[hist-analytic] General v. specific definitions of analyticity

Roger Bishop Jones rbj at rbjones.com
Fri Apr 10 10:46:56 EDT 2009


Steve raised a few problems in relation to "general semantics",
which I will try to respond to here.

Firstly, on whether Quine "called" for it,
we have done this already, and my intention
was only to refer to the previously cited passage
of "Two Dogmas" and the notion he there describes
(by contrast I think with Steve's notion of general
definition in which analyticity is supposed to
be defined completely independently of language).
I can't comprehend how this can be thought of
as hyperbole, though Quine did not so much
call for such a definition as state that it
would be required in order to make sense of a
language specific definition.

Here it is again:

      In short before we can understand a rule
      which begins 'a statement S is analytic for
      language L_0 if and only if ...' we must
      understand the general relative term
      'analytic for': we must understand
      'S is analytic for L' where 'S' and 'L'
      are variable.

"we must" is a pretty strong call, 
and Carnap, possibly in response to this, later
delivered such a definition.
Of course, I don't deny that Quine was alleging
that this call was unanswerable (despite providing
a very nice answer later in the paper).

However, what Carnap did, and what I agree with
him was the right thing to do, was to give the
general definition as the sole definition of
analyticity, and to abandon the use of the term
'analytic' in defining the semantics of particular
languages (or in giving a general account of such
definitions).

As a result of the various misunderstandings we
have had in attempting to discuss the difference
between general and specific definitions of analyticity
it may now be useful to make the following more
elaborate classification:

(a) A general definition of analyticity in which
    S and L are variable

such as Quine spoke of, and Carnap offered in the
Schilpp volume.

(b) A general definition of analyticity in which
    no specific mention is made of the language.

this is the usual form of a definition of analyticity.
It is what Hume does in describing his "fork",
what Kant does in his definitions of analyticity,
and what other philosophers such as Frege, Carnap,
Ayer, Quine, Kripke, and in fact every definition
of analyticity I have ever come across apart from
the language specific ones discussed by Carnap.

The fact that these do not explicitly mention
the language does not mean that they are definitions
which are independent of language, but to make this
case to someone who doubts this has to be done
on a case by case basis.  The most popular rendition
in the 20th Century was "true in virtue of meaning",
and this clearly applies principally to things which
have meaning, and clearly results in a concept which
is dependent on meaning, and hence on the semantics
of the language in question.

(c) A general definition of analyticity in which
    the concept is independent of language

This possibility had simply not occurred to me until
I realised that this is what Steve was expecting from
a general definition.
This I would suggest is only possible for things which
*are* meanings rather than things which *have* meanings,
e.g. for propositions rather than sentences.
In which case Carnap and I would agree that the concept
in question is synonymous with logical necessity.

(d) Language specific definitions of analyticity in
    which the set of analytic sentences in some particular
    language is defined.

Personally I think these cause confusion and should be
avoided.  I am acquainted with none but those of Carnap
who appears to have abandoned them by the time of the
Schilpp volume.  They are of course the principle
object of criticism in Quine's "Two Dogmas".
Their abandonment does not in itself disarm Quine's
criticisms, for to apply a general definition of
analyticity to some specific language one would
need to know something about the semantics of the
language.

Steve seems to think that "truth" does not suffer
from so severe a problem as "analyticity".
It is of course just as dependent on the semantics
of the language in question, and Quine's arguments
if accepted are equally devastating against truth
as they are against analyticity.
The idea that one can determine truth without
knowing meaning reduces language to a formal
calculus.  Of what avail is it to know (if one
could) that a judgement is true, if we do not
know what it means?

Now we come to Steve's nominalistic qualms.

If we are to come to a good understanding of
how descriptive language works, of which an
elementary part is the discussion of concepts
such as analyticity, then it is valuable to
devise precise models of language.
The use of mathematics for modelling the real
world is pervasive in science, without it science
would be devastated.
Scientists do this without the slightest ontological
qualm.  It works. I am aware of no defect in
empirical science which has ever been traced to
the ontological premises on which mathematics is
based.
Mathematics has always been considered far more
rigorous and reliable in its conclusions than
philosophy.  So for philosophers to spurn
mathematics because of its ontology in the name
of rigour seems the greatest hypocrisy.

In the theoretical study of language, just as
we can reason about syntax by arithmetisation,
we can model semantics in similar ways.
For general semantics, set theory suffices.
Whatever abstract structures we may desire for
modelling language, isomorphic structures will
be available in the ontology of set theory
(i.e. as sets).

So the first step out of the mystery of entities
such as "meanings" or "propositions" is to come
up with set-theoretic models of language.
These enable us to reason about language in a 
general way, and support the demonstration of
elementary results such as that expressing the
relationship between analyticity and necessity.

If nominalistic qualms persist, it is possible
by known techniques (discussed by Boolos for
example) to eliminate the ontological presumptions
and obtain a formal theory independent of ontology.
The resulting theory may not mean exactly the
same as the set theoretic version (the translation
after all is intended to eliminate supposedly
undesirable ontological implications or presumptions)
but will serve the same purposes in clarifying
the issues at stake.

