Parts of a Logical System
by Juan
February 20, 2024 1 minutes • 190 words
While going through Newton’s Principia, Spninoza’s Ethics, and Euclid’s Elements, we found an arrangement of chapters that turned out to be commmon among mathematicians.
These used the following words, which we defined as:
| Word | Meaning |
|---|---|
| Axiom | A self-evident fundamental statement considered true by definition. It serves as a starting point for other statements. |
| Corollary | A resulting statement that directly follows from a theorem |
| Definition | A statement that gives the properties of a new idea |
| Lemma | A resultant statement proven true, used to prove another, more important theorem |
| Postulate | An axiom with a specific scope |
| Proposition | A statement that can be either true or false |
| Theorem | A resultant statement proven true through other statements |
| Truth | the equality of perceptions from different perceiving identitites |
To make these easy to understand, we visualize it as a house.
| Word | House Part |
|---|---|
| Axioms | The foundation, accepted as true without question. |
| Definitions | The material type that define the building’s components. |
| Propositions | Individual materials that contribute to the structure. |
| Theorems | Major structural elements based on the foundation. |
| Postulates | Additional supporting beams. |
| Corollaries | Smaller structures built directly on a larger theorem. |
| Lemmas | Scaffolding, for new theorems, later removed. |
