Universal generalization

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Universal generalization

Type	Rule of inference
Field	Predicate logic
Statement	Suppose ${\displaystyle P}$ is true of any arbitrarily selected ${\displaystyle p}$, then ${\displaystyle P}$ is true of everything.
Symbolic statement	${\displaystyle \vdash \!P(x)}$, ${\displaystyle \vdash \!\forall x\,P(x)}$

In predicate logic, generalization (also universal generalization, universal introduction,^[1][2][3] GEN, UG) is a valid inference rule. It states that if ${\displaystyle \vdash \!P(x)}$ has been derived, then ${\displaystyle \vdash \!\forall x\,P(x)}$ can be derived.