Sum
Examine whether the following statement pattern is a tautology or a contradiction or a contingency.
[p → (∼ q ∨ r)] ↔ ∼ [p → (q → r)]
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Solution
| p | q | r | ∼ q | ∼ q ∨ r | p → (∼ q ∨ r) | q → r | p → (q → r) | ∼[p → (q → r)] | [p → (∼ q ∨ r)] ↔ ∼ [p → (q → r)] |
| T | T | T | F | T | T | T | T | F | F |
| T | T | F | F | F | F | F | F | T | F |
| T | F | T | T | T | T | T | T | F | F |
| T | F | F | T | T | T | T | T | F | F |
| F | T | T | F | T | T | T | T | F | F |
| F | T | F | F | F | T | F | T | F | F |
| F | F | T | T | T | T | T | T | F | F |
| F | F | F | T | T | T | T | T | F | F |
All the entries in the last column of the above truth table are F.
∴ [p → (∼ q ∨ r)] ↔ ∼ [p → (q → r)] is a contradiction.
Concept: Statement Patterns and Logical Equivalence
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