Question

If two formulas, p and q, both entail a third formula, then p and q must be logically equivalent.

A. True
B. False

Answer 1

The statement that two formulas, p and q, which both entail a third formula must be logically equivalent is false. Two different causes can lead to the same effect without being equivalent.

Step-by-step explanation:

If two formulas, p and q, both entail a third formula, it does not necessarily mean that p and q are logically equivalent. Two formulas can lead to the same conclusion without being interchangeable in all scenarios. For example, consider two different premises that lead to the same outcome; they can both be true in their contexts yet may not be equivalent in their applications or implications.

Considering two premises:

1. If it is raining, then the ground is wet.
2. If the sprinkler is on, then the ground is wet.

Both premises lead to the conclusion that the ground is wet, yet one does not necessarily imply that it is raining (p), nor does the other necessarily imply that the sprinkler is on (q). They are not logically equivalent despite leading to the same conclusion. Thus, the statement is False.