Denying the Antecedent

1. If I made it to class, that means my car is working well.

I didn't make it to class.

So I guess my car isn't working well.

2. If you're a born-again Christian, then you will go to heaven.

You aren't a born-again Christian.

/.. You won't go to heaven

[Note: as originally stated, the first premise claims that being a Christian [=the antecedent in the "if A, then C" phrase] is a sufficient condition for going to heaven [=the consequent] --

but finding yourself in heaven is only a necessary condition for guaranteeing that you are a Christian.

This leaves open the logical (if not theological...) possibility that others may find themselves in heaven, who are not born-again Christians. Hence the conclusion is not certainly true.

BUT: this argument becomes valid if the first premise is rephrased to make the stronger claim:

You will go to heaven if and only if you're a born-again Christian.

This premise is logically equivalent to:

If you're a born-again Christian, then you will go to heaven, and

If you go to heaven, then you are a born-again Christian.

This says that being a born-again Christian is sufficient condition for going to heaven, and that being in heaven is a sufficient condition to guarantee that you are a Christian.

So only those who are born-again Christians will find themselves in heaven -- and those who are not born-again Christians will not.

That is, the conclusion becomes certainly true (IF the premise is true...).

4. If you're crying, you must be upset.

You are not crying.

So you must not be upset.

5. If you can afford expensive lawyers, then you can escape justice.

The average person cannot afford expensive lawyers.

So the average person cannot escape justice.

6. Schools that are meant to educate the nation fail to take a critical look at what harm they are doing in pushing multi-culturalism.

In Africa, multi-culturalism is the source of genocidal strife in Rwanda and Burundi, where the Tutus and the Hutus are deadly enemies.

Multi-culturalism obviously doesn't produce unity.

So if you push multi-culturalism, you won't get unity.

So if we stop pushing multi-culturalism, we will get unity.