July 18, August 21, 2003
The Formal Equivalence of Grue and Green and How It Undoes the New Riddle of Induction

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
jdnorton@pitt.edu

The hidden strength of Goodman's ingenious "new riddle of induction" lies in the perfect symmetry of grue/bleen and green/blue. The very same sentence forms used to define grue/bleen in terms of green/blue can be used to define green/blue in terms of grue/bleen by permutation of terms. Therein lies its undoing. In the artificially restricted case in which there are no additional facts that can break the symmetry, grue/bleen and green/blue are merely notational variants of the same facts; or, if they represent different facts, the differences are ineffable, and no account of induction should be expected to pick between them. This still obtains in the more interesting case in which we embed grue/bleen in a grue-ified total science; the grue-ified and regular total sciences are merely equivalent descriptions of the same facts. In the most realistic case, we allow additional facts that break the symmetry and then we can also evade Goodman's new riddle by  employing an account of induction rich enough to exploit these facts. Unaugmented enumerative induction is not such an account and it is the primary casualty of Goodman's new riddle.
1. Introduction
	Nelson Goodman's ingenious problem of grue—the new riddle of induction—is widely presumed to have revealed a significant limitation on inductive inference. The strength of Goodman's argument lies in the perfect symmetry of grue/bleen and green/blue. Each is defined in terms of the other by the same formulae. In Goodman's example, the evidence can be described equally as the observation of green emeralds or of grue emeralds. Thus, using the symmetry, we may assert a meta-claim that applies to all accounts of induction: any account of inductive inference that allows the evidence to support the hypothesis that all emeralds are green must also allow exactly equal support for the incompatible hypothesis that all emeralds are grue. 
	What I will seek to establish here is that this essential symmetry requirement is actually the undoing of Goodman's example. To do this, I will recall that, in other contexts, perfectly symmetrical accounts are routinely understood to be merely variant descriptions of the same facts. The most familiar example is the symmetric descriptions from different frames of references in physical theories that obey the principle of relativity. One could insist that there are factual differences separating the accounts, but that would be at the cost of accepting that the factual differences are inexpressible in the theory.
	Applying this construal of perfectly symmetrical accounts to grue/bleen and green/blue, we arrive at a very different view of them. As long as the descriptions fully respect the symmetry, then we have strong reasons for believing that they are merely notationally variant descriptions of the same physical facts; anyone insisting that the worlds described are factually different must resort to distinguishing facts that are literally inexpressible. That is, the supposed  difference is ineffable.
	The perfect symmetry of the descriptions can be sustained very artificially if we restrict ourselves to a highly contrived, impoverished emerald-world (to be described below), bereft of all but a very few physical facts. More plausibly, we might imagine that we preserve the symmetry by somehow grue-ifying our total science. In either case it will turn out that the propositions that all emeralds are green or that they are grue describe the same physical facts; or, if there is a factual difference, it is ineffable. So we should not expect an account of induction to distinguish between them. Indeed we should be suspicious of one that does. 
	The new riddle of induction depends essentially on the symmetry of descriptions. If we forgo it, then solutions of the riddle come quickly. Once there is an asymmetry, we can readily augment our accounts of induction to exploit the difference and to give results in accord with our intuitive expectations. This is how ordinary inductive practice actually circumvents Goodman's riddle; it presumes that there are facts that favor green over grue, so that the evidence of green emeralds supports just the hypothesis that all emeralds are green.
	What I hope to show is that we have nothing to fear if we drop the assumption that there are facts that break the symmetry of green and grue. If we drop it, we also eradicate the physical difference between green and grue and Goodman's riddle with it. The real import of Goodman's riddle lies in showing the failure of accounts of induction that cannot exploit these asymmetries when they are available. The most prominent casualty is a simple form of enumerative induction that places no restriction on the predicates over which induction may proceed.
	In the following, Goodman's new riddle of induction is sketched in Section 2 and the importance of the symmetry displayed. Section 3 contains a brief excursion into intertranslatable descriptions. I give special attention to symmetric descriptions and urge that they can only fail to represent the same physical systems, if we presume ineffable factual differences. In Section 4, I return to grue and look at two ways that the symmetry of its description can be retained, in the impoverished emerald-world and in a grue-ified total science; and I also consider the consequences of failing to retain the symmetry. A transformation essentially similar to the grue-ification of our total science will be seen to have been effected by the nocturnal expansion described in Section 3.  Conclusions are in Section 5.
