Eduprobe
Question:
The statement ¬(p ↔ ¬q) is equivalent to p ↔ q. Is it a tautology or a fallacy?
a tautology
a fallacy
equivalent to p ↔ q
equivalent to ¬(p ↔ q)
Solution:
Let's analyze the statement ¬(p ↔ ¬q). We'll use a truth table to determine its equivalence to p ↔ q.
| p | q | ¬q | p ↔ ¬q | ¬(p ↔ ¬q) | p ↔ q |
|---|---|---|---|---|---|
| T | T | F | F | T | T |
| T | F | T | T | F | F |
| F | T | F | T | F | F |
| F | F | T | F | T | T |
As you can see from the truth table, the columns for ¬(p ↔ ¬q) and p ↔ q are identical. Therefore, the statement ¬(p ↔ ¬q) is equivalent to p ↔ q. Since they are equivalent, it is not a fallacy. However, it's not a tautology either, as it's not always true. It is only true when p and q have the same truth value.
Therefore, the correct answer is that ¬(p ↔ ¬q) is equivalent to p ↔ q. It is neither a tautology nor a fallacy; it's a logical equivalence.