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

Roger Bishop Jones rbj at rbjones.com
Tue Feb 24 09:46:17 EST 2009

On Tuesday 24 February 2009 12:58:57 Richard Grandy wrote:

> What do we need to represent
> syntax?It may  be natural or habitual to think about ontology or
> domains of discourse in this context, but if we are analyzing what
> is required we need to think more carefully.


> To put it more directly, I am arguing that what is required for a
> metalanguage M to provide resources to analyze the syntax of language
> L is that the syntax  of M can represent the syntax of L.

However, what Steve was seeking was not a minimalist account,
but an intelligible explanation, and this is best done by
calling a spade a spade (and by talk about numbers rather
than numerals).

The metalanguage is for *talking about* syntax (inter alia)
and semantics is of the essence, without it the metalanguage
expresses nothing.

Our most tangible example is Godel's use of arithmetisation in the
proof of his "incompleteness" theorem.
It is said that Godel arrived at the incompleteness result via
the liar paradox, a semantic paradox, but carefully recast the
matter as a syntactic result because of a prejudice against semantics
which is still alive today in some quarters.
However, even though his result is strictly proof theoretic,
his description of how arithmetisation works is openly semantic
in character.

Here are some snippets from the second paragraph of the 1931 paper.

  "Of course, for metamathematical considerations it does not
   matter what objects are chosen as primitive signs, and we
   shall assign natural numbers to this use [that is, we map
   the primitive signs one-to-one onto some natural numbers].
   Consequently, a formula will be a finite sequence of
   natural numbers..."

  "The metamathematical notions (propositions) thus become
   notions (propositions) about natural numbers or sequences
   of them; therefore they can, at least in part, be expressed
   by the symbols of PM itself."

This is pretty semantic.

My explanation was generic with respect to the metalanguage,
and so instead of talking specifically of natural numbers,
I talk of "the ontology of the metalanguage" or its "domain
of discourse".

Roger Jones

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