A monotonic rule is an expression of the following form or its reverse (using =>>, where , are sentences.
Such an expression should be distinguished from an implication like the following.
Athough sentences can be monotonic rules, monotonic rules are not sentences; they are similar to inference rules, familiar from elementary logic. If, for instance,
consists of some sentences
and one rule (<<=
are sentences without free variables, then the set of sentences entailed by
is the smallest set of sentences which (i) contains
, (ii) is closed under logical entailment, and (iii) contains
provided that it contains
. It is not generally true that this set contains the implication (<=
The rationale for using monotonic rules in knowledge representation, instead of implications, is twofold. On the one hand, the ``directed'' character of rules can simplify the task of developing efficient inference procedures. On the other hand, in some cases, replacing <<= by <= would be semantically unacceptable. For instance, the rules
(<<= (status-known ?x) (citizen ?x)) (<<= (status-known ?x) (not (citizen ?x)))
allow us to infer (status-known Joe) only if one of the sentences
(citizen Joe), (not (citizen Joe))
can be inferred. Replacing the rules by implications would make (status-known ?x) identically true.