This page should eventually list various types of fallacy -- a popular sport of debaters and logic text authors -- but just to introduce the notion of a logical fallacy, a few examples should suffice. There are many other fallacies than those listed here. Whole books have been written cataloguing fallacies, and [traditional logic]? texts typically list a few dozen.
Here is an example of a bad argument. Suppose James wanted to argue for the claim that all killing is wrong. Suppose he was giving this argument to a group of people who supported the death penalty: they think that some killing is fine, as punishment of the worst murderers. So James argues as follows:
The supporters of the death penalty would not be impressed by this argument. It commits the logical fallacy of begging the question. In the argument, James says that one should absolutely never kill. But to prove that, he would have to prove that all killing is wrong -- which is what he is trying to argue for. Anyone who disagrees with the conclusion will disagree with the premise that one should absolutely never kill. One might maintain to the contrary that, indeed, in some cases one actually should kill: it is our grim duty, an unfortunate yet necessary part of justice.
The argument presupposes? its conclusion: one of the premises assumes that the conclusion is true. This is an error in arguing. The kind of error has a name: begging the question. If James' argument begs the question, then in his argument he assumes the very thing that he is trying to argue for. Of course an argument that begs the question will not, or should not, convince anyone.
Here is another example of a fallacy. Suppose Barbara argues like this:
Here the problem is that the word "good?" has different meanings, which is to say that it is an ambiguous word. In the premise, Barbara says that Andre is good at some particular activity, in this case tennis. In the conclusion, she says that Andre is a morally good person. Those are clearly two different senses of the word "good." So, of course, the premise might be true while the conclusion would still be false: Andre might be the best tennis player in the world but a rotten person morally speaking. Appropriately, since it plays on an ambiguity, this sort of fallacy is called the Fallacy of Equivocation.
Some fallacies are used freely in the media and politics. For example, the argumentum ad hominem is used when instead of refuting an statement, the person who made that statement is attacked. Every time a politician says to another politician, "You don't have moral authority to say that" is using that fallacy, not attacking the argument, but the person who uses it. Strictly speaking, this is a fallacy; but, arguably, the politician is not even making an argument, but is instead offering a moral rebuke. This is an example of the difficulty of helpfully and respectfully identifying fallacies as such; it is more difficult than it might at first appear, e.g. to a student armed with a list of fallacies.
In the opposite direction is the fallacy of argument from authority. A classic example of this is the [master dixit]? -- "the master said it" -- used through the Middle Ages in reference to Aristotle. A modern use of this is "celebrity spokepersons" in advertisements: that product is good because your favorite celebrity endorses it.
Sometimes, however, an appeal to an authority is best construed not as a fallacy but as an appeal to [expert testimony]? -- a type of [inductive argument]?. This is another example of the difficulty of identifying fallacies as such.
Typically, logical fallacies are invalid, but they can often be written or rewritten so that they follow a valid argument form; and in that case, the challenge is to discover the false premise, which makes the argument unsound.
There are some argument forms that are themselves invalid, however. One of the best-known examples is affirming the consequent.
See also: fallacies of definition, good argument, validity, soundness, cogency, college logic, [informal logic]?
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