[hist-analytic] Frrom AUNE: Analytic and A Priori
Roger Bishop Jones
rbj at rbjones.com
Fri Mar 20 04:34:10 EDT 2009
Here is a third response to Bruce Aune's comment
on my proposed definition of analyticity:
On Wednesday 18 March 2009 11:12:36 Bruce Aune wrote:
> But as I said several times in response
> to Roger, a useful notion of analyticity should provide an a priori means
> of deciding on the truth-value of sentences, and this is something a mere
> appeal to necessity cannot do.
My first two responses were:
1. I have argued that my definition is consistent with
the definition "true in virtue of meaning", and that
my definition is in this respect no better or worse
than its principle alternative.
2. The requirement that a definition should yield a
decision procedure should be rejected.
However, by way of a more constructive response here is an
account of how one could produce formal proofs
of specific claims about analyticity of sentences.
I have not previously offered this because the role
of the definition of analyticity in this is rather
trifiling, all the action is in reasoning about the
semantics.
Before such specific proofs can be attempted one must
have a definition of the semantics of the relevant
language, and this must provide enough information
about the truth conditions to establish the desired
result. (this is in contrast to general reasoning
about analyticity, which can be done without any
specific definition of truth conditions)
It suffices to have an abstract semantics, and for this
purpose set theory (or something of similar expressive
power) is the best metalanguage to use.
The techniques for defining an abstract semantics
involve something like arithmetisation, except that
sets rather than numbers are used to represent syntax,
and are also used to represent the domain of discourse,
viz: "possible worlds".
Any method of definition which is acceptable in set
theory is acceptable, for formulating the semantics.
I can and will offer examples at a later date.
The main difficulty in formalising the semantics lies
in deciding what the semantics of the language is, and
in pathologies in natural languages which makes it
doubtful that there can be a coherent semantics which
meets all that one might expect of it. This is related
to the existence of the semantic paradoxes, which
suggest that no semantics for a natural language can
be consistent with usual presumptions about sound
inference in these languages.
These difficulties to not apply to formal languages,
for which it is normal to produce something equivalent
to such a definition of semantics (in the course of
establishing the consistency of the deductive system).
One may then conduct proofs that some sentence is
or is not analytic in the context of the two definitions,
i.e. the definition of analyticity and the definition
of the semantics of the language in question.
Informal proofs can be conducted in the normal manner
of mathematicians reasoning informally in an axiomatic
set theory.
Of course, conjectures about what is or is not analytic,
even in a context in which the semantics has been formally
defined, may be arbitrarily hard to obtain, or may not
exist even if the sentence is analytic, since set theory
as a deductive system is incomplete.
At some point I may come up with a fully worked example.
This is as close as one can get to an "a priori" method
of deciding the truth value of claims about analyticity.
Does it satisfy your request, or might it if the details
were spelt out a bit more?
Roger Jones
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