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  1. $\forall x(P(x)) \vee \forall x(Q(x))$

I am currently reading a logic book by Patrick J. Hurley, and in the book the author says that we can't universally instantiate a statement like statement 1. Specifically, he says that universal instantiation must be applied only to whole lines but he never defines what he means by whole lines and he never explains why universal instantiation is applicable only to whole lines.

I don't know why it might be wrong to instantiate statement 1 at just one step. For example, what is wrong with using John as our instance and then saying that either John has the property P or John has the property Q?

Brian Tung
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desoana
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  • That's not what that says. That says (rather broadly) that "all $x$'s are $P$ or all $x$'s are $Q$ (or both)." – Brian Tung Sep 25 '22 at 00:02
  • @BrianTung so, what might be wrong with saying that John has the property P or Q or both? – desoana Sep 25 '22 at 00:05
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    Ahh, I think I understand what you're asking. I'll delete some of my earlier comments and start over. First of all, it is still important that each $\forall$ only applies to the statement in its scope. Statement 1 is logically equivalent to $\forall x(P(x)) \vee \forall y(Q(y))$. You shouldn't use the same instantiation for both of those. You can, however, break that sentence into two separate sentences, instantiate each, and then recombine them. – Brian Tung Sep 25 '22 at 00:12
  • I don't understand why if the x doesn't "carry over" to then other scope, then I won't be able to use the same instance? Both of the disjuncts are universal statements, so the same instance can be used for both them – desoana Sep 25 '22 at 00:14
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    Not having that text, I don't know exactly what Hurley is getting at. But it may be that there's concern about whether the instance that you use for $P$ is quantified in $Q$ somewhere. – Brian Tung Sep 25 '22 at 00:16
  • No, it is actually the first premise in the argument – desoana Sep 25 '22 at 00:17
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    I meant, perhaps Hurley gives some motivation for that elsewhere in the text that you haven't found yet. I don't have the text so I don't know. Generally the only restriction is on instances that are already quantified in the predicate. – Brian Tung Sep 25 '22 at 00:19
  • Thanks for help Mr. Brian – desoana Sep 25 '22 at 00:22
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    Nevertheless, the usual rule says that you can't "cross scopes." That is likely what Hurley means by "the whole line." There is no connection between the two $x$'s. Of course, I don't know what problem that rule is supposed to avoid, since one can split the conjunction and rejoin after instantiation, provided each instantiation is OK. – Brian Tung Sep 25 '22 at 00:24
  • Mr. Brian, I don't know what justifies splitting the disjunction. If you split it, then wouldn't you be asserting that one of the disjuncts is true? But what justifies that? – desoana Sep 25 '22 at 00:36
  • You also said that we can split and then instantiate and then join the two instances. I am afraid that you misread statement 1. That statement contains disjunction not conjunction. – desoana Sep 25 '22 at 00:39
  • Ahh, yes. I noticed at first it was disjunction, and then forgot it. Sorry, I've made a hash of my explanation! – Brian Tung Sep 25 '22 at 01:03
  • Let us continue this discussion in chat. – Brian Tung Sep 25 '22 at 01:11

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