Theorems generally have a set of conditions. Then they came to a conclusion-a mathematical statement that was established under certain conditions. Usually, they are written as "If conditions, then conclusions". Writing them with symbolic logic means conditions → conclusions. The proof is not considered as a part of the theorem.
Law is the expression of objective facts and the conclusion drawn through a large number of specific objective facts.
Law is a theoretical model used to describe the real world in a specific situation and scale, which may be invalid or inaccurate in other scales. No theory can describe all the situations in the universe, and no theory can be completely correct.
Axiom is a self-evident truth, and other knowledge must depend on it, and other knowledge is also established from it. In this case, an axiom can be known before you know any other proposition. Not all epistemologists admit the existence of any axiom in this sense.
In logic and mathematics, axioms are not necessarily self-evident truths, but formal logical expressions used to produce further results in deduction. Axiomatization of a knowledge system is to prove that all its propositions can be deduced from a small group of independent sentences. This does not mean that they can learn independently. There are usually many ways to axiomatize a given knowledge system (such as arithmetic). Mathematicians distinguish between two types of axioms: logical axioms and illogical axioms.
Rules are rules that people assume to describe something.
Perhaps from different English words, we can learn the following differences:
Definitions, rules, theorems, laws and axioms are in English.