Let p, q, r and z be four primitive statements. Consider the following arguments:

P: [(p v q) ^ (p - r) ^ (q - r)] -...

Question

Let p, q, r and z be four primitive statements. Consider the following arguments:

P: [(p v q) ^ (p -> r) ^ (q -> r)] -> r
Q: [(p -> q) ^ (p -> ~q)] -> ~p
R: [(p -> q) ^ ~p] -> ~q
S: [(p -> q) -> r] -> (q -> r)

Which of the above arguments are valid?

A. P and Q only
B. P and R only
C. P and S only
D. P, Q, R and S

Accepted Answer

P and S only (Option C) An argument is valid if and only if the conditional (premises -> conclusion) is a tautology — true under every truth assignment. We test each of P, Q, R, S by checking whether any truth assignment makes the premises all true while the conclusion is false. P uses proof by cases (disjunction elimination): if both p and q independently lead to r, and at least one of p, q holds, then r must hold. Q captures the principle of contradiction — if a hypothesis implies both q and ~q, the hypothesis must be false (~p). R is the classic fallacy of denying the antecedent: from p -> q and ~p we cannot conclude ~q (q might hold for another reason). S is valid because if (p -> q) -> r, setting p = q = True makes (p -> q) = True, so r = True; and (q -> r) = (True -> True) = True — it holds in all cases.