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 contains a fallacy is invalid. However, this may not impact the truth of the argument since validity and truth are separate in formal logic. For example, there could be a correlation between the number of times it rains and whenever a day is Tuesday, which could lead one to believe that "Tuesdays are days when it rains." However, in this case one would commit the ad hoc fallacy because there is no causal link between a day of the week and how often it rains. So, although it may be true in one's own perception it is impossible to validate using logic.
"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. Recognizing fallacies can develop reasoning skills to expose the weaker links between premises and conclusions to better discern between what appears to be true and what is true.
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, so violations of these rules are fallacies.
Fallacies are used in place of valid reasoning to communicate a point with the intention to persuade. 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."