Geometry classes today tend to treat axioms and postulates as synonyms; however, there were distinctions between these two ideas. What were the distinctions made by Greek mathematicians?

Axioms were applied in all sciences while postulates were for specific studies.

Aristotelean logic begins with the familiar grammatical distinction between subject and predicate. A subject is typically an individual entity, for instance a man3 or a house or a city. It may also be a class of entities, for instance all men. A predicate is a property or attribute or mode of existence which a given subject may or may not possess. For example, an individual man (the subject) may or may not be skillful (the predicate), and all men (the subject) may or may not be brothers (the predicate).

The fundamental principles of predication are:

Identity. Everything is what it is and acts accordingly. In symbols: is. For example, an acorn will grow into an oak tree and nothing else. Non-contradiction. It is impossible for a thing both to be and not to be. A given predicate cannot both belong and not belong to a given subject in a given respect at a given time. Contradictions do not exist. Symbolically: and non - cannot both be the case. For example, an honest man cannot also be a thief. Either-or. Everything must either be or not be. A given predicate either belongs or does not belong to a given subject in a given respect at a given time. Symbolically:Either or non-. For example, a society must be either free or not free. These principles have exercised a powerful influence on subsequent thinkers. For example, the 20th century intellectual Ayn Rand titled the three main divisions of her best-selling philosophical novel Atlas Shrugged4[18 ] after the three principles above, in tribute to Aristotle.