[hist-analytic] The Two Color Problem:Putnam/Aune
Baynesr at comcast.net
Baynesr at comcast.net
Fri Jan 1 12:14:58 EST 2010
PRELIMINARY REMARKS ON THE “TWO COLOR PROBLEM”
A. One problem with Putnam's approach is that because in order
to "prove" his case that the nothing can be red and blue all
over he introduces the right postulates to get what he wants.
And since he CAN get those postulates, he argues that it
follows that we can regard the proposition just as analytic
as 'All ba chelors are unmarried."
One aspect of this is that he must use postulates to get
one of the RST properties of the equivalence relation, transitivity,
and he uses something like exact resemblance to get the other
properties, viz. reflexivity and symmetry. So the postulate has
to be stuck in there and then the other superadded and voila (!)
he gets the relation "right." Just consider how a relation can
under one set of circumstances be transitive but under another
not. Take the relation 'has exactly the same hammer as'. Ok,
now since it is possible that even though x has the same hammer
as y and y has the same hammer as z, nevertheless x does not
have the same hammer as z. In other words x and z may have
different hammers. The relation, then, is not transitive. But
now suppose that there is only one hammer; then if x has the
same hammer as y and y as z then x has the same hammer as z,
so the relation IS transitive. The relation 'has the same hammer
as' is nontransitive. But nontransitivity will not afford
Putnam what he wants, an equivalence relation.
B. Suppose a thing were allowed to be both red and blue all over.
Would it be blue? Partly blue? Partly blue all over? Clearly, we are
faced with more than logical or ontological problems, problems
I don’t think can be met by stipulating a meaning postulate or
constructing a system of postulates with an objective in mind of
ruling out some things and other things in.
C. Putnam has simply made up a “word” meaning “exactly
the same Color as” then endowed it by fiat with a meaning
that rules out the offending synthetic a priori.
D. I do not discover that this is the same color as that the
way I discover that Hesperus is Phosphorus. There is much
here I cannot digress to examine. If I know both colors I
know they are not the same; but this is not the case with
Hesperus and Phosphorus.
E. An object can’t have two surfaces; what does this have to
do with being two colors “all over.” The argument against
synthetic a priori with respect to colors may depend on
admitting it in the case of surfaces.
F. I will address Bruce’s argument; the one he sent to the
list separately. I don’t think it does the job, but it does deserve
careful attention. More soon.
G. Sometimes it seems to me that Bruce thinks you can
solve certain problems, the ones under discussion having
to do with defining, etc. “determinate colors” by simply
going to the paint store and asking to view a color wheel.
This sounds cynical, but it sure looks that way to me. I
invite Bruce to reflect on a couple of quotes from Wittgenstein
which I would introduce in opposition to his view as I see it.
“Let us imagine that someone were to paint something
from nature and in its natural colors. Every bit of the surface
of such a painting has a different color. What color? How do I
determine its name? Should we, e.g. use the name under which
the pigment applied to it is sold? But mightn’t such a pigment
look completely different in its special surrounding than on the
palette? (Remarks on Colour, para 68)
“People might have the concept of intermediate colours or
mixed colours even if they never produced colours by mixing
(in whatever sense). Their language-games might only have to
do with looking for or selecting already existing intermediary or
blended colours.” (op. cit. para 8).
“If I say a piece of paper is white and then place snow next to
it and it then appears grey, in normal surroundings and for
ordinary purposes I would call it white and not light grey. I could
be that I’d use a different and, in a certain sense, more refined
concept of white in, say, the laboratory, (where I sometimes also
use a more refined concept of ‘precise’ determination of time.” (
op. cit. para 160)
“Couldn’t there be people who understand our way of speaking
when we say that orange is reddish-yellow and who were inclined
to say this in cases in which orange occurs in an actual transition
from red to yellow? And for such people there might very well be
a reddish green.
Therefore, they couldn’t “analyse blends of colours” nor could they
learn our use of X-ish.” (op. cit. para 129).
