* Postulates are fundamental assumptions in a system of logic or mathematics that are accepted as true without proof. They are the starting points for building a logical structure.
* Statements are simply assertions that can be true or false. They don't necessarily have the same level of foundational importance as postulates.
Let's break down your examples:
* All cats walk on four legs. This is a generalization that is generally true, but not a postulate. There could be exceptions (e.g., a cat with a disability).
* The sun is a hot ball of gas. This is a scientific fact, supported by evidence and observation. It's not a postulate; it's a conclusion based on scientific investigation.
* Mammals are animals that produce milk and have hair. This is a definition of mammals, not a postulate. It's a description that helps us categorize animals.
Here are some examples of postulates:
* In geometry, Euclid's postulates: These are fundamental statements about points, lines, and planes that are accepted as true without proof.
* In set theory, the axiom of choice: This postulate asserts the existence of a way to choose one element from each set in a collection of sets, even if there are infinitely many sets.
The key difference is that postulates are foundational assumptions used to build a system of logic or theory, while statements are simply assertions that can be evaluated for truth or falsity.