List of rules of inference

This is a list of rules of inference, logical laws that relate to mathematical formulae.

Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.

Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation

indicates such a subderivation from the temporary assumption ${\displaystyle \varphi }$ to ${\displaystyle \psi }$.

In the following rules, ${\displaystyle \varphi (\beta /\alpha )}$ is exactly like ${\displaystyle \varphi }$ except for having the term ${\displaystyle \beta }$ wherever ${\displaystyle \varphi }$ has the free variable ${\displaystyle \alpha }$.

Restriction 1: ${\displaystyle \beta }$ is a variable which does not occur in ${\displaystyle \varphi }$.
Restriction 2: ${\displaystyle \beta }$ is not mentioned in any hypothesis or undischarged assumptions.

Restriction: No free occurrence of ${\displaystyle \alpha }$ in ${\displaystyle \varphi }$ falls within the scope of a quantifier quantifying a variable occurring in ${\displaystyle \beta }$.

Restriction: No free occurrence of ${\displaystyle \alpha }$ in ${\displaystyle \varphi }$ falls within the scope of a quantifier quantifying a variable occurring in ${\displaystyle \beta }$.