Mathematical Foundation of Computer Science

DHARM

PROPOSITIONAL LOGIC 115

R

F

F

T

T

C

F*

T

F*

T

RC→

T

T

F

T

Corresponding to above, symbolic logic of two premises & conclusion are shown below,

(i)R → C where, R : it rains

(ii) R C : I will not go to Institute

∴ C
From given set of premises and the conclusion, we can justify that argument is valid or
invalid by formal proof. An argument is a valid argument if truth values of all premises is
true (T) and truth value of conclusion must also be true (T). Consequently, if particular set of
premises derived the conclusion then it is a valid conclusion.
If truth value of all premises is true (T) but truth value of the conclusion is false (F) then
argument is invalid argument. Consequently, if we find any one interpretation which makes
the premises true (T) but conclusion is false (F) then argument is an invalid argument. Simi-
larly, if set of premises not derived the conclusion correctly then it is an invalid conclusion.
To investigate the validity of an argument we take the preveious example, i.e.,
(i)R → C
(ii)R
∴ C
Form the table shown in Fig. 5.18 we find conclusion (C) is F
when,
(i) R is F and R → C is T; or
(ii) R is T and R → C is F
Thus, we find no interpretation so that argument is invalid.
Hence, we have a valid conclusion and the argument is a valid argu-
ment.
Validity of an argument is also justified by assuming that particular set of premises and
the conclusion construct a formula (say X). Rule for constructing the formula X is as follows,
Let premises are S 1 , S 2 , S 3 ,.........Sk that derives the conclusion C then formula X will
be,
X : ((.........((S 1 ∧ S 2 ) ∧ S 3 )..........∧ Sk) → C)
Here formula X is an implication formula that will be obtained by putting the conjunc-
tion of all premises as the antecedent part and the conclusion as the consequent part.
It means we have the only conclusion,
∴ ((.........((S 1 ∧ S 2 ) ∧ S 3 )..........∧ Sk) → C)
· If antecedent part is T and also consequent part is T i.e.,
⇒ T → T ⇒ T
Hence, argument is a valid argument.
· If antecedent part is F and consequent part is T i.e.,
⇒ F → T ⇒ T
Again, argument is a valid argument.
· If antecedent part is T and consequent part is F i.e.,
⇒ T → F ⇒ F
Hence, argument is invalid.
Thus, we conclude that formula X must be tautology for valid argument.

Fig. 5.18

R

ST