4. Affirming the Consequent

4. Affirming the Consequent

“Affirming the Consequent” is the name of an invalid conditional argument form. You can think of it as the invalid version of modus ponens.

Below is modus ponens, which is valid:

1. If A then B
2. A
Therefore, B

Now, below is the invalid form that you get when you try to infer the antecedent by affirming the consequent:

1. If A then B
2. B
Therefore, A

No matter what claims you substitute for A and B, any argument that has the form of I will be valid, and any argument that AFFIRMS THE CONSEQUENT will be INVALID.

Remember, what it means to say that an argument is invalid is that IF the premises are all true, the conclusion could still be false. In other words, the truth of the premises does not guarantee the truth of the conclusion.

Here’s an example:

1. If I have the flu then I’ll have a fever.
2. I have a fever.
Therefore, I have the flu.

Here we’re affirming that the consequent is true, and from this, inferring that the antecedent is also true.

But it’s obvious that the conclusion doesn’t have to be true. Lots of different illnesses can give rise to a fever, so from the fact that you’ve got a fever there’s no guarantee that you’ve got the flu.

More formally, if you were asked to justify why this argument is invalid, you’d say that it’s invalid because there exists a possible world in which the premises are all true but the conclusion turns out false, and you could defend this claim by giving a concrete example of such a world. For example, you could describe a world in which I don’t have the flu but my fever is brought on by bronchitis, or by a reaction to a drug that I’m taking.

Another example:

1. If there’s no gas in the car then the car won’t run.
2. The car won’t run.
Therefore, there’s no gas in the car.

This doesn’t follow either. Maybe the battery is dead, maybe the engine is shot. Being out of gas isn’t the only possible explanation for why the car won’t start.

Here’s a tougher one. The argument isn’t written in standard form, and the form of the conditional isn’t quite as transparent:

“You said you’d give me a call if you got home before 9 PM, and you did call, so you must have gotten home before 9 PM.”

Is this inference valid or invalid? It’s not as obvious as the other examples, and partly this is because there’s no natural causal relationship between the antecedent and the consequent that can help us think through the conditional logic. We understand that cars need gas to operate and flus cause fevers, but there’s no natural causal association between getting home before a certain time and making a phone call.

To be sure about arguments like these you need to draw upon your knowledge of conditional claims and conditional argument forms. You identify the antecedent and consequent of the conditional claim, rewrite the argument in standard form, and see whether it fits one of the valid or invalid argument forms that you know.

Here’s the argument written in standard form, where we’ve been careful to note that the antecedent of the conditional is what comes after the “if”:

1. If you got home before 9 PM, then you’ll give me a call.
2. You gave me a call.
Therefore, you got home before 9 PM.

Now it’s clearer that the argument has the form of “affirming the consequent”, which we know is invalid.

The argument would be valid if the you said that you’d give me a call ONLY IF you got home before 9 PM, but that’s not what’s being said here. If you got home at 9:30 or 10 o’clock and gave me a call, you wouldn’t be contradicting any of the premises.

If these sorts of translation exercises using conditional statements are unfamiliar to you then you should check out the tutorial course on “Basic Concepts in Propositional Logic”, which has a whole section on different ways of saying “If A then B”.