# Non sequitur (logic)

Not to be confused with Non sequitur (literary device).

Non sequitur (Latin for "it does not follow"), in formal logic, is an argument in which its conclusion does not follow from its premises.[1] In a non sequitur, the conclusion could be either true or false, but the argument is fallacious because there is a disconnection between the premise and the conclusion. All invalid arguments are special cases of non sequitur. The term has special applicability in law, having a formal legal definition. Many types of known non sequitur argument forms have been classified into many different types of logical fallacies.

## In everyday speech

In everyday speech, a non sequitur is a statement in which the final part is totally unrelated to the first part, for example:

Life is life and fun is fun, but it's all so quiet when the goldfish die.

—West with the Night, Beryl Markham[2]

It can also refer to a response that is totally unrelated to the original statement or question:

Mary: I wonder how Mrs. Knowles' next-door neighbour is doing.

Jim: Did you hear that the convenience store two blocks over got robbed last night? Thieves got away with a small fortune.[3]

## Fallacy of the undistributed middle

The fallacy of the undistributed middle is a fallacy that is committed when the middle term in a categorical syllogism is not distributed. It is thus a syllogistic fallacy. More specifically it is also a form of non sequitur.

The fallacy of the undistributed middle takes the following form:

1. All Zs are Bs.
2. Y is a B.
3. 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:

` (B=humans; Y=Mary; Z=men) `
1. All men are humans.
2. Mary is human.
3. Therefore, Mary is a man.

Note that if the terms (Z and B) were swapped around in the first co-premise then it would no longer be a fallacy, although it could still be unsound. This also holds for the following two fallacies which are similar in nature to the fallacy of the undistributed middle and also non sequiturs.

## Affirming the consequent

Any argument that takes the following form is a non sequitur

1. If A is true, then B is true.
2. B is true.
3. Therefore, A is true.

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

An example of affirming the consequent would be:

1. If Jackson is a human (A) then Jackson is a mammal. (B)
2. Jackson is a mammal. (B)
3. Therefore, Jackson is a human. (A)

While the conclusion may be true, it does not follow from the premises: 'Jackson' could be another type of mammal without also being a human. The truth of the conclusion is independent of the truth of its premises - it is a 'non sequitur'.

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:

1. If A is true, then B is true.
2. A is false.
3. 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:

1. If I am Japanese, then I am Asian.
2. I am not Japanese.
3. Therefore, I am not Asian.

While the conclusion may be true, it does not follow from the premises. For all the reader knows, the declarant of the statement could be Asian, but for example Chinese, in which case the premises would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true.

## Affirming a disjunct

Main article: Affirming a disjunct

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

1. A is true or B is true.
2. B is true.
3. Therefore, A is not true.*

The conclusion does not follow from the premises 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:

1. I am at home or I am in the city.
2. I am at home.
3. Therefore, I am not in the city.

While the conclusion may be true, it does not follow from the premises. 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
1. I am either at home or I am in the city.
2. I am at home.
3. Therefore, I am not in the city.

## Denying a conjunct

Main article: Denying a conjunct

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

1. It is not the case that both A is true and B is true.
2. B is not true.
3. Therefore, A is true.

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

An example of denying a conjunct would be:

1. I cannot be both at home and in the city.
2. I am not at home.
3. Therefore, I am in the city.

While the conclusion may be true, it does not follow from the premises. 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 premises would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true.