[FOM] The rule of generalization in FOL, and pseudo-theorems
Victor Makarov
viktormakarov at hotmail.com
Wed Sep 8 20:12:33 EDT 2004
Sandy Hodges on Mon, 30 Aug 2004 13:29:00 -0700 wrote:
>The problem is the 'rule of
>generalization.' This says that if we have proved some formula:
> Z(a)
>where 'a' represents an arbitrary variable, and "Z(a)" represents any
>formula containing that variable, then we may conclude:
> (Forall x) Z(x)
>
>This rule only conforms to the above definition of a logical theory, if
> Z(a)
>is a theorem. But the claim that "Z(a)" is a theorem causes
>difficulties for the philosophy of logic.
We can avoid these difficulties if 'a' is considered as a new constant
name (see "Set Theory and Continuum Hypothesis" by Paul Cohen, pp. 10-11).
More precisely, the 'rule of generalization' can be formulated in the
following way:
Let T1 is the extension of a (first-order) theory T by adding to T a new
constant c. If Z(c) is a theorem in T1, then (Forall x) Z(x) is a theorem in
T.
Victor Makarov
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