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

Suppose $P$ is true of any arbitrarily selected $p$, then $P$ is true of everything.

Symbolic statement

$\vdash \!P(x)$, $\vdash \!\forall x\,P(x)$

Generalization with hypothesesedit

The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume $\Gamma$ is a set of formulas, $\varphi$ a formula, and $\Gamma \vdash \varphi (y)$ has been derived. The generalization rule states that $\Gamma \vdash \forall x\,\varphi (x)$ can be derived if $y$ is not mentioned in $\Gamma$ and $x$ does not occur in $\varphi$.

These restrictions are necessary for soundness. Without the first restriction, one could conclude $\forall xP(x)$ from the hypothesis $P(y)$. Without the second restriction, one could make the following deduction:

This purports to show that $\exists z\,\exists w\,(z\not =w)\vdash \forall x\,(x\not =x),$ which is an unsound deduction. Note that $\Gamma \vdash \forall y\,\varphi (y)$ is permissible if $y$ is not mentioned in $\Gamma$ (the second restriction need not apply, as the semantic structure of $\varphi (y)$ is not being changed by the substitution of any variables).

Example of a proofedit

Prove:$\forall x\,(P(x)\rightarrow Q(x))\rightarrow (\forall x\,P(x)\rightarrow \forall x\,Q(x))$ is derivable from $\forall x\,(P(x)\rightarrow Q(x))$ and $\forall x\,P(x)$.

In this proof, universal generalization was used in step 8. The deduction theorem was applicable in steps 10 and 11 because the formulas being moved have no free variables.