This is a term in logic
that refers to an agument where the truth of the premises gaurantees the truth of the conclusions. The most common forms are modus ponens
and modus tollens
. These are what are called sylogisms, argument forms that are known to be valid. In the early (medieval
) days of logic it was held that there are only a certain number of valid argument forms, and so an attempt to codify these sylogisms was made. There are some very interesting texts describing the hundred or so that they came up with.
In the modern era it has been discovered there is an easier way to check the validity of an argument, other than consulting one's logic textbook. You simply create a truth table that shows all the statements, the premises, and the conclusion. As in any truth table you go through every possible combination of truth values for the statements. If you are dealing with three or less statements this isn't terribly hard, but anything over about five gets a litlle hard. You then find the critical rows, the one's where all the premises are true. If the conclusion is true in all these rows, then the argument form is valid.
The thing to keep in mind is that this only checks the validity of the form, whether each of the premises is true is another matter entirely. An argument can be placed in valid argument form and still be completely wrong, as long as one or more of the premises is wrong. An argument with both true conclusions and valid form is called a sound argument.
The most interesting proof if the proof using a contradiction. This is not the method of proof where you assume something is true, and then prove it false by showing that this entails a contradiction. This ia a little more off the wall. Consider the argument:
P and not P
This is valid argument form. The premise is a contradiction, so it is never true. The conclusion is true in every row where the premise is, i.e. none of them, therefore it is valid. If you were to prove a contradiction true, then anything is provable. Then again this also invalidates all mathematics and logic, and since that, as far as we can tell, the underpinnings of any possible reality, this is both unlikely and extremely undesireable.