Final answer:
The three argument patterns presented in the question are disjunctive syllogism, modus ponens, and modus tollens. For argument A, the disjunctive syllogism pattern is used to prove its validity. Argument B follows the modus tollens pattern, and argument C can be considered a combination of universal instantiation and existential generalization. All three arguments are valid.
Step-by-step explanation:
The three argument patterns presented in the question are disjunctive syllogism, modus ponens, and modus tollens.
A. To prove the validity of argument A, we can use the disjunctive syllogism pattern. Rewrite the argument in the form:
- P → (Q ∧ R)
- P ∨ S
- ¬S ∧ T
- ∴ R
The third premise (¬S ∧ T) states that ¬S is true. Since P ∨ S is also true (second premise), we can conclude that ¬S is false. Therefore, S is true. By using the first premise (P → (Q ∧ R)), we know that if P is true, then Q ∧ R must also be true. Since P is true (from the second premise), we can deduce that Q ∧ R must also be true. From Q ∧ R, we can conclude that R is true, which matches the conclusion of the argument.
B. The argument pattern in B is modus tollens. Rewrite the argument in the form:
- (¬T ∧ S) → ¬R
- (¬Q ∨ P) → S
- ¬T → Q
- ∴ ¬R ∨ U
By using the first premise (¬T ∧ S) → ¬R and the third premise (¬T → Q), we can deduce that Q must be true (since ¬T is true). From the second premise (¬Q ∨ P) → S, we know that since Q is true, ¬Q must be false, which leads to P being true. Therefore, S must also be true. Finally, using the first premise (¬T ∧ S) → ¬R, we know that since S is true, ¬R must be true as well. Therefore, ¬R ∨ U is true, matching the conclusion of the argument.
C. The argument pattern in C is not mentioned explicitly in the question, but it can be considered a combination of universal instantiation and existential generalization. Rewrite the argument in the form:
- ∀x(P(x) → Q(x))
- ∃x(R(x) ∧ P(x))
- ∴ ∃x(R(x) ∧ Q(x))
By using universal instantiation, we can assign a specific value to x, such as a. Therefore, we have P(a) → Q(a) (from the first premise). By using existential generalization, we can conclude that there exists an x (in this case, a) that satisfies both R(x) and Q(x) (the conclusion). Therefore, the argument is valid.