16 J;;i Theory ofComputer Science
We give in Table 1.13 some of the important rules of inference. Of course,
we can derive more rules of inference and use them in valid arguments.
For valid arguments, we can use the rules of inference given in
Table 1.13. As the logical identities given in Table 1.11 are two-way
implications. we can also use them as rules of inference.
TABLE 1.13 Rules of Inference
Rule of inference
Implication form
RI 1 : Addition
P
:. PvQ
Rh Conjunction
p
Q
:. P A Q
Rh Simplification
PAQ
P
Rh: Modus ponens
P
P=>Q
~
F~I5: Modus tollens
-,Q
P=>Q
:. ~P
RIs: Disjunctive syllogism
-,P
PvQ
~
RI 7 : Hypothetical syllogism
P=>Q
Q=>R
:. P => R
RIa: Constructive dilemma
(P => Q) /" (R => 8)
PvR
:. Qv S
RIg: Destructive dilemma
CJ => Q) 1\ (R => 8)
~Qv--,S
:. P v R
P => (P v Q)
(P A Q) => P
(P 1\ (P => Q)) => Q
(-, Q 1\ (P => Q)) => -, Q
(-, P 1\ (P V Q)) => Q
((P => Q) 1\ (Q => R)) => (P => R)
((P => Q) 1\ (R => 8) 1\ (P v R)) => (Q v 8)
((P => Q) 1\ (R => 8) 1\ (-, Q v -,8)) => (--, P v -, R)
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