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

Richard Grandy rgrandy at rice.edu
Wed Feb 25 00:38:14 EST 2009


OK,  can we separate two questions?

1.  What is the minimal metalanguage we need to represent syntax of 
another language.  My answer stands to that.  I hope/don'tbelieve 
that I have a prejudice against semantics, but I do have a deep 
commitment to keeping track of what can (and cannot)  be done 
syntactically.  That is a mathematical question independent of one's 
preferences for/against syntax/semantics/pragmatics.

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).

REG

>
>
>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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