Determine whether the following argument is valid

In this post, we will determine whether the following argument is valid or invalid. Lets consider the following two examples.

Question 1: Check whether the argument is valid or not:

p∨q, q→r, p→s, ∼s ⇒ r. 

Answer:

We will proceed as follows:

1. p∨q (given)

2. q∨p (commutativity)

3. q→r (given)

4. ∼q∨r (equivalent to 3)

5. p∨r (applying Resolution to 2 and 4)

6. p→s ≡ ∼p∨s (given)

7. r∨s (applying Resolution to 5 and 6)

8. ∼s (given)

Now applying Disjunctive Syllogism to 7 and 8, we deduce r.

Therefore, the argument p∨q, q→r, p→s, ∼s ⇒ r is valid.

Question 2: Check whether the argument is valid or not:

p∨q, p∨q→∼w, ∼w→u∧∼v, u∧∼v→r∨s ⇒ r∨s. 

Answer:

We will proceed as follows:

1. p∨q (given)

2. p∨q→∼w (given)

3. ∼w (applying Modulus Ponens to 1 and 2)

4. ∼w→u∧∼v (given)

5. u∧∼v (Modulus Ponens to 3 and 4)

6. u∧∼v→r∨s (given)

7. r∨s (applying Modulus Ponens to 5 and 6)

Therefore, r∨s is true, that is, we have shown that the argument p∨q, p∨q→∼w, ∼w→u∧∼v, u∧∼v→r∨s ⇒ r∨s is valid.