A logical fallacy is a failure in an argument (or something that is supposed to be an argument) that is committed with intent to deceive. A fallacious argument may appear to be correct, but on closer examination can be shown to be false.
In common speech, the word "fallacy" is often used to describe any incorrect belief, but in mathematics and in logic, the word "fallacy" refers to a false argument, not necessarily a false conclusion. A fallacy in an argument does not necessarily mean that the conclusion is false—just that the conclusion is not implied by the premises of the argument.
The topic of fallacies overlaps with topics that are more philosophical than mathematical in nature; however, I believe that the ability to think logically and critically is important for the mathematician as well.
Aristotle was the first to identify fallacies. In On Sophistical Refutations, he identified thirteen. He classified six as linguistic fallacies and seven as non-linguistic fallacies. Aristotle's six linguistic fallacies were:
and his seven non-linguistic fallacies were:
According to Aristotle, all logical fallacies are a form of ignoratio elenchi, or ignorance of what a valid argument is. He wrote: "for it is possible to resolve all the kinds of fallacy which we have mentioned into violations of the definition of refutation."
According to Aristotle, a fallacy (or, as Aristotle would have called it, a "sophistical refutation") was a deliberately deceptive tactic of argumentation that could be used to trick someone with whom you were debating. Following Aristotle, however, the concept of fallacies was viewed in a somewhat more broad manner, such that it could encompass almost any sort of error that occurs in dialogue. More modern research (see the sources listed below) suggests that this viewpoint is flawed (it is inconsistent with Aristotle, and it makes creating a theory of fallacy more difficult). This research also suggests that a more coherent account of fallacy requires that
According to this point of view, a fallacy is not simply a weak argument or an inadvertent error or blunder made in reasoning; rather, it is a fairly serious error in argumentation, usually found in the course of a dialogue.
Since then, over than 100 different fallacies have been identified, many of which have been given impressive-sounding names, often in Latin. They can be classified in a variety of ways; I have used a somewhat traditional method here, wherein they can be classified into two categories: formal fallacies, and informal fallacies (the latter of which is further divided). These names do not have to do with how well-dressed the fallacies are; formal fallacies are errors in the form or structure of an argument, and informal fallacies are errors that do not relate to an argument's form or structure.
Sources used (see bibliography page for titles corresponding to numbers): 6, 19, 27.