Logic Strand Lecture 3

Rule of Substitution of Equivalence: If in a tautology we
replace any part of a statement by a statement equivalent to
that part, the result is still a tautology.
Example:
• Determine if P ⇒ (~ Q ∨ P)
is a tautology.
We know: P ⇒ ( Q ⇒ P)
is a tautology and
( P ⇒ Q)
≡~
P ∨ Q
By the rule of substitution
( Q ⇒ P)
≡~
Q ∨ P
Thus, by the rule of substitution of equivalence,
P ⇒ ( Q ⇒ P)
≡ P ⇒ (~ Q ∨ P)
, and hence
P ⇒ (~ Q ∨ P) is also a tautology.
Exercise:
• ~ T ∨ (~ S ∨ T ) a tautology? Yes.
We know ( P ⇒ Q)
≡~
P ∨ Q. So, ( S ⇒ T ) ≡~
S ∨ T and
T ⇒ (~ S ∨ T ) ≡~
T ∨ (~ S ∨ T ) (by RoS).
Hence, ~ T ∨ (~ S ∨ T ) ≡ T ⇒ ( S ⇒ T ) (by SoE).
P ⇒ ( Q ⇒ P) is a known tautology, thus (by (SoE)
T ⇒ ( S ⇒ T ) is a tautology, and since
~ T ∨ (~ S ∨ T ) ≡ T ⇒ ( S ⇒ T ), ~ T ∨ (~ S ∨ T ) is a
tautology.
WUCT121 Logic 54