Non sequitur in formal logic, is an argument in which its conclusion does not follow from its premises. 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 formal fallacies

Formal fallacy

In philosophy, a formal fallacy is a pattern of reasoning that is always wrong. This is due to a flaw in the logical structure of the argument which renders the argument invalid...

 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.

Non sequitur 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:

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

Fallacy of the undistributed middle

The fallacy of the undistributed middle

Fallacy of the undistributed middle

The fallacy of the undistributed middle is a logical fallacy, and more specifically a formal fallacy, that is committed when the middle term in a categorical syllogism is not distributed in the major premise...

 is a logical fallacy that is committed when the middle term

Middle term

The middle term must distributed in at least one premises but not in the conclusion of a categorical syllogism. The major term and the minor terms, also called the end terms, do appear in the conclusion.Example:...

 in a categorical syllogism is not distributed. It is thus a syllogistic fallacy

Syllogistic fallacy

Syllogistic fallacies are logical fallacies that occur in syllogisms. They include:Any syllogism type :*fallacy of four termsOccurring in categorical syllogisms:*related to affirmative or negative premises:...

. 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.

Note that if the terms were swapped around in the first co-premise

Co-premise

A co-premise is a premise in reasoning and informal logic which is not the main supporting reason for a contention or a lemma, but is logically necessary to ensure the validity of an argument...

 or if the first premise was rewritten to "Only Zs can be Bs" then it would no longer be a fallacy, although it could still be unsound

Soundness

In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word...

. This also holds for the following two logical fallacies which are similar in nature to the fallacy of the undistributed middle and also non sequiturs.

An example can be given as follows:

1. Men are human.
2. Mary is human.
3. Therefore, Mary is a man.

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

Affirming the consequent

Affirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form:#If P, then Q.#Q.#Therefore, P....

.

An example of affirming the consequent would be:

1. If I am a human (A) then I am a mammal. (B)
2. I am a mammal. (B)
3. Therefore, I am a human. (A)

While the conclusion may be true, it does not follow from the premises: I 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 the conclusion can indeed be false, this cannot be linked to the premise since the statement is a non sequitur. This is called denying the antecedent

Denying the antecedent

Denying the antecedent, sometimes also called inverse error, is a formal fallacy, committed by reasoning in the form:The name denying the antecedent derives from the premise "not P", which denies the "if" clause of the conditional premise....

.

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

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 have her home 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. However, this statement is false because the initial premise is false, there are many possible places other than home or 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 premises as it could be the case that A and B are both false.

An example of denying a conjunct would be:

1. It is not the case that both I am at home and I am 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.

See also

• Ignoratio elenchi
  Ignoratio elenchi
  Ignoratio elenchi is the informal fallacy of presenting an argument that may in itself be valid, but does not address the issue in question...
  
  
• Modus tollens
  Modus tollens
  In classical logic, modus tollens has the following argument form:- Formal notation :...
  
  
• Modus ponens
  Modus ponens
  In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
  
  
• Post hoc ergo propter hoc
  Post hoc ergo propter hoc
  Post hoc ergo propter hoc, Latin for "after this, therefore because of this," is a logical fallacy that states, "Since that event followed this one, that event must have been caused by this one." It is often shortened to simply post hoc and is also sometimes referred to as false cause,...
  
  
• Regression fallacy
  Regression fallacy
  The regression fallacy is an informal fallacy. It ascribes cause where none exists. The flaw is failing to account for natural fluctuations. It is frequently a special kind of the post hoc fallacy.-Explanation:...
  
  
• Fallacy of many questions
  Fallacy of many questions
  A loaded question is a question which contains a controversial assumption such as a presumption of guilt.Such questions are used rhetorically, so that the question limits direct replies to be those that serve the questioner's agenda...