Relational Proof Walkthrough
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A relational proof carries two variables at once, which makes the order of instantiation and the restriction on universal generalisation both matter at the same time.
Prove that if there is someone everyone loves, then everyone loves someone. Premise: there exists a y such that for all x, Lxy. Goal: for all x, there exists a y such that Lxy. Line two: for all x, Lxa, from line one by existential instantiation, with a fresh. Existentials first, as always. Line three: Lba, from line two by universal instantiation to a new name b. Line four: there exists a y such that Lby, from line three by existential generalisation. Line five: for all x, there exists a y such that Lxy, from line four by universal generalisation. Check the restriction aloud before writing line five: b must not appear in any premise and must not have come from an existential instantiation. It came from universal instantiation on line three, so it stands for an arbitrary individual, and the generalisation is legitimate. The converse direction has no proof at all, and the next video shows why.
Universal generalisation on a name that arrived by existential instantiation turns a claim about one unnamed thing into a claim about everything. The line looks identical either way, so the check has to be made against where the name came from, not against how it looks.
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