Axioms and rules of inference are fundamentally different:<p>(0) An axiom is an <i>internal</i> statement to a mathematical theory that is assumed to be true. That is, inside of a mathematical theory, you don&#x27;t need to prove that its axioms hold. However, if you want to construct a <i>model</i> of a mathematical theory, you need to prove <i>externally</i> that the axioms hold. In return, you get the theory&#x27;s theorems (suitably interpreted) for free.<p>(1) A rule of inference exists in an external <i>metatheory</i>, where the original mathematical theory we were studying is treated as a syntactic object (also known as <i>object language</i>), in very much the same way a compiler treats the program being compiled as a (possibly annotated) syntax tree. A rule of inference defines a class of valid <i>syntactic transformations</i>, but doesn&#x27;t concern itself with the impact of these transformations have on the <i>meaning</i> of the phases in the object language.<p>It is perfectly sensible to consider the effect of changing the rules of inference, on an axiomatic system.