Argument

In mathematics, argument is used in at least three senses:
- A parameter or independent variable that is the input to a function. So if f(x) is the value of a function, x is the argument.
- In complex analysis, one component of the mod-arg form of that number. See complex number.
- A proof of a proposition.
In computer science, an argument is an informal term for actual parameter, which can be variable or value passed into a function, subroutine, or computer program. The usage is analogous to that in mathematics.

An argument is an attempt to provide a demonstration of the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. Arguments can be valid or invalid, although how arguments are determined to be in either of these two categories can often itself be an object of much discussion. Informally one should expect that a valid argument should be convincing in the sense that it is capable of convincing someone about the truth of the conclusion. This validity criterion, however, is inadequate or even misleading since it depends more on the skill of the person constructing the argument to manipulate the person who is being convinced and less on the argument itself. Less subjective criteria for validity of arguments are clearly desirable, and in some cases we should even expect an argument to be rigorous, that is adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to be a formal proof.

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.

Arguments are commonly used in philosophy and other disciplines when a logical approach is required. A discussion of the types of arguments used in philosophy may be considered meta-philosophy.

In mathematics, an argument can be formalized using symbolic logic. 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 axioms using rules of inference. The last statement in the list is the conclusion. Most arguments used in mathematical proof are rigorous, but not formal. In fact, strictly formal proofs of all but the most trivial assertions are extremely hard to construct and hard to undrestand without some assistance from a computer. One of the goals of automated theorem proving is to design computer programs to produce and check formal proofs.

In general usage, however, arguments are rarely formal or even have the rigor of mathematical proofs. The logical relationship between the premises and the conclusion may not be explicitly stated, and sometimes the conclusion itself is left to the reader to supply.

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.