3.3: Conditional Arguments

Conditional Arguments that Affirm

Arguments that have a conditional as one premise and either the antecedent or the consequent of that very conditional as the second premise are called conditional arguments. The first type of conditional argument we will study has the antecedent of the conditional as the second premise.

1. If you own a Switch, then you must buy Animal Crossing.
2. You own a Switch.
3. Therefore, you must buy Animal Crossing.

The first premise of this argument is a conditional and the second premise says that the antecedent of that conditional is true. The second premise just repeats and affirms the antecedent in the first premise. We say that such arguments affirm the antecedent. All arguments that affirm the antecedent are deductively valid. It is impossible for an argument with this format to have all true premises and a false conclusion. This format is sometimes known by its Latin name modus ponens.

In case you are skeptical that this argument structure is valid, remember we can always use the method of counterexample from chapter 2 to check them. Assume that if you own a Switch you must buy Animal Crossing (maybe some strange law is in place) and assume that you own a Switch. Can you think of any way that you don’t have to buy Animal Crossing? No. If you didn’t buy it while owning a Switch, then the first premise would be false, but we were assuming it was true. So, if the premises are true then the conclusion must be true, showing that affirming the antecedent is valid.

We can also have arguments where the second half of the conditional – the consequent – is repeated as a premise:

1. If Norman is in Oklahoma, then Norman is south of Kansas.
2. Norman is south of Kansas.
3. Therefore, Norman is in Oklahoma.

The first premise is a conditional and the second premise says that the consequent of the conditional is true. Such arguments affirm the consequent. Each and every argument that has this format is deductively invalid. It is possible for such arguments to have all true premises and a false conclusion.

Just to be safe, let’s check again, using the method of counterexample. Assume that if Norman is in Oklahoma, then Norman is south of Kansas, and that Norman is south of Kansas. Can you think of any way these premises can be true and it not be the case that Norman is in Oklahoma? Yes. Plenty of places are south of Kansas without being Oklahoma. Norman could be in Texas or Mexico or Brazil (or any number of other places better than Oklahoma). It doesn’t matter that we know Norman is in Oklahoma, because validity is just asking us to take for granted that the premises are true and check to see if the conclusion follows from them.

Conditional Arguments that Deny

Negations

We have studied one kind of sentence, the conditional. Now we need to introduce another kind, the negation. The negation of a sentence is another sentence which says that the first sentence is false. It says the opposite of what the first sentence says; it denies it. We could express the negation of the sentence:

It is raining.

by any of the following sentences:

1. It is not the case that it is raining.
2. It is not true that it is raining.
3. It is not raining.
4. It isn’t raining.
5. It ain’t raining.
6. Ain’t rainin’.

Arguments that have a conditional as one premise and either the negation of that conditional’s antecedent or the negation of the conditional’s consequent are also conditional arguments. So, there are two conditional arguments that affirm and two more that deny, for a total of four.

Here is a conditional argument in which the second premise is the negation of the antecedent of the first premise:

1. If Norman is in Oklahoma, then Norman is south of Kansas.
2. Norman is not south of Kansas.
3. Therefore, Norman is not in Oklahoma.

The first premise is a conditional and the second premise says that the consequent of the conditional is false. Such arguments deny the consequent. Each and every argument that has this format is deductively valid. This format is sometimes known by its Latin name, modus tollens.

Let’s use the method of counterexample again to double check that this structure is valid. Assuming that if Norman is in Oklahoma, then Norman is south of Kansas, and that Norman is not south of Kansas, could it be possible for Norman to be in Oklahoma? No. If being south of Kansas is necessary for Norman to be in Oklahoma, and we know that Norman isn’t south of Kansas, then there is no way Norman is in Oklahoma.

By contrast, consider this argument:

1. If Norman is in Oklahoma, then Norman is south of Kansas.
2. Norman is not in Oklahoma.
3. Therefore, Norman is not south of Kansas.

The first premise is a conditional and the second premise says that the antecedent of the conditional is false. Such arguments deny the antecedent. All arguments having this format are deductively invalid. Denying the antecedent is always a fallacy.

For completion, let’s go back to the method of counterexample one more time. Assume that if Norman is in Oklahoma then Norman is south of Kansas, and that Norman is not in Oklahoma. Does this mean that Norman can’t be south of Kansas? No. Just like before it could be in Texas, etc.

Here are two more examples:

1. If he builds it, they will come. But they didn’t come. So, he didn’t build it.

We can repackage the argument into standard form like this:

1. If he builds it, they will come.
2. They didn’t come.
3. Therefore, he didn’t build it.

It is impossible for both premises of this argument to be true while its conclusion is false, and so is deductively valid. The argument denies the consequent.

2. If the sawdust is the work of carpenter ants, then we’ll need something stronger than Raid to fix the problem. But fortunately, it’s not the work of carpenter ants, so we won’t need anything stronger than Raid.

In standard form:

1. If the sawdust is the work of carpenter ants, then we’ll need something stronger than Raid.
2. The sawdust is not the work of carpenter ants.
3. Therefore, we won’t need anything stronger than Raid.

This argument denies the antecedent. Hence, it is invalid. But we should be able to see this without knowing the label: if you knew that the two premises were true, you still could not be sure whether the conclusion was true or not. The sawdust might be the work of termites (in which case we’ll definitely need something stronger than Raid.)

It is important to remember that you will always be able to work through the argument using the method of counterexample to determine validity and invalidity. But, thinking in terms of shortcuts, you are going to save yourself a lot of time of you memorize these rules.

Recap

Affirming the antecedent and denying the consequent are always valid. Denying the antecedent and affirming the consequent are always invalid.