I have no idea why (2) is an invalid argument.
(1)
If A, then B
Not B
Therefore not A
(2)
If A, then B
Not A
Therefore not B
2
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1See the truth tabel for the conditional: as you can see "if A, then B" is T(rue) in both lines F-T and F-F. Thus, if A is F(alse), i.e. Not A is T, we have no guarantee that B is also F. – Mauro ALLEGRANZA Oct 19 '17 at 8:02
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11. "If I walk through the park, then I will get home before 3pm" 2. "I did not walk through the park" 3. "therefore, I did not get home before 3pm" This is not necessarily true, I could have gotten a ride that took even less time than walking through the park. The point is that (A -> B) doesn't mean that A is the only time that b can appear, that is what (A <-> B) means. (A -> B) can be true and B can still show up without A, what it says is that if A shows up B will always be there too. – Not_Here Oct 19 '17 at 9:47
3
Let's try with some examples.
- If Alex is walking in the forest, Alex wears shoes.
- If Alex is walking on the street, Alex wears shoes.
- Alex is not walking in the forest.
if (2) were valid, then it would follow that Alex is not wearing shoes. But clearly we haven't shown that Alex is not wearing shoes, because there are other conditions under which he wears shoes.
(1), however, can remain valid, if
- If Alex is walking in the forest, Alex wears shoes.
- If Alex is walking on the street, Alex wears shoes.
- Alex is not wearing shoes.
In this case, we can infer both that Alex is not walking in the forest and that Alex is not walking on the street.
Now let's make it a bit more formal. Valid in logic means that if the premises happened to be true, then the conclusion must also be true.
By the counter example above, we have shown that the pattern you refer to as (2) can have a false conclusion with true premises. This pattern is the fallacy called "denying the antecedent."
