Question

from the premises: (p → ¬q) ∧ q, ¬r → p. show that r is a valid conclusion by providing the argument.

Answer 1

To prove that r is a valid conclusion, we can use the deduction rule called modus tollens. Modus tollens states that if the conditional statement ¬r → p is true, and p is false, then ¬r must be true. By applying this rule to the premises given, we can conclude that r is a valid conclusion.

Step-by-step explanation:

Premises:

1. (p → ¬q) ∧ q
2. ¬r → p

Argument:

To prove that r is a valid conclusion, we can use the deduction rule called modus tollens. Modus tollens states that if the conditional statement ¬r → p is true, and p is false, then ¬r must be true. In our premises, we have (p → ¬q) ∧ q, so using modus tollens on this premise, we can conclude that if p is false, ¬q must be true. Combining this with the second premise ¬r → p, we can apply modus tollens again to conclude that if ¬r is true, p must be false. Therefore, we have shown that if ¬r is true, then p is false, which means r must be true.