These quotes are pertinent to my disagreement with Bruce. I hope to
elaborate, but for now I just want to post this stuff in such a way that will
get the ba ll rolling again.
THE SIGNFICANCE OF THE “TWO COLOR PROBLEM”
By the “two color problem” I mean the problem of determining
the status of the proposition: “No two things can be the same
color all over at the same time.” If we take the proposition
synthetic, then we will either regard it as contingent or synthetic
a priori. Why shouldn’t we regard it as synthetic a posteriori?
What is the argument that it is not. Now it would appear that this
is an issue for neither Aune nor Putnam, since both are prepared
to regard the above proposition as in some sense a priori. We
won’t discuss what sense, but we should take note that Putnam
will speak of propositions which “feel” analytic. (p. 74). I think there
is something to this, but if there is we have to contend with a
number of issues not discussed.
One such issue is whether such “feelings” make a difference;
perhaps not, but when we reflect on the above proposition it is
difficult to maintain that if there are such feelings, then while ‘A
ba chelor is an unmarried man’ may “feel” analytic, it sure doesn’t
sound as if ‘Nothing can be red and green all over at the same
time’ “feels” analytic in the same way. This is not a trivial point,
especially given the emphasis some philosophers have placed
on the idea of “epistemic counterparts” in dealing with the “feeling
” or “appearance” of contingency with respect to propositions
that seem to be necessary, such as ‘Phosphorous is Hesperus’.
Without such “feelings” philosophy pro ba bly doesn’t come
into play; the rest may just be word games about common
sense discourse, games that resolve no puzzles, in particular
puzzles that such “feelings” engender. So I think this is important.
THE TWO COLOR PROBLEM AS ILLUSTRATING METHODS
A number of philosophers have said that the problem of
the synthetic a priori may not be very important. Sometimes
these philosophers, then, go about the task of discussing
the matter at great length. Is there a justification for this? Yes,
I think so. The “two color problem” brings this issue into play
and how we deal with it tells a lot about our methods of analysis.
There are at least four approaches to the two color problem.
1. The method of formal language where constructing the
language is ba sed on an intuitive or scientific understanding
of some pre-existing nature of the subject under examination.
2. The method of formal language where constructing the
language is ba sed on pure construction; that is, where the
world is not understood as preexisting the construction but,
rather, is itself a notion to be constructed.
3. The informal method dispensing with formal language all
4. A mixed approach involving (1) and (3) or (2) and (3).
There are other approaches but for the problem at hand these will suffice.
Putnam adopts the first approach; it would appear that Aune
subscribes to (4). Nelson Goodman is, perhaps, the best example
of an advocate of (2). Of Goodman’s approach, Putnam once said this:
“This brings me to perhaps my most important remark about
Goodman’s philosophical methods and attitudes…by rejecting
the most fashionable problems of philosophy, he is totally free
of the “now philosophy is over” mood that haunts much of
twentieth century philosophy. If there isn’t a ready made world,
then let’s construct worlds, says Goodman. If there aren’t
objective standards, then let’s construct standards! Nothing
i s ready made, but everything is to be made.” (Forward to
_Fact, Fiction and Forecast_. Harvard, 1983, p. xv)
Of particular interest is Putnam’s remark about the “now
philosophy is over mood.” He doesn’t tell us what he means
but there is room for speculation.
EPISTEMOLOGY vs. ONTOLOGY
One thing that makes the “two color problem” important as
well as interesting is not only that different methodologies
are tested, but there has been a contest going on for some
time between epistemology and ontology, one that is
clarified by attention to this issue. On the view I take, the
“now philosophy is over” crowd to be mainly epistemologists
who eschew metaphysics as it is both traditionally conceived
as revealing the nature of the world, and as identifiable with
ontology. I now offer a partisan comment on this conflict.
