Logical Reasoning by Contradiction
Idea of Contradiction
If a set of assumptions leads to a contradiction (something impossible like R ∧ ¬R),
then at least one assumption must be false.
Contradiction Form
If:
(P ∧ Q) ⇒ (R ∧ ¬R)
Then the assumptions P and Q cannot both be true.
Logical Equivalence
A contradiction implication simplifies to:
((P ∧ Q) ⇒ (R ∧ ¬R)) ⇔ (¬P ∨ ¬Q)
Key Insight
If an assumption leads to contradiction, reject at least one assumption.
If only one assumption exists, it must be false.
Truth Table Insight
Both expressions behave the same logically:
| P | Q | P ∧ Q | R ∧ ¬R | (P ∧ Q) ⇒ (R ∧ ¬R) | ¬P ∨ ¬Q |
| T | T | T | F | F | F |
| T | F | F | F | T | T |
| F | T | F | F | T | T |
| F | F | F | F | T | T |
Interpretation
Both columns match, meaning the contradiction implication is equivalent to ¬P ∨ ¬Q.