Question

If p, q, and r are three formulas, then if p and q both entail each other, and q and r both entail each other, then p and r are equivalent.

A. True
B. False

Answer 1

The statement is true; if two formulas, p and q, entail each other, and q and r also entail each other, then by logic, p and r are equivalent.

Step-by-step explanation:

The initial question asks whether if two formulas, p and q, entail each other, and q and r also entail each other, then p and r must be equivalent. This statement is true. If p entails q, it means that whenever p is true, q must be true, and vice versa; this implies that p and q are equivalent. Similarly, if q entails r, then q and r are equivalent. Since p is equivalent to q, and q is equivalent to r, by the transitive property of equivalence, p must also be equivalent to r. This forms a deductive argument structure, much like the examples provided, where the entailment of two items to a common third item leads to their equivalency to each other.