In philosophy, a formal fallacy (also called deductive fallacy) 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. An argument that is formally fallacious is always considered wrong. 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.
The presence of a formal fallacy in a deductive argument does not imply anything about the argument's premises or its conclusion. Both may actually be true, or even more probable as a result of the argument, 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.
"Fallacious arguments usually have the deceptive appearance of being good arguments." Recognizing fallacies in everyday arguments may be difficult since arguments are often embedded in rhetorical patterns that obscure the logical connections between statements. Informal fallacies may also exploit the emotional, intellectual, or psychological weaknesses of the audience. Having the capability to recognize fallacies in arguments is one way to reduce the likelihood of such occurrences.
Argumentation theory provides a different approach to understanding and classifying fallacies. In this approach, an argument is regarded as an interactive protocol between individuals that attempts to resolve their disagreements. The protocol is regulated by certain rules of interaction, and violations of these rules are fallacies.
Such fallacies are used in many forms of modern communications where the intention is to influence behavior and change beliefs. Examples in the mass media today include but are not limited to propaganda, advertisements, politics, newspaper editorials and opinion-based "news" shows.
Formal logic is not used to determine whether or not an argument is true. Formal arguments can either be valid or invalid. A valid argument may also be sound or unsound:
Ideally, the best kind of formal argument is a sound, valid argument.
Formal fallacies do not take into account the soundness of an argument, but rather its validity. Premises in formal logic are commonly represented by letters (most commonly p and q). A fallacy occurs when the structure of the argument is incorrect, despite the truth of the premises.
As modus ponens, the following argument contains no formal fallacies:
A logical fallacy associated with this format of argument is referred to as affirming the consequent, which would look like this:
This is a fallacy because it does not take into account other possibilities. To illustrate this more clearly, substitute the letters with premises:
Although it is possible that this conclusion is true, it does not necessarily mean it must be true. The street could be wet for a variety of other reasons that this argument does not take into account. However, if we look at the valid form of the argument, we can see that the conclusion must be true:
This argument is valid and, if it did rain, it would also be sound.
If statements 1 and 2 are true, it absolutely follows that statement 3 is true. However, it may still be the case that statement 1 or 2 is not true. For example:
In this case, statement 1 is false. The particular informal fallacy being committed in this assertion is argument from authority. By contrast, an argument with a formal fallacy could still contain all true premises:
Though, 1 and 2 are true statements, 3 does not follow because the argument commits the formal fallacy of affirming the consequent.
An argument could contain both an informal fallacy and a formal fallacy yet lead to a conclusion that happens to be true, for example, again affirming the consequent, now also from an untrue premise:
"Some of your key evidence is missing, incomplete, or even faked! That proves I'm right!"
"The vet can't find any reasonable explanation for why my dog died. See! See! That proves that you poisoned him! There's no other logical explanation!"
"Adolf Hitler liked dogs. He was evil. Therefore, liking dogs is evil."