FOM: Contextual definitions
urquhart at cs.toronto.edu
Mon Mar 26 22:01:15 EST 2001
Thanks to Charles Parsons and Richard Heck for their
comments on my postings. In particular, thanks to
Charles for answering my question about the proof theory
of second-order arithmetic. I thought that was the
answer, but I felt I might have overlooked some subtleties.
As for the whole question of the "contextual definition"
terminology, it seems that both Parsons and Heck are in
agreement with my criticisms. So, the only remaining question
is whether the phrase occurs in the literature of Frege studies.
First, let me say that my criticisms of George Boolos were off
the mark. On looking over "Logic, Logic and Logic" again,
I realized that the passage defining "contextual definitions"
comes from an introductory section, and is not by Boolos
himself. When Boolos talks of "contextual definitions", he
uses scare quotes consistently. I apologize for this error.
Boolos also quotes Crispin Wright on p. 311 as thinking the
terminology to be unfortunate.
However, the notion of contextual definition is quite prominent
in the literature of the subject, as can be verified by
consulting the index of Demopoulos's fine anthology.
Charles Parsons rightly takes me to task for my references to
the Arche web site. The notion
of "contextual definition" does not appear there, but what
does figure importantly is the notion of "implicit definition."
I am not sure what is meant by this.
So, let me close with a philosophical question, and try to get
away from these terminological wrangles. Given that second
order logic + the cardinality principle is proof-theoretically
equivalent to second order arithmetic, what do we learn from
this equivalence ("Frege's Theorem") from a philosophical/foundational
point of view?
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