[hist-analytic] What do we need to represent syntax?

Roger Bishop Jones rbj at rbjones.com
Wed Feb 25 03:33:14 EST 2009


On Wednesday 25 February 2009 05:38:14 Richard Grandy wrote:
> OK,  can we separate two questions?
>
> 1.  What is the minimal metalanguage we need to represent syntax of
> another language.

Depends on what you want to do with the representation.

For the kind of metatheory which I engage in I would state the
minimal requirement as:

    It must be possible to define in the metalanguage
    an algebraic structure isomorphic to the abstract syntax
    of the object language.

This entails that there is an injection from the abstract syntax
of the object language into the things over which one can
quantify in the metalanguage.

It does not entail that there is an injection from the syntax
of the object language into that of the metalanguage.
This may be seen in my own most recent technical work
(see: http://www.rbjones.com/rbjpub/pp/doc/t024.pdf)
which involves formalising the syntax and semantics of
an infinitary set theory using as metalanguage
a higher-order set theory. In this case the syntax of
the metalanguage is countable, but that of the object language
is inaccessible, so there is no possiblity of an injection
from the object language syntax into the metalanguage syntax.

> 2.  What is an intelligible/psychological/historical  explanation  of
> the above.  On that I defer to Steve and you (both to explain the
> phrase an answer it).

Too many questions there!
I was attempting only the "intelligible" and I seem to have failed
so far. I may try again.

Roger Jones




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