A proof is a logical argument demonstrating that a specific statement, proposition, or mathematical formula is true. It consists of a set of assumptions (also called premises) that are combined according to logical rules in order to establish a valid conclusion. This validation can take one of two forms. In a direct proof, a given conclusion can be shown to be true. In an indirect proof, a given conclusion can be shown not to be false and, therefore, presumably to be true.
A direct proof begins with one or more axioms or facts. An axiom is a statement that is accepted as true without being proved. Axioms are also called postulates. Facts are statements that have been proved to be true to the general satisfaction of all mathematicians and scientists. In either case, a direct proof begins with a statement that everyone can agree with as being true. As an example, one might start a proof by saying that all healthy cows have four legs. It seems likely that all reasonable people would agree that this statement is true.
The next step in developing a proof is to develop a series of true statements based on the beginning axioms and/or facts. This series of statements is known as the argument of the proof. A key factor in any proof is to be certain that all of the statements in the argument are, in fact, true statements. If such is the case, one can use the initial axioms and/or facts and the statements in the argument to produce a final statement, a proof, that can also be regarded as true.
As a simple example, consider the statement: "The Sun rises every morning." That statement can be considered to be either an axiom or fact. It is unlikely that anyone will disagree it.
One might then look at a clock and make a second statement: "The clock says 6:00 A.M. " If we can trust that the clock is in working order, then this statement can be regarded as a true statement—the first statement in the argument for this proof.
The next statement might be to say that "6:00 A.M. represents morning." Again, this statement would appear to be one with which everyone could agree.
The conclusion that can be drawn, then, is: "The Sun will rise today." The conclusion is based on axioms or facts and a series of two true statements, all of which can be trusted. The statement "The Sun will rise today" has been proved.
Situations exist in which a statement cannot be proved easily by direct methods. It may be easier to disprove the opposite of that statement. For example, suppose we begin with the statement "Cats do not meow." One could find various ways to show that that statement is not true—that it is, in fact, false. If we can prove that the statement "Cats do not meow" is false, then it follows that the opposite statement "Cats meow" is true, or at least probably true.