During the 1920s the idea was still around that metaphysics
could provide some understanding of the logical structure
of the world and experience. The ascent of Tarskian
semantics let to the overthrow of this point of view, at least
in large measure. Epistemology felt the effect. No longer did
epistemology trade in sense data; and no longer was the unity
of consciousness, the nature of mental acts and questions
related to the idealist tradition more generally the focus of
concern. Instead, there was a move towards “meta-epistemology”;
defining ‘knowledge’; induction, pro ba bility theory,
confirmation theory, etc. These issues unrelated to
traditional metaphysics supplanted the old regime. Philosophy
was seen more in terms of mathematical considerations on
pro ba bility, Bayesian analysis, and such notions as “reliable
belief producing mechanism began to make it appear that
science and mathematics not metaphysics was central.
What had begun with the logical positivists as a critic of
metaphysics became a critic of “philosophy” as traditionally
understood. Epistemology was no longer, in other words, the
“ontology of the knowing situation” as it had been for people
like Broad and the early Russell, to take two prominent
Some of this was good, but not all. It is now made to
appear, at least for some of us in the metaphysics camp
that if you want to know whether there are physical objects,
then “open your eyes stupid” is to be a significant part of
the new “epistemology.” And if you want to know what a
color is, just go to the paint store and they will be happy
to show you a color chart; if you want to know the structure
of the world ask a physicist; as for essences, you can refute
them too; just find a three legged lion and you are all set! If
there is anything left we’ll just sweep in under the rug of
“common sense.” Philosophy, then, has gone “casual,”
a occupation free from labor, a form of relaxation for the
well to do with an inordinate sense of their own self importance.
Now this is, admittedly, a very bias characterization.
I bring it up because the “two color problem” on the
view I take is a metaphysical problem possessing many
dimensions, each one a sort of handle one might take hold
of in order in searching for the best grip. One reason
this is a metaphysical issue is the nature of color is very
much at issue, if I’m right. Putnam sees this, but it doesn’t
appear to be very important to Aune; let me briefly
elaborate one concern.
COLORS, OBJECTS AND IDENTITY
A great deal of the “two color problem” depends on
what we take colors to be, ontologically. If we take
colors to be particulars we will get one solution (and
it will be quick); if we take colors as objects, like physical
objects, then the nature of identity statements is something
different from what it would be were we to regard colors
as universals or mental entities. Let me supply a contrast
that may be sufficiently illuminating to obviate any need for
protracted discussion. Let us consider the identity
Hesperus = Phosphorus
How many times have we been told this was a significant
discovery? Quite a few. We are told that it was significant
at least in part because it was truly informative. Why was
it truly informative? Well, in the morning people would see
one star and as the sun rose it disappeared and later they
saw another star. Later it was discovered that these two
stars are identical. What happened was that two terms were
*discovered* to be co-referential. This is what made the
identity “significant.” But not all contingent identities are
similarly significant, color providing a case in point.
Suppose that instead of pointing to a star in the morning and
one in the evening and later finding out that they are the same
I point to a color in one room; then I go into the other room
and point to a color and say that
This color is exactly the same color as that
Now what I would argue is that there is a sense in which this is
no discovery at all. As long as I am talking about the color(s)
and not the object which have the color(s) then one plausible
view is that the identity is contingent and a posteriori.
Alternatively, if we take this as a true identity it must be
a necessary truth. The first option is, itself, ambiguous.
If colors are tropes then if the identity holds it is necessary;
if colors are universals, then we are faced with problems of
individuation of properties etc. Now we have three options
and if we add Putnam’s relevant sentence
‘x is exactly the same color as y’
where ‘x’ and ‘y’ refer to objects, then we have
four possibilities where much depends on how
“exactly the same” and “being identical” are related.
If you are unconvinced of this sort of variation among
identity sentences, that is whether there are different
conditions for identity depending on the objects asserted
to be identical, consider another problem, one which I think
can be related to the “two color problem.”
Suppose someone claims the following to be the case.
No object can have two surfaces at the same time.
Is this true? I think it is. Is it analytic? I don’t think so but, to use
Putnam’s expression, it “feels” more analytic than
No object can be red and green all over.
