# 4.6: Formal Fallacies

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## 23 Formal Fallacies^{41}

Affirming the consequent

- If A is true, then B is true.
- B is true.
- Therefore, A is true.

Even if the premise and conclusion are all true, the conclusion is not a necessary consequence of the premise. This sort of *non sequitur* is also called affirming the consequent.

An example of affirming the consequent would be:

- If Jackson is a human (A), then Jackson is a mammal. (B)
- Jackson is a mammal. (B)
- Therefore, Jackson is a human. (A)

While the conclusion may be true, it does not follow from the premise:

- Humans are mammals
- Jackson is a mammal
- Therefore, Jackson is a human

The truth of the conclusion is independent of the truth of its premise – it is a 'non sequitur', since Jackson might be a mammal without being human. He might be an elephant.

Affirming the consequent is essentially the same as the fallacy of the undistributed middle, but using propositions rather than set membership.

Denying the antecedent

Another common *non sequitur* is this:

- If A is true, then B is true.
- A is false.
- Therefore, B is false.

While B can indeed be false, this cannot be linked to the premise since the statement is a *non sequitur*. This is called denying the antecedent.

An example of denying the antecedent would be:

- If I am Japanese, then I am Asian.
- I am not Japanese.
- Therefore, I am not Asian.

While the conclusion may be true, it does not follow from the premise. For all the reader knows, the statement's declarant could be another ethnicity of Asia, e.g. Chinese, in which case the premise would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true.

Affirming a disjunct

Affirming a disjunct is a fallacy when in the following form:

- A is true or B is true.
- B is true.
- Therefore, A is not true.*

The conclusion does not follow from the premise as it could be the case that A and B are both true. This fallacy stems from the stated definition of *or* in propositional logic to be inclusive.

An example of affirming a disjunct would be:

- I am at home or I am in the city.
- I am at home.
- Therefore, I am not in the city.

While the conclusion may be true, it does not follow from the premise. For all the reader knows, the declarant of the statement very well could be in both the city and their home, in which case the premises would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true.

*Note that this is only a logical fallacy when the word "or" is in its inclusive form. If the two possibilities in question are mutually exclusive, this is not a logical fallacy. For example,

- I am either at home or I am in the city.
- I am at home.
- Therefore, I am not in the city.

Denying a conjunct

Denying a conjunct is a fallacy when in the following form:

- It is not the case that both A is true and B is true.
- B is not true.
- Therefore, A is true.

The conclusion does not follow from the premise as it could be the case that A and B are both false.

An example of denying a conjunct would be:

- I cannot be both at home and in the city.
- I am not at home.
- Therefore, I am in the city.

While the conclusion may be true, it does not follow from the premise. For all the reader knows, the declarant of the statement very well could neither be at home nor in the city, in which case the premise would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true.

Fallacy of the undistributed middle

The fallacy of the undistributed middle takes the following form:

- All Zs are Bs.
- Y is a B.
- Therefore, Y is a Z.

It may or may not be the case that "all Zs are Bs", but in either case it is irrelevant to the conclusion. What is relevant to the conclusion is whether it is true that "all Bs are Zs," which is ignored in the argument.

An example can be given as follows, where B=mammals, Y=Mary and Z=humans:

- All humans are mammals.
- Mary is a mammal.
- Therefore, Mary is a human.

Note that if the terms (Z and B) were swapped around in the first co-premise then it would no longer be a fallacy and would be correct.