Non sequitur (logic)

A non sequitur (Latin for "it does not follow"), in formal logic, is an invalid argument – an argument whose conclusion does not follow from its premises.^[1] In a non sequitur, the conclusion could be either true or false (because there is a disconnect between the premises and the conclusion), but the argument nonetheless asserts the conclusion to be true and is thus fallacious. While a logical argument is a non sequitur if, and only if, it is invalid (and so, technically, the terms 'invalid argument' and 'non sequitur' are equivalent), the word 'non sequitur' is typically used to refer to those types of invalid arguments which do not constitute logical fallacies covered by particular terms (e.g. affirming the consequent). In other words, in practice, 'non sequitur' is used to refer to an unnamed logical fallacy. Often, in fact, 'non sequitur' is used when an irrelevancy is showing up in the conclusion. The term has special applicability in law, having a formal legal definition.

Logical constructions

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 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:

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 premise:

1. Humans are mammals
2. Jackson is a mammal
3. 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, say, 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:

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 premise. For all the reader knows, the declarant of the statement could be Asian, but for example 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:

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 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:

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 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,

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

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 premise 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 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 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, where B=mammals, Y=Mary and Z=humans:

1. All humans are mammals.
2. Mary is a mammal.
3. 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.

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]

See also

• Ignoratio elenchi
• Modus tollens
• Modus ponens
• Post hoc ergo propter hoc
• Regression fallacy
• Relevance logic
• Fallacy of many questions

References

1. Barker, Stephen F. (2003) [1965]. "Chapter 6: Fallacies". The Elements of Logic (Sixth ed.). New York, NY: McGraw-Hill. pp. 160–169. ISBN 0-07-283235-5.
2. Quoted in Hindes, Steve (2005). Think for Yourself!: an Essay on Cutting through the Babble, the Bias, and the Hype. Fulcrum Publishing. p. 86. ISBN 1-55591-539-6. Retrieved 2011-10-04.