User talk:68.81.2.180

The example below that purports to show why the second restriction is needed is wrong because \forall u\,(u\not =x) would also be faulty universal generalization, so overloading the name x is not the only logical error in the example. The first restriction needs to be restated because it seems to allow the following invalid reasoning:\\ Assume ∃z P(z) \\ P(x)\\ ∀y P(y) \\ The second restriction needs a new example where the only error is overloading the bound variable name, e.g.,\\ Let x be arbitrary\\ Assume ∀z P(z,y)\\ P(x,y)\\ ∀y P(y,y)\\

Without the second restriction, one could make the following deduction:\\ {\displaystyle \exists z\,\exists w\,(z\not =w)} {\displaystyle \exists z\,\exists w\,(z\not =w)} (Hypothesis)\\ {\displaystyle \exists w\,(y\not =w)} {\displaystyle \exists w\,(y\not =w)} (Existential instantiation)\\ {\displaystyle y\not =x} y \not = x (Existential instantiation)\\ {\displaystyle \forall x\,(x\not =x)} {\displaystyle \forall x\,(x\not =x)} (Faulty universal generalization)\\

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