[FOM] The rule of generalization in FOL, and pseudo-theorems

[Victor Makarov]

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