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I've recently read some of the article on first order logic from http://mathworld.wolfram.com/First-OrderLogic.html and I'm puzzled by the explanation of universal instantiation that the author gives. It says

the following rule holds provided that F(r) is the result of substituting variable r for the free occurrences of x in sentential formula F and all occurrences of r resulting from this substitution are free in F,

∀x F(x) 
-------
  F(r)

I don't understand how F(r) is the result of substituting variable r for the free occurrences of x when x is bound by the universal quantifier. How exactly is x free?

  • F(r) is the result of subst the term r for the free occurrences of the variable x into F(x), and not (forall x F(x) ...). You can see this post for more details. – Mauro ALLEGRANZA Dec 28 '14 at 14:29
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It says that (∀x)Fx ⇒ Fr

where F(x) is any sentential formula in which x occurs free, r is a term, F(r) is the result of substituting r for the free occurrences of x in sentential formula F, and all occurrences of all variables in r are free in F.

Note that that way of putting it doesn't say that x is free in (∀x)F(x). It says that x is free in F(x) — in the sentence that the universal quantifier in question applies to. In other words, that substitution of r for that x works because that x is not bound in F by any other quantifier.

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