I therefore recommend, if an understanding of language
is required, the use of mathematical modelling,
and suggest that nominalistic qualms can and should
be set aside.

On Sunday 29 March 2009 20:08:44 Baynesr at comcast.net wrote:

...

>I will explain my point about Carnap's view
>of analyticity and L-truth. He begins the tradition,
>more or less, of assuming logical truths are
>analytic.

I don't think it is correct to call this an assumption.
He used the two terms synonymously, but this was
more a deliberate and considered usage than
an assumption.
In this I agree with him, it seems to me unreasonable
to deny the title "logical truth" to any proposition
which is "logically necessary" and this is the
position in which those who urge that logical truth
should be construed more narrowly find themselves.

>Restricting ourselves for purposes of
>illustration to Boolean logic of propositions,
>we say that a proposition of logic is analytic
>if it is true in all state descriptions - there
>are qualifications to this, such as the propositions
>have to be in normal form. But the point is that
>the whole idea of L-truth and, therefore, analyticity
>in Carnap relies on state descriptions, and
>being "true" in all state descriptions (this relates
>to but is not exactly the same as worlds). But
>now analyticity depends on truth in Carnap.
>Since all other analytical truths, 'analytic'
>in the broad sense, depend on L-truths it follows
>that Carnap's position on analyticity, generally,
>depends on the analyticity of L-Truths.

I don't really understand your point here.
The terms "L-Truth" and "analytic" as used by
Carnap are synonymous.  Why is that a problem? 

>"the domain is part of the interpretation, and
>therefore the sentence must be true in interpretations
>with any cardinality"
>
>If you interpret the sentence '(Ex)(Ey)(z)[f(x) &
>f(y) -> f(z)]' in a domain containing more than
>two individuals it is not true.

Not necessarily.
In any interpretation in which f is somewhere false
that sentence will be true.
(it is equivalent to:
    (Ex)(Ey)[f(x) & f(y) -> (z) f(z)]
which will be true if:
      (Ex)(Ey) (not(f(x) & f(y))
in which x and y need not be distinct, i.e. if
      (Ex)(not f(x))

>Similarly, if you
>the sentence '(Ex)(Ey)(z)[z=x v z=y & ~(x=y)' in
>any domain greater than or less than 2 it, too,
>is false.

This one is OK.

What you are talking about here is
"truth in an interpretation" i.e. your point is
that "truth in an interpretation" depends in general
on the size of the domain of the interpretation.
The reason why I complained is that this is not
what you said.  You said that "logical truth"
depends on the domain of the interpretation,
and the term "logical truth" is usually taken
in modern logic as synonymous with "valid",
i.e. true in all interpretations, so it can't
be relativised to an interpretation.

>This is pretty much standard stuff. Similarly,
>in order to do the semantics for any system capable
>of handling transfinite numbers your are going to
>need more than a countable domain. Anyway, the real
>philosophical point is that the individuals
>constituting the domain enter into any philosophical
>account of a rational reconstruction of cardinal
>arithmetic.

I still don't understand the point here.

>With respect to your other comments, I take a Quinean
>position. I understand the concept 'true' (consider
>Tarski "reliance" on Aristotle); I don't understand
>'analytic' ab initio. So I don't see much value in
>trying to define it with all the "mumbo logics" etc.
>The term, unlike 'true', is a technical term. Tell me
>why I need it and I'll wade through more of Carnap's
>efforts to do whatever it was he was trying to do.
>But outside the notion of a "judgment" I see little
>use for it, unlike 'logically true'.

Well I'm somewhat dumbfounded.
How can one of a pair of synonyms be useful and the
other useless?

>A final point.
>It has been argued, vigorously, (and here I'd mention
>Marian David's paper "Analyticity, Carnap, and Truth"
>_Philosophical Perspectives_, 10, 1996) that Quine
>will have to reject 'true' for the same reasons he
>rejects 'analytic'. Otherwise, the entire enterprise
>initiated by the "linguistic turn" is in danger.
>I agree with this assessment, but I don't agree that
>Quine is wrong for the reasons he gives. The question
>we might pursue is this: can we do without 'true' as
>well as 'analytic'? I think not. The problem runs
>deeper than constructed languages can carry us. A
>general concept of truth is what we get when we use
>Tarski as providing a criterion but find the definition
>to be "value added." The same might be said of 'analytic'.

The arguments which Quine uses in "Two Dogmas" are
rooted in a radical scepticism about meaning.
They are Pyrrhonean in their scope and impact.
You cannot accept those arguments and consistently
continue to engage in philosophical debate or in
science or in any other enterprise in which descriptive
language is involved.

>Finally, I like Quine have a problem with meanings; no
>one has a convincing argument so far as I can tell
>that we need them. We may need essences, but that
>depends on how we reconcile them with nominalism
>in an empiricist epistemology.

You don't need meanings as entities.
But why tie one hand behind your back?
Understanding semantics is not so easy.
Its harder for a radical nominalist.
However, it is very much easier than Quine would
have us believe, even for dogmatic nominalists.

As to essences, I don't know whether they are useful.

If we are to continue we probably need to focus down
a bit so we aren't chasing too many hares at once.

Roger Jones




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