Foundations of Mathematics

Axiomatic System

Are there different types of proofs?

There are several different types of proofs in mathematical logic. Direct proofs are based on rules that result in one true proposition from two propositions. They show that a given statement is true by simply combining existing theorems with or without some mathematical manipulations. For example, if you have two sides of a triangle with the same length, a definition and theorem show that a line bisecting their vertex produces two congruent triangles—a direct proof that the angles at the other two vertices have the same size.

In logic, indirect proofs are also called “proofs by contradiction,” and are known in Latin as reductio ad absurdum (“reduced to an absurdity”). This type of proof initially assumes that the opposite of what you are trying to prove is true; from this assumption, certain conclusions can be drawn. One then searches for a conclusion that is false because it contradicts given or known information. Sometimes, a given piece of information is contradicted, which shows that, since the assumption leads to a false conclusion, the assumption must be false. If the assumption is false (the opposite of the conclusion one is trying to prove), then it is known that the goal conclusion must be true. All of this has therefore been shown “indirectly.”

Finally, a disproof is a single instance that contradicts a proposition. For example, the disproof of “all primes are odd” is the true statement “the number 2 is a prime and not odd.” If a disproof exists for a proposition, then the statement is false.


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