2. Goodman's "New Riddle of Induction"
Grue and bleen
	Goodman (1983, ch.III, pp.74, 79) introduced two new predicates, grue and bleen, defined as
Grue applies to all thing examined before [some future time]t 
just in case they are green
but to other things just in case they are blue.
Bleen applies to all things examined before t 
just in case they are blue 
but to other things just in case they are green.
The new predicates immediately present fatal problems for any account of induction that licenses the inference:
Enumerative induction: Some As are B confirms all As are B. 
where we may freely substitute any individual term for A and any predicate for B. The reason is that the observation of green emeralds prior to t can equally be described as the observation of grue emeralds prior to t. Thus, by enumerative induction, the same observations equally confirm that all emeralds are green and that all emeralds are grue. That is, the same observations equally confirm hypotheses one of which entails that emeralds examined after t will be green and another that entails that they will be blue. Our past history of observation of green emeralds ends up assuring us that all emeralds we observe in the future will be blue. An account of induction that allows such an assurance is surely defective.
The importance of the formal equivalence of grue and green
	The natural response is to dismiss predicates like "grue" as somehow bogus because of the explicit mention of time t in their definitions. This amounts to a plausible restriction on the scheme of enumerative induction to predicates B that are not grue-ified. Goodman's ingenious rejoinder is to notice that there is a perfect formal symmetry between green/blue and grue/bleen, so that if we take grue/bleen as our primitives, green and blue are now the grue-ified predicates:
Green applies to all thing examined before [some future time]t 
just in case they are grue
but to other things just in case they are bleen.
Blue applies to all things examined before t 
just in case they are bleen 
but to other things just in case they are grue.
What makes Goodman's rejoinder apparently impregnable is the perfect symmetry of the two sets of definitions. They use the same sentences up to a permutation of terms. The original definitions of grue/bleen becomes the definitions of green/blue simply by applying the transformation
green  grue      blue  bleen      grue green      bleen  blue
The same transformation (or equivalently, its inverse) converts the definitions of green/blue into the definitions of grue/bleen.
	The symmetry allows a general argument that there is no property of grue that allows us to deprecate it in comparison to green. For any formal property of green, there will be a corresponding property of grue; and conversely. For example, our natural intuition is that a green emerald before t is the same qualitatively as a green emerald after t. But a grue emerald before t is not the same qualitatively as a grue emerald after t, since the former is really green and the latter really blue. We might like some second order property "same" that can express this difference. However the perfect symmetry of the definitions defeats this. For any predicate "same" that can be defined in a system which takes green and blue as primitives, there will be a grue-ified analog in a system that takes grue and bleen as primitives. Any virtue that the former attributes to the green/blue system will be attributed by the latter to the grue/bleen system.
2. Symmetric Descriptions and Physically Equivalence
	The perfect symmetry of green/blue and grue/bleen is reminiscent of equivalences that arise in other areas of philosophy of science. In physics it we find many cases of apparently distinct physical systems whose descriptions are intertranslatable; that is, there is a transformation of the terms in the first description that turns it into the second and conversely. As a result, the two are often judged to be just the same system, described in two different ways. These cases were first made prominent through the principle of relativity. They are now often discussed under the label of gauge freedoms and the mathematical transformation that takes us between the different descriptions of the same system is called a gauge transformation.
	One of the simplest cases of intertranslatable descriptions arises in Newtonian gravitation theory. The gravitational field is described by a potential , which generates certain motions for planets, comets and falling stones. We get an alternative description of the same field by a gauge transformation that just adds a constant amount K to yield a new field '=+K. The new field ' is mathematically distinct from the original field  since they assign different values to the same point in space. The differences between the two mathematical fields corresponds to nothing physical. For example, the fields support identical motions, since the motions are governed solely by differences of potential, and, while ' and  disagree in absolute values, they agree in all differences of values. The two fields  and ' are physically equivalent.
