You now have all four quantifier rules (UI, EG, EI, UG) and quantifier negation (QN). This lesson puts them together into a strategy for predicate proofs, building on Universal Generalization (UG).
Key idea
Why it works
| The conclusion is … | Plan |
|---|---|
| about a name, like $Ga$ | UI with that name; propositional rules |
| existential, like $\exists x Gx$ | get an instance with some name, then EG |
| universal, like $\forall x(Fx \to Hx)$ | UI with a variable, then UG at the end |
| a negated quantifier | aim for the QN form, then QN |
Worked examples
Worked example: Start with QN
- Line 3: $\exists x\neg Gx$ by QN 2.
- Line 4: $\neg Ga$ by EI 3.
- Line 5: $Fa \to Ga$ by UI 1.
- Line 6: $\neg Fa$ by MT 5, 4.
- Line 7: $\exists x\neg Fx$ by EG 6.
Answer: QN, EI, UI, MT, EG.
Worked example: A universal conclusion
- Line 3: $Fy$ by UI 1. Line 4: $Gy$ by UI 2.
- Line 5: $Fy \land Gy$ by Conj 3, 4.
- Line 6: $\forall x(Fx \land Gx)$ by UG 5.
Answer: Variables throughout, then UG.
Common mistake
Why does the second example use variables instead of names? Because the conclusion is universal and must end with UG, which only works on a variable. Had line 3 been $Fa$, the proof would reach $Fa \land Ga$ and then be stuck, since UG never applies to a name.
If a proof seems impossible, it may be invalid. Before searching too long, try to build a counterexample world: if you find one, no proof exists. If every attempt at a world fails, that is a good sign a proof is there to be found.
- Negated quantifier in a premise? QN.
- Existential premises: EI, each with its own new name.
- Universal premises: UI, with those names (or a variable if the conclusion is universal).
- Propositional rules in the middle.
- EG or UG to finish.