Is there any justification for this intuition? Two objects may
have the same color (unless they are tropes) but they cannot
have the same surface. It seems to be the case that the very
concept of a physical object precludes having more than one
surface, whereas it does not seem to me to be the case that
it is part of the very concept of a physical object that it can
have both the color of that thing over there and this thing
here at the same time all over, which is a Putnamian way
of saying that nothing can have the same color all over at
the same time. If an object can have two surfaces at the
same time, then it is not absurd to suppose that an object
can have two surfaces each with a different color. So, maybe,
nothing’s being the same color all over at the same time
depends on no object having more than one surface
at the same time. But now what of
No surface can have two shapes at the same time?
Now it would almost seem that this is false for reasons we
may touch on later; but for now the point is simply this: how
we solve the two color problem, or address it, will tell us
something about our ontology; if the claim that no object
can be red and green all over is analytic, like ‘all ba chelors
are unmarried’ then there is nothing informative in the
assertion beyond the way we use our words. Both Aune
and Putnam hold this to be an analytic and so just as trivial as
Hesperus is Hesperus.
I think this is a reduction of their position, precisely because I think
it is informative in a way that such sentences as these are not.
Now I don’t think Putnam shares Goodman’s form of “constructivism.”
Putnam is “modeling” the world, so to speak in the medium of formal
l anguage; Goodman is creating worlds. If he admits to understanding
what Putnam is saying, then he’ll pro ba bly have some view on the
subject as to the sense in which he does or does not align himself
with this trend. The point of bringing this up is that it raises the important
question of whether the two color problem has a solution; that
is, whether it is a problem or a mere figment of a problem; what
Schlick first called a “pseudo-problem.” (1920).
Putnam maintains that to such questions that lead to such problems
“there is never a final answer.” (Sumner and Woods p. 77) One
can see the reasoning here in Goodman’s case. We create worlds;
there is no “final world” so there can never be a final answer to any
questions that go beyond any one constructed world. For Putnam,
the case is less clear. However, here is what I think Putnam is
suggesting: while there are no final answers, some answers are
better than others; that is, we may construct any one of a number
of answers, but no answer is “right.” What he may mean is that there
is a cost associated with any one of a number of answers and we
have to decide what price to pay. Since philosophers vary on what
is valuable etc. the price to pay varies; there being no “natural” price;
we are in the realm of “exchange value” so to speak. So we have it
that there may be a number of answers but no right answer. What
is Aune’s view on this?
Aune will sometimes rely on informal methods as central to
solving a problem. Other times he seems to rely on formal
methods, such as introducing “meaning postulates,”
something we will discuss shortly. Given this “mixed”
approach – that is, “mixed” in a way that Putnam’s is
not – with respect at least to the problem under investigation
there is not deciding whether for Aune there are in fact
solutions to ANY philosophical problems. Clearly, he wouldn’t
propose them if he didn’t believe they were indeed solutions,
but this does not exclude the possibility of other, equally good,
solutions. It would be interesting to know his feelings on this matter.
We have been talking about the a priori in the context of
Bruce’s formulation of an empiricist theory of knowledge.
I think it is important to see where we may disagree on the
importance of what I’ve been calling the “two color problem”
in arriving at some conclusion as to the viability of empiricism.
Some such views are more radical than others.
Although we have been discussion judgments and
whether or not Kant has been refuted it is important t o
keep in mind that for Kant there is more to the issue
than judgments. For Kant not only are some judgments
a priori some concepts are a priori as well. The empiricist,
traditionally, has held that there are no a priori concepts, but
there may be a priori judgments. This is possible as long
as the judgments that are a priori are analytic. The reason
for this is that the empiricist wants to rule out knowledge of
necessary connection between worldly objects. He can
have an a priori but only if it is analytic, that is, as long as its
truth (the judgment, that is) does not describe the world.
In the case of the judgment that no thing can be two
colors all over at the same time, we have an interesting
case. It is interesting because it seems to say something
about the world and yet it appears necessary. This is
just the sort of thing that will cause most empiricists to
recoil. So the empiricist must analyze the nature of
color etc. in such a way the judgment becomes analytic.
