
axiomatic theory preserves all theorems of the theory. In other words, any
theorem of the initial theory is a theorem of the enlarged theory as well.
Because of default assumptions in reasoning problems, the monotonicity
property does not hold good in many real world problems. The following
example is used to illustrate this phenomenon.
Example 7.1: Let us consider the following information about birds in an
axiomatic theory T :
Bird (tweety)
Ostrich (clide)
∀ X Ostrich (X) → Bird (X) ∧ ¬ Fly (X).
Further, we consider the default assumption R, which can be stated as
R: ( T |~ Bird (X) ) ,
(T |~/ ¬ Fly(X)) → (T |~ Fly (X))