But do note that an axiom can also be viewed as an inference rule, simply by considering it as an inference rule with zero premises. This means that to an extent, axioms and rules of inference are interchangeable. read more
Nominally, an axiom is simply a proposition that is given (i.e. assumed true of the domain of discourse), whereas an inference rule is a computation rule that allows you to combine propositions, called the premises and get a result, called the conclusion of the inference rule. read more
In both cases, one may express each instance of an axiom and/or inference rule as a string in a particular formal language. A theory consists of axioms that one adds to the logical axioms that come from the logic. Theories don't contain any proper rules of inference of their own. read more
In logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). read more