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Mathematics uses logic, paper, and calculator. These things are used to create general rules, which are an important part of mathematics. These rules leave out information that is not important so that a single rule can cover many situations. By finding general rules, mathematics solves many problems at the same time as these rules can be used on other problems.

A proof gives a reason why a rule in mathematics is correct. This is done by using certain other rules that everyone agrees are correct, which are called axioms. A rule that has a proof is sometimes called a theorem. Experts in mathematics perform research to create new theorems. Sometimes experts find an idea that they think is a theorem but can not find a proof for it. That idea is called a conjecture until they find a proof.

Sometimes, mathematics finds and studies rules or ideas in the real world that we don't understand yet. Often in mathematics, ideas and rules are chosen because they are considered simple or neat. On the other hand, sometimes these ideas and rules are found in the real world after they are studied in mathematics; this has happened many times in the past. In general, studying the rules and ideas of mathematics can help us understand the world better.

The same syllogism can be written in a notation:

is read like "and", meaning both of the two. is read like "or", meaning at least one of the two. is read like "implies", or "If ... then ...". is read like "not", or "it is not the case that ...". Parentheses (,) are added for clarity and precedence; this means that what is in parenthesis should be looked at before the things outside.

This is the same example using logic symbols:

And this is the same example using general terms:

Finally, those talking about logic talk about statements. A statement is simply something like "Aristole is human" or "all humans are mortal". Statements have a truth value; they are either true or false, but not both. Mistakes in logic are called "fallacies".