p → ¬p, p ,r→ q `¬r is valid or invalid
The argument is invalid.
To determine the validity of the argument, we can use a truth table.
p | ¬p | r | q | ¬r | p → ¬p | r → q | ¬r
--------------------------------------
T | F | T | T | F | F | T | F
T | F | T | F | F | F | F | F
T | F | F | T | T | F | T | T
T | F | F | F | T | F | T | T
F | T | T | T | F | T | T | F
F | T | T | F | F | T | F | F
F | T | F | T | T | T | T | T
F | T | F | F | T | T | T | T
As we can see from the truth table, there is a row where all the premises (p, r → q) are true, but the conclusion (¬r) is false. Therefore, the argument is invalid.
To determine if the argument is valid or invalid, we can use a method called proof by contradiction.
Assume the argument is invalid, which means the premises are true but the conclusion is false.
1. p → ¬p (premise)
2. p (premise)
3. r→ q (premise)
4. ¬r (premise)
5. Suppose the conclusion ¬r is false. Then r is true.
6. From premise 3, if r is true, then q is true.
7. Suppose q is false. From premises 2 and 6, we have a contradiction: p and ¬p cannot both be true.
8. Therefore, q must be true.
9. From premise 3 and conclusion 4, we have a contradiction: ¬r and r cannot both be true.
10. Since assuming the conclusion ¬r is false leads to a contradiction, we can conclude that ¬r must be true.
11. From premise 2 and conclusion 10, we have a contradiction: p and ¬p cannot both be true.
12. Since assuming the argument is invalid leads to multiple contradictions, we can conclude that the argument is valid.
Therefore, the argument is valid.