[FOM] More on first-order logic with operation constants

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Mon Dec 6 11:03:34 EST 2004

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

> John Corcoran has asked a series of questions about the history of 
> thinking about FOL:
> >In Hilbert-Ackermann 1938 and in Church 1956, addition is is treated as
> >a 3-place relation. Kleene 1952 and Mendelson 1956, on the other hand,
> >treat addition as a 2-place operation.
> >Q1.When did the logic community, as opposed to the algebra community,
> >come to recognize the "legitimacy" of taking operation symbols as
> >primitive symbols?
> ---> Did they ever NOT recognize the legitimacy of function symbols? 
> To give one example, Herbrand's 1931 paper (the one that comes AFTER 
> Gödel in Van Heijenoort's "From Frege to Gödel") gives a logical 
> treatment of an  axiomatization of arithmetic with = as the only 
> predicate but with assorted function symbols.

Hilbert 1922 has  + as a primitive symbol. Although H calls 1 and + 'parts 
of the number signs', he clearly treats + as an operation, not as a 



Panu Raatikainen

Helsinki Collegium for Advanced Studies
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