[hist-analytic] Methods of Proof: Re: Clarity Is Not Enough
Roger Bishop Jones
rbj at rbjones.com
Mon Feb 23 10:10:32 EST 2009
On Monday 23 February 2009 13:46:00 steve bayne wrote:
> "In general, to reason about language A in language B one must
> be able to represent the syntax of A in the ontology of B."
>
> I agree with most all of what you have said here, but I'm not
> sure about the above. Some maintain no ontology involved in
> logic at all. Now I don't agree with this, but we have to be
> clear before passing on to other matters. Can you give me an
> example of how syntax is represented in ontology. Now a point
> on structure.
Its convenient to assume the existence of the things you
want to talk about, technically it might be possible to
avoid this but it would be cumbersome.
Typically we chose a metalanguage which does not assume
the existence of syntactic objects, but presumes an
ontology sufficient to represent them.
The most common examples are arithmetic, in which the
existence of the natural numbers is presumed, and set
theory in which the existence of a rich variety of
sets is presumed. In the former case we speak about
syntax via arithmetisation, in the latter using some
way of coding up syntactic objects as sets (it would
also be possible to do this via arithmetisation given
that we have ways of representing numbers as sets).
> My interest here is redundancy in a logical system. When you
> speak of "languages" I take it you mean canonical or formal
> languages, if so then I think there is a difference in how
> redundancy is to be regarded. For example, in making the
> grammar of a natural language explicit lack of redundancy
> has always been considered a virtue. So when you are setting
> up principles or parameters that will "generate" (as in
> "generative syntax") all on only the sentences of a given
> language alternatives must be evaluated and redundancy in the
> *application* of a rule becomes paramount. Now in logic there
> is a difference. Let me give an example. In some proofs of
> both up and down versions of Lowenheim/Skolem the occurrence
> of vacuous quantifiers makes no difference, so that there is
> nothing really wrong with '(Ex)(Ey)(Ez)Fxy', but in standard
> linguistic theory, where natural languages are the "object
> language" there is nothing wrong with this. So my point was
> this: if you set up an two axiomatic systems, and one involves
> less redundancy then that is the better one. Redundancy would
> be determined by how many ways a theorem can be proven; the
> more ways of doing it the more redundancy. The quantificacional
> example I've given above would if duplicated in standard
> linguistic theory would yield massive over generation. In logic
> this doesn't lead to inconsistency but my point was, in part,
> to raise the question whether the linguistic case and the
> logic case are similar. One other thing: If you take redundancy
> to mean repeated application of a rule, then it might be that in
> the sense I intend, a system like Hilbert's would be less
> redundant than, say, Frege's. I suppose formalization of
> arithmetic and the formalization of natural language may
> differ here.
I can certainly see certain kinds of redundancy which it is
usual to avoid if possible.
The classic example in first order theories is the preference
for axiomatisations of a theory in which none of the axioms
can be proven from any combination of the others.
Similar considerations would apply to rules.
As far as syntax is concerned some redundancy is common.
For example, repetition is often achieved by recursion,
and this will often lead to harmlessly ambiguous parse trees,
Similarly with proofs, where for example a linear conception
of forward proof leaves open the order in which lemmas are
proven.
> Finally, on topology.
> There is a connection here. Tarski wrote on it briefly and there
> have been others. It might be argued, and I think not without
> reason that topology is a branch of model theory; the connection,
> notwithstanding the flawed suggestion of Heine-Borel (perhaps) seems
> pretty clear.
I'm sure there are connections.
For example, one way of constructing models for positive set theory
is topological.
> Anyway, I think we need to get clear on what, precisely, you
> take to be ontology and its relation to syntax.
I hope the above helps in that.
regards,
Roger
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