Putnam points out just as we might analyze ‘All ba chelors
are unmarried” in such a way as to make it analytic, similarly
we want to analyze ‘No object can be two colors at the
same time all over’ in such a way that it, too, comes out
analytic. It is not important whether two colors cannot be had
by one object at the same time all over. What is important is
the status of this judgment. The empiricist may argue that it is
analytic that this is analytic or, alternatively, that is merely
contingently true. Bruce and Putnam arrive at similar arguments,
but there are crucial differences. Let’s begin with Putnam and
then go to Aune.
We begin with the idea that logical truths, such as ‘Ba v ~Ba’, are
analytic. But such truths are not the only ones considered to be
analytic. Recall how it is, typically, shown that ‘All ba chelors are
unmarried’ is analytic, although it is not a logical truth. What we
do is show that it can be converted to a logical truth by way of
definitions. We may consider what is required as definitions or
as “meaning postulates.” There is a difference but let’s give the
illustration and go from there. We begin with:
1. All ba chelors are unmarried.
We, then, introduce the meaning postulate or definition:
2. ‘ ba chelor’ means ‘unmarried man’.’
From this the following logical truth is derived by substitution:
3. All unmarried men are unmarried.
Now something very similar is afoot in attempting to
argue that the sentence ‘Nothing is red all over and
green all over at the same time’. What we need to do i n
order to show this analytic is to derive a logical truth
from it along with definitions. In attempting to demonstrate
the analyticity of ‘Nothing is red all over and green all over
at the same time’, Putnam will engage the task of showing
that ‘Nothing is the same color as A and the same color as
B at the same time where A and B are not exactly the same
color. Mimicking the approach to the analyticity of (1) he will
introduce a couple of properties of the relation ‘exactly the
same color as’. One property is that it is a stronger relation
than ‘indistinguishable from’ since it entails this relation without
being entailed by it. Later he will provide a definition (Sumner
and Woods p. 79) which depends on this weaker relation. The
other property which goes into defining ‘exactly the same color
as’ is transitivity. So the claim will be that that original sentence,
‘Nothing can be red all over and green all over’, can be shown
to be analytic by being shown to be a logical truth once these
definitions are added to the language.
There are three claims essential to Putnam’s paper that
I want to focus our attention on. They are:
9) (x)(~Ex[A, B] -> . Ex[x, A] -> ~Ex[x, A} -> ~Ex[x, B])
10) (x)(~Ex[x, B] & (x)(~Ex[x, B])
11) ~Ex(A, B)
Where ‘(x)(~Ex[x, y]’ reads “x is exactly the same color as y.”
In connection with this relation Putnam lays down the first two of a
number of postulates he will, eventually, require to make his point. These are:
1. It is an equivalence relation
2. ‘Ex’ implies ‘indistinguishable from’, not vice versa.
(Summer and Woods p. 75)
Logically, his point will be that from (9) the equivalence of (10)
and (11) follows. So it will turn out that, given, (9), if two things
have different colors (i.e., not exactly the same colors), then
no thing has them both. But this will not give us what he wants,
namely, analyticity of ‘Nothing can be two colors all over’
(hereafter “S”). How, then, we do we get the analyticity
and, thereby deny the rationalist factual necessities, such
as S? What he does is point out that (9) is equivalent to (12):
12. (x)(Ex[x, A] ->. Ex[x, B] -> Ex[A, B])
He points out that (12) “expresses the transitivity” of “EX”’,
keeping in mind that transitivity is guaranteed by the postulates.
In this way, he arrives, eventually, at the conclusion that S is
analytic. A couple of formal observations are in order.
It it is important to notice that he can derive (10), but he can’t
do it from (9) alone; he needs (11). (11) is not analytic. So (10)
may not be analytic after all, IF you mean by ‘analytic’, in part
at least, that analytical statements do not depend, essentially,
on contingent facts. Suppose there is a way around this. There
is something I think is more interesting. Although (10) may
be inferred from (9) and (11), as a matter of fact (9) follows
f rom (10)! If so, then, given (11), (9) and (10) are equivalent;
and, since (9) is equivalent to a postulate, (10), and so S, will
have (11), alone, as the single premise upon which the argument
depends. (We don’t include postulates among the premises).
