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Davis [4], however, explained the above phenomena as follows.
{ P → ¬ Q , Q → ¬ P } |~ (¬ P ∨ ¬ Q )
and { P→ ¬ P } |~ falsum.
Problems encountered in NML I
McDermott and Doyle identified two basic problems in connection with
reasoning with NML I. These are
i) A cannot be inferred from (A ∧ B)
ii) T = { P → Q, ¬ Q } |~ falsum,
which means that the axiomatic theory T is inconsistent in NML I.
7.5 Non-Monotonic Reasoning Using NML II
In order to overcome the above difficulties, McDermott and Doyle recast nonmonotonic
logic with the help of another modal [4] operator like consistency,
called the necessitation operator and symbolized by . This operator is related
to modal operator by the following manner:
P ≡ ¬ ¬ P
or P ≡ ¬ ¬ P
where the former notation denotes that P is necessary could be described
alternatively as negation of P is not consistent. The second definition
implicates that P is consistent could be written alternatively as the negation of
P is not necessary.
The significance of the necessitation operator can be understood from the
following example.
Example 7.3: Given that T |~ A, i.e., A is consistent with the derived
consequences from T. Then it can be inferred that
T |~ ¬ ¬ A,
which means that it is not necessary to assume that ¬ A is derivable from T.
This is all about the example.