FOM: second-order logic lives... Vedasystem at
Wed Mar 24 07:28:07 EST 1999

In a message dated 3/23/99 8:57:55 PM Eastern Standard Time,
holmes at writes:

<<In any case, no language has semantics in the absence
 of a given interpretation or interpretations>> 

 It seems to be a standard assumption of classical mathematical logic, but 
I would like to cite the following statement of Robert Kowalski:

"The assumption ... that there exists a reality composed of individuals, 
functions, and relations, separate from the syntax of language, is both
unnecessary and unhelpful". (R. Kowalski. Logic without model theory. In
"What is a logical system?", ed. by D.M. Gabbay, 1994, p.38).

Holmes writes:


First-order logic has proved to be fruitful because it has helped 
to solve some important long - standing mathematical problems.
What are some significant mathematical problems solved with 
the help of "second-order-logic"?

Holmes writes:

<< How do we see that the definition captures our pre-formal
notion of what a natural number is? >>

Holmes tacitly assumes that everybody has the same 
"pre-formal notion of what a natural number is".
But it is not the case!. Sazonov, for example, may have
 a different such  notion than Holmes has. 
Or Holmes may have a different notion of set that Zermelo had.
A great benefit of formalization is that it allows to write down
 in a formal language a formal analog of a pre-formal notion . 
There are may be several such analogs. 
A lot of debates go on because different people mean 
under the same name a different notion. 
But the formal language used must be really formal
 (it means that actually it must be a formal logical system).
The first-order ZFC is a formal language in that sense. 
Is "second order logic" a formal language in that sense? 

Victor Makarov
Brooklyn, New York

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