The expressions p implies q (p → q), q implies p (q → p), and p iff q (p ↔ q) can be shown to be logically equivalent by analyzing their truth tables.
Let’s define the truth values:
- True (T)
- False (F)
Now, we can construct a truth table for each of the expressions:
| p | q | p → q | q → p | p ↔ q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |
From the truth table, we see the following results:
- For p → q: it is only false when p is true and q is false.
- For q → p: it is only false when q is true and p is false.
- For p ↔ q: it is true when both p and q are either true or false.
Next, we notice that:
- p → q is true unless we have a true p and a false q, which is also consistent with p iff q when p and q differ.
- q → p behaves similarly, making it logically dependent on the same conditions.
Because both implications (p → q and q → p) contribute to the biconditional relationship (p ↔ q), the three expressions indeed exhibit logical equivalence. This makes sense as all three encapsulate the relationship between p and q in terms of their truth values. In a sense, they provide different perspectives on the same underlying conditions between the two propositions.