Universal generalization (UG) puts a $\forall$ back on. It is how you prove a conclusion like $\forall x(Fx \to Hx)$. Like EI (Existential Instantiation (EI)), it has a restriction, and the restriction is the whole point of the rule.
Key idea
Why it works
Worked examples
Worked example: Allowed
- Line 2: $Fy \land Gy$ by UI 1 (a variable).
- Line 3: $Fy$ by Simp 2.
- Line 4: $\forall x Fx$ by UG 3.
Answer: UG is fine: $y$ is arbitrary.
Worked example: Not allowed: a name
- $a$ is a name, not a variable.
- One thing being $F$ does not make everything $F$.
Answer: No.
Worked example: Not allowed: from EI
- $a$ was introduced by EI: it is a particular thing.
Answer: No.
Common mistake
A familiar comparison: a geometry teacher proves a fact about "any triangle" by drawing one triangle and using only what holds for every triangle. That drawn triangle plays the role of $y$. If the proof used something special about it, such as its being isosceles, the conclusion would hold only for triangles like it.
- Universal conclusion? Plan to finish with UG.
- Instantiate universal premises with a variable, not a name.
- Check that the variable did not come from EI and is not in an open assumption.
- Generalize: put $\forall x$ in front and $x$ for the variable everywhere.
UI and UG work as a pair: UI takes a universal claim down to an arbitrary object, the propositional rules work on it, and UG lifts the result back to a universal claim. EI and EG are the other pair, for existential claims.