In logic and philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur (; ) is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.
While a logical argument is a non sequitur if, and only if, it is invalid, the term "non sequitur" typically refers to those types of invalid arguments which do not constitute formal fallacies covered by particular terms (e.g., affirming the consequent). In other words, in practice, "non sequitur" refers to an unnamed formal fallacy.
A special case is a mathematical fallacy, an intentionally invalid mathematical proof, often with the error subtle and somehow concealed. Mathematical fallacies are typically crafted and exhibited for educational purposes, usually taking the form of spurious proofs of obvious contradictions.
A formal fallacy is contrasted with an informal fallacy which may have a valid logical form and yet be unsound because one or more premises are false. A formal fallacy; however, may have a true premise, but a false conclusion.
Taxonomy
Prior Analytics is Aristotle's treatise on deductive reasoning and the syllogism. The standard Aristotelian logical fallacies are: Fallacy of four terms (Quaternio terminorum); Fallacy of the undistributed middle; Fallacy of illicit process of the major or the minor term; Affirmative conclusion from a negative premise.
Other logical fallacies include: The self-reliant fallacy
In philosophy, the term logical fallacy properly refers to a formal fallacy—a flaw in the structure of a deductive argument, which renders the argument invalid.
It is often used more generally in informal discourse to mean an argument that is problematic for any reason, and encompasses informal fallacies as well as formal fallacies—valid but unsound claims or poor non-deductive argumentation.
The presence of a formal fallacy in a deductive argument does not imply anything about the argument's premises or its conclusion (see fallacy fallacy). Both may actually be true, or even more probable as a result of the argument (e.g. appeal to authority), but the deductive argument is still invalid because the conclusion does not follow from the premises in the manner described. By extension, an argument can contain a formal fallacy even if the argument is not a deductive one; for instance an inductive argument that incorrectly applies principles of probability or causality can be said to commit a formal fallacy.
Affirming the consequent
Any argument that takes the following form is a non sequitur:
If A is true, then B is true. B is true. Therefore, A is true.
Even if the premise and conclusion are both 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. 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.