Consider
Statement‑1: (p ~ q) (~ p q) is a fallacy.
Statement‑2: (p q) (~ q ~ p) is a tautology.
Statement‑I: (p ~ q) (~ p q) ~ (p q) ~ (q p)
Statement‑I is true.
Statement‑II: (p q) (~ q ~ p)
(~ q ~ p) is equivalent to (p q)
(p q) (p q)
Hence, (p q) (~ q ~p) is tautology.
Statement-II is true but not correct explanation Statement-I.
We are given two logical statements and need to determine their truth values and whether Statement-II explains Statement-I.
Statement‑I: (p ∧ ~q) ∧ (~p ∧ q) is a fallacy (i.e., it is always false).
Let's construct a truth table for (p ∧ ~q) ∧ (~p ∧ q):
| p | q | ~p | ~q | p ∧ ~q | ~p ∧ q | (p ∧ ~q) ∧ (~p ∧ q) |
|---|---|---|---|---|---|---|
| T | T | F | F | F | F | F |
| T | F | F | T | T | F | F |
| F | T | T | F | F | T | F |
| F | F | T | T | F | F | F |
As we can see, the final column is false for all combinations of p and q. Therefore, Statement-I is true; it is indeed a fallacy (a contradiction).
Statement‑II: (p → q) ↔ (~q → ~p) is a tautology (i.e., it is always true).
Recall that ~q → ~p is the contrapositive of p → q. In logic, an implication and its contrapositive are logically equivalent. Therefore, (p → q) ↔ (~q → ~p) should always be true. Let's verify this with a truth table.
| p | q | p → q | ~q | ~p | ~q → ~p | (p → q) ↔ (~q → ~p) |
|---|---|---|---|---|---|---|
| T | T | T | F | F | T | T |
| T | F | F | T | F | F | T |
| F | T | T | F | T | T | T |
| F | F | T | T | T | T | T |
The final column is true for all combinations of p and q. Therefore, Statement-II is also true; it is a tautology.
Statement-II is a fundamental law of logic (the equivalence of an implication and its contrapositive). Statement-I is a specific compound proposition that is a contradiction. The truth of Statement-II (a law about implications) does not directly explain why the specific conjunction in Statement-I is always false. They are two separate, unrelated logical facts. Therefore, Statement-II is not a correct explanation for Statement-I.
Statement‑I is true, Statement‑II is true, Statement‑II is not a correct explanation for Statement‑I.