FOM: Simple Types in BSL

Steve Awodey awodey at
Thu Jun 6 12:13:15 EDT 2002

Dear FOM,

With regard to the article in the current BSL:

"Types in Logic and Mathenmatics", F. Kamareddine, T. Laan, and R.
Nederpelt, BSL 8(2) 2002.

I am surprised to find several glaring historical errors in an historical
article published in this journal.  For example, on p. 213 the authors

"There is no definition of "type" in the *Principia*, only a definition of
"being of the same type": ... "

But as anyone who knows some early Russell can tell you, a type is the
range of significance of a propositional function.  Indeed in section *12
of the *Principia* on p. 161 (sentence 4), we find:

"A "type" is defined as the range of significance of some function."

The purpose of this posting is not just to point out the errors in this
article, however, but to ask the FOM readers for a correction.  Several
times, the authors credit the simple theory of types to Ramsey (1926) and
Hilbert-Ackermann (1928).  I suppose this follows custom, but it is again
not historically accurate.  Ramsey (1926) is no more than a sketch,
certainly much less precise than what was already in Frege.  And
Hilbert-Ackermann (1928), I believe, used ramified types, simplifying only
in the second edition from 1938 (and then erroneously crediting Russell's
second edition of PM for the simplification).

I think the first published source of simple type theory may have been
Carnap's *Abriss der Logistik* (1929) - which Carnap says was finished in
1927.  Carnap was of course a student of Frege's in Jena, and so could
plausably have gotten simple types from that source.

Does anyone else have information (or opinions) on the origins of simple
type theory?


Steve Awodey
Department of Philosophy
Carnegie Mellon University

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