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#### Mathelogician

##### Member

- Aug 6, 2012

- 35

In Natural deduction in Predicate logic we have a rule which says [assume the set of hypotheses to be H)

In other words, if we want to proof a universal quantified formula, we need only to prove the formula without any quantifier (x occurs free in all hypotheses ; or arbitrarily chosen in other words) and then assert it in (universally) quantified form.

Now i need an example of real case in math. For example let the structure contain the set R\{0} of real numbers without 0 and the natural ordering < ; we want to prove : for all x x^2>0.

Now i don't get what 'the occurrence of x as free' means here when we aim to prove the theorem for an arbitrary x [ I myself tried to solve it for two cases x>0 and x<0 ; but i know that here we have x as free in both hypotheses!! ]

Anyway I'm a bit confused about the real application of the rule; And any help is welcome!

Regards

HTML:

`if H implies phi(x) then H implies [for all x phi(x)] such that x doesn't belong to FV(psi) for all psi in H [indeed such that x occurs free in no one of formulas in H]`

Now i need an example of real case in math. For example let the structure contain the set R\{0} of real numbers without 0 and the natural ordering < ; we want to prove : for all x x^2>0.

Now i don't get what 'the occurrence of x as free' means here when we aim to prove the theorem for an arbitrary x [ I myself tried to solve it for two cases x>0 and x<0 ; but i know that here we have x as free in both hypotheses!! ]

Anyway I'm a bit confused about the real application of the rule; And any help is welcome!

Regards

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