Monotonic Rules

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## Monotonic Rules

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 (<<=

), where

and

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.

Vishal I. Sikka
Wed Dec 7 13:23:42 PST 1994