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An argument is an attempt to provide a compelling, rigorous demonstration of the truth of a conclusion, based on the truth of any number of premises. If the argument is valid, the premises together entail or imply the conclusion.

The ways in which arguments go wrong fall into certain patterns, called logical fallacies, meaning false notation of logic.

In mathematics, an argument can be formalized using symbolic logic of discrete mathematics. In that case, an argument is seen as an ordered list of statements, each one of which is either one of the premises or derivable from the combination of some subset of the preceding statements and one or more axiom. The last statement in the list is the conclusion.

Often, however, arguments are much more informal. Often the logical relationship between the premises and the conclusion is not explicitly stated, and sometimes the conclusion itself is not stated either, but left to the reader to supply.

The sorts of arguments used in philosophy are of a very interesting character, studied in many various ways by many philosophers in their writings about meta-philosophy.

In recent decades one of the more influential discussions of philosophical arguments is that by the prolific University of Pittsburgh philosophy professor Nicholas Rescher[?] in his book The Strife of Systems[?]. Rescher models philosophical problems[?] on what he calls aporia[?] or an aporetic cluster[?]: a set of statements, each of which has initial plausibility but which are jointly inconsistent. The only way to solve the problem, then, is to reject one of the statements. If this is correct, it constrains how philosophical arguments are formulated.

Arguments vs. explanations

There are other kinds of sets of statements besides arguments, such as explanations. Logic does not, except in its applications, concern itself with explanations. For example, suppose James offers an explanation for why there are tides: he talks about the gravitational effect of the Moon and the Sun on the oceans, and so on. That is not an argument; it is an explanation. In that case, James explains why there are tides. He is not trying to convince anyone that there are tides. It is already agreed that there are tides. The question the explanation answers is why there are.

On the other hand, suppose the response to James was "I don't believe you, everybody knows that tides are caused by Poseidon". He could respond by collecting information, such as the position of the moon and sun and the height of the tide and using this to show why he is more likely correct. Then he will have produced an argument, irrespective of whether he manages to convince anybody.

The difference between an argument and an explanation should be clear. On the one hand, the function or purpose of an argument is to convince people who might be doubting the conclusion. On the other hand, the function or purpose of an explanation is to give the cause of some phenomenon which we observe, or are willing to assume actually occurs. To put it even more briefly, the purpose of an argument is to persuade, while the purpose of an explanation is to explain.

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