In this notation, the axioms are simply rules without hypotheses such as in 1.7.1(i):
(a    (0 --> a))
, one)
7
j .   .i ,< i   - ■ ,    -       -    < .        i        ...An are
(Nonmonotonic formal systems):    Let E/ be a set (of proposition*!
Example 7.2:   Let U --- (a. A 7> and let
a : 3
/*(.  ^ HI" fli"lit6 lists of 6I6JT
•en in the form
«1,... ,«„ : Aii • • •) An
U.
We call at,. • •, an the premise,? of the rule r and Ai      Am its restraint1;.
Note that either P or G or both may be empty,
i'ii) If P = G = 0, then the rule r is called an aaxora,
(iii) A nonmonotonic formal system is a pair  (D", " > where U is a nonempty
set (of prepositional letters) and N is a set of nonmonotonic rules.
(iv) A subset S of l7 is deductively closed in the system {£/, iV} if, for each rut r of JV such that all the premises ai;,... ,«„ of r axe in 5 and none of it restraints Ai, • • • 1 Am are in 5, the conclusion <p of r is in S.
Defc ■