# [hist-analytic] The Two Color Problem:Putnam/Aune

Baynesr at comcast.net Baynesr at comcast.net
Fri Jan 1 12:14:58 EST 2010

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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

OF ANALYSIS

A number of philosophers have said that the problem of

the synthetic a priori may not be very important. Sometimes

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

together.

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

then let’s construct worlds, says Goodman. If there aren’t

objective standards, then let’s construct standards! Nothing

_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

examples.

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

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.

PUTNAM’S PROGRESS

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

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.

Putnam says,

“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.

Regards

STeve Bayne
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