Definition:Rule of Inference

Definition
Let $\LL$ be a formal language.

Part of defining a proof system $\mathscr P$ for $\LL$ is to specify its rules of inference or proof rules.

A rule of inference is a specification of a valid means to conclude new theorems in $\mathscr P$ from given theorems and axioms of $\mathscr P$.

Often, the formulation of rules of inference also appeals to the notion of provable consequence to be able to deal with assumptions as part of a proof.

Also see

 * Definition:Axiom (Formal Systems), the other part in specifying a proof system
 * Definition:Derived Rule, which are often convenient in working with a proof system
 * Examples of proof rules are gathered in Category:Proof Rules

Also known as
Rules of inference are also known as rules of transformation or transformation rules.

Further alternatives are rule of derivation and rule of proof.

With all these, literature might have a specific meaning attached, so be careful before treating any of these as synonyms.

On, proof rule and rule of inference are the terminology of choice and are used interchangeably.

Example
In the context of propositional logic, an example of a rule of inference is:


 * If $p$ is a theorem, and $p \implies q$ is a theorem, then $q$ is a theorem.

which expresses Modus Ponendo Ponens.