So now the burden is, largely, on the transitivity of ‘exactly the
same color as y’. I reject the transitivity and, therefore, reject
the idea that ‘Ex’ is transitive. If I can sustain this claim,
Putnam is refuted. But can I?
Bruce Aune’s approach will, critically, depend on the
notion of being a “determinate color’; Putnam’s approach
will not. The reason, I think, is this: Putnam believes, as do
I, that there are no determinate colors. Putnam remarks
that “ …when we think of the color concepts, the most
striking fact we observe is that they form a continuum.”
(op. cit. p. 78). This will prove very important to Putnam’s
case. He will introduce a lot of postulates to accommodate
this fact. So many that I am inclined to include him among
the “American Postulate Theorists”! But setting this aside,
consider why he might do this and, at one point, even admit
that people may suspect him of “smuggling” in some stuff.
One of the cleverest of his many clever proposals is related
to this and brings into the picture views, originally, expressed
by Henri Poincare. Poincare was a Kantian. He was loathe to
admit to actual infinities. He was, therefore, adverse to the idea
of a “continuum” in the sense of being a real structure. We
construct it. The mathematical continuum on his view is
derived from what he called the “physical continuum.” The
idea is both ingenious and believable, even if it is wrong.
I’m not sure that it is. Here is what he said:
“It has, for instance, been observed that a weight A of 10
grammes and a weight B of 11 grammes produced identical
sensations, that the weight B could no longer be distinguished
from a weight C of 12 grammes, but that the weight A was
readily distinguished from the weight C. Thus the rough results
of the experiments may be expressed bythe following relations:
A=B, B=C, A<C, which may be regarded as the formula of the
physical continuum. But here is an intolerable disagreement
with the law of contradiction, and the necessity of ba nishing
this disagreement has compelled us to invent the mathematical
continuum.”_Science and Hypothesis_ Dover. P. 22.
The main problem with Aune’s approach is he assumes
that the colors of things are determinate. He, also,
appears to accept the idea that colors, themselves, are
determinate. He offers no reason for believing either of
these two assumptions.
If there are determinate colors, then on any meaning I can
attach to ‘determinate’, this will entail that the number of colors
is finite. I’m not so sure this is true, unless we resort to certain
limitations on the eye; but let’s leave eyes and light waves out
of the picture, for now.
Crucially, Aune’s proof depends on the transitivity of identity,
but when it comes to colors as objects of visual awareness
I don’t believe this. Suppose two things are the same color
(I am looking at them). But there is a third thing that is identical
in color to the second but not the first. Then it would appear
that I have at least three colors, although only two are
distinguishable. How often can I repeat this procedure?
No one can tell for sure, but in theory it looks like it could
go on forever. This sort of thing seems to be assumed by
Putnam as we shall see. In other words, if Bruce admits to
“determinate” colors, then by any definition I can see the
number of colors must be finite. I see no reason to believe
this. In addition, this will prove irrelevant because there is
another problem with his account related to the alleged
transitivity of identity between colors. But before harping
on that theme, let’s take a look at what his formal proof
actually proves, for if I am right, using the same methods
one can prove that it is impossible for one man to hold
two hammers at the same time.
“Consider the conditions under which we would say that
that two objects are exactly the same color. These do
not always coincide with the conditions uner which we
would say that they are indistinguishable, e.g., let A and B
be indistinguiahable but supposre that C is indistinguishable from
B but distinguishable from A. Then we would say that the color of
B is *between* that of A and that of C. In other words, that every
thing that is indistinguishable from A be indistinguishable
f rom B (and vice versa) is a necessary condition for A an B being
exactly the same color.” (p. 79)
Let's take a close look at this; more on Aune. Better yet, let's get off this topic
and move on to the rest of Aune's book. Sorry for all the errors here.
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