Question

12. Show that the argument form with premises (p ∧ t) → (r∨s), q→(u∧t), u→p, and ¬s and conclusion q → r is valid by first using Exercise 11 and then us- ing rules of inference from Table 1.11.Show that the argument form with premises p1,p2,...,pn and conclusion q → r is valid if the argument form with premises p1,p2,...,pn,q, and conclusion r is valid.

Answer 1

The argument is shown to be valid through a series of logical inferences applying modus tollens, modus ponens, and disjunctive syllogism, leading to the conclusion that if q then r must be true.

Step-by-step explanation:

To show that the argument form with premises (p ∧ t) → (r∨s), q→(u∧t), u→p, and ¬ s and the conclusion q → r is valid, we need to use rules of inference and the fact that if an argument with additional premise q and conclusion r is valid, then the argument with conclusion q → r is also valid.

From the premise ¬ s and (p ∧ t) → (r∨s), we can apply modus tollens to infer ¬(p ∧ t).

Given u→p, we apply modus ponens with the premise u (from q→(u∧t) assuming q) to infer p.

Since we have ¬(p ∧ t) from step 1 and p from step 2, we can deduce that ¬ t must be true.

Now that we have q implies both u and ¬ t (from steps 1 and 3), but since we also have u → p, we know p must be true. Therefore, from q we have p and ¬t.

Finally, given (p ∧ t) → (r∨s) and knowing that s is false, r must be true if p and t are both true. But since t is not true, p alone guarantees r. Therefore, if q then r, validating the conclusion q → r.

Answer 2

The validity of the argument is shown through deductive reasoning, applying logical rules such as modus ponens, modus tollens, and disjunctive syllogism, which collectively affirm the conclusion q → r from the given premises.

Step-by-step explanation:

The validity of a deductive argument using given premises and a conclusion. According to the information provided and rules of inference, we must first assert that if the argument with premises p1, p2, ..., pn, q, and conclusion r is valid, then the argument with premises p1, p2, ..., pn and conclusion q → r is also valid.

The premises offered are (p ∧ t) → (r ∨ s), q → (u ∧ t), u → p, and ¬s. The conclusion is q → r. By using the rules of inference such as modus ponens, modus tollens, and disjunctive syllogism, we can proceed to validate the argument. By asserting ¬s, we can apply modus tollens to the first premise to affirm r in the presence of q, as the disjunction (r ∨ s) erodes due to the negation of s, leaving r as the sole outcome of that implication. Additionally, u → p and q → (u ∧ t) collectively affirm p once q is affirmed, which in turn, given (p ∧ t) → (r ∨ s), completes the chain leading to r being the necessary conclusion from q.