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)\\
Welcome to this talk page
Talk pages are where people discuss how to make content on Wikipedia the best that it can be. Use this page to start a discussion about the edits made from this IP address. What you say here will be public for others to see. Many IP addresses change periodically, and are often shared by several people.
 | This is the discussion page for an IP user, identified by the user's IP address. Many IP addresses change periodically, and are often shared by several users. If you are an IP user, you may create an account or log in to avoid future confusion with other IP users. Registering also hides your IP address. |