[FOM] reducing higher to second order?

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Sun Dec 21 05:06:34 EST 2003

Todd Wilson <twilson at csufresno.edu>:

> I have seen passing references in the literature to a reduction of
> higher-order logic to second-order logic, but none of the sources I
> have at hand make any mention of it.  Can anyone supply a statement of
> this reduction and/or point me to where it was first (or most
> perspicuously) established?

Dear Todd

The issue was actually discussed here in FOM in September 2002. Here is how 
it went:

"A.P. Hazen" <a.hazen at philosophy.unimelb.edu.au> wrote:

> There is an old result (it MAY be due to Hintikka, "Reductions in 
> the Theory of Types," an  "Acta Philosophica Fennica" monograph (?) 
> from the 1950s) that, just  as Second-Order Logic (with the "standard" 
> -- non-Henkin -- interpretation) can characterize the natural number 
> series categorically, it can characterize the "standard model" of Third-
> Order Logic, or of full Finite Type Theory, up to isomorphism.  

Yes, it is:

K. Jaakko K. Hintikka: Two Papers in Symblic Logic (Form and content in 
quantification theory. Reductions in the theory of types.) Acta 
Philosophica Fennica VIII (1955). 



Panu Raatikainen
Ph.D., Docent in Theoretical Philosophy
Fellow, Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki

E-mail: panu.raatikainen at helsinki.fi

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