[FOM] reducing higher to second order?

[STEWART SHAPIRO]

The relevant resutls (or at least some relevant results) are presented in my book, *Foundations without foundationalism:  a case for second-order logic*, OUP, 1991, Chapter 6.  I don't have a copy here, and so can't give exact page references.

----- Original Message -----
From: praatika at mappi.helsinki.fi
Date: Sunday, December 21, 2003 5:06 am
Subject: Re: [FOM] reducing higher to second order?

> 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). 
> 
> 
> Best
> 
> Panu
> 
> Panu Raatikainen
> Ph.D., Docent in Theoretical Philosophy
> Fellow, Helsinki Collegium for Advanced Studies
> P.O. Box 4
> FIN-00014 University of Helsinki
> Finland
> 
> E-mail: panu.raatikainen at helsinki.fi
> 
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