Intertranslatability and physical equivalence: Four claims
	I will state and briefly defend some of the principal claims concerning intertranslatable descriptions, in preparation for their application to grue/green.
	As long as two descriptions are intertranslatable but formally or mathematically distinct, we cannot rule out the possibility that the formal or mathematical differences in some way represent something factual so that the descriptions are not physically equivalent. We can readily generate examples of intertranslatable descriptions that pertain to distinct systems by finding cases of distinct systems with similar properties. Both Newtonian gravitational fields and electrostatic fields are governed by an inverse square law, for example. Thus the descriptions of many gravitational fields are intertranslatable with those of electrostatic fields; yet the fields are physically distinct. Hence:
A. Intertranslatable descriptions are not assuredly of the very same facts; that is, they are not assuredly physically equivalent.
	When no other considerations enter, whether two intertranslatable descriptions are physically equivalent depends just on our stipulation over what their terms mean. They mean what we choose them to mean. Therefore, because of the intertranslatability, we can choose them to represent the same facts, in which case they are physically equivalent; or we can chose them to represent intertranslatable but distinct facts, in which case they are not physically equivalent.
	In practice other considerations almost invariably enter and these realistic cases are the ones I will consider henceforth. In them, the meaning of terms in intertranslatable descriptions is already set by the role these terms play in a larger theory. Thus whether two intertranslatable descriptions represent the same facts is no longer open to stipulation and is already decided by the physical content of the broader theory. That the differences between two intertranslatable descriptions corresponds to no factual differences can be a physical result of some importance in the broader theory. For example, that the two Newtonian gravitational fields  and '=+K represent the very same facts depends on the factual assumption that all that the absolute value of the field corresponds to no physical property of the field; these properties are fully captured in the differences of values of the potentials. Hence:
B. That particular intertranslatable descriptions describe the very same facts, that is, are physically equivalent, must be established by argumentation that proceed from factual assumptions.
This need for factually based argumentation to establish the physical equivalence of intertranslatable descriptions is one of the most important outcomes of the recent discussion of the "hole argument" in philosophy of space and time. (See Earman and Norton, 1987; Norton, 1999.) 
	The facts that decide whether two intertranslatable descriptions are physically equivalent can enter in two ways, internally or externally. I shall say they enter internally if those facts are expressed in the descriptions themselves. For example, the gravitational potential g of a unit mass and the electrostatic potential e of a unit positive charge both vary inversely with distance r from the mass or charge. The two formulae are intertranslatable with g=–1/r translating to e=+1/r merely by a transformation that multiplies by –1: e=–1xg. (I set the constants to one by the selection of units.) These intertranslatable descriptions do not describe the same physical systems. The facts that distinguish gravity from electrostatics are at least partly expressed in the formulae themselves. The minus sign of g=–1/r expresses the fact that gravitational forces are attractive, whereas the positive sign in e=+1/r expresses the fact that electrostatic forces, between like charges, are repulsive.
	I shall say that the distinguishing facts enter externally if they are not expressed within the descriptions themselves. A simple example is afforded by a three dimensional Euclidean space. Consider two parallel, flat two dimensional surfaces in the space. Their descriptions are identical since their geometries are the same. So they could represent the same two dimensional surface. That they do not is not expressed as a fact within the description of the geometry of the two dimensional surface. Rather it is expressed by facts outside the description of the geometry of the surfaces; that is, by facts that specify how the surfaces are embedded in the larger three dimensional space.
	In this last case, the facts supporting the failure of physical equivalence of the two descriptions had to be external for the simple reason that the geometries of the two surfaces are intrinsically identical, agreeing in all geometrical facts within the surfaces. As a result, their descriptions contain exactly the same sentences and formulae. They are examples of symmetrical descriptions, which I define as pairs of descriptions that employ exactly the same sentences and formulae. Since they are exactly the same, there is usually some sort of indexing of terms or quantities so we can keep track of which description is which. For example, symmetrical descriptions of the geometry of a three dimensional Euclidean space are supplied by describing the one geometry in two Cartesian coordinate systems, say (x,y,z) and another (x',y',z'), produced from the first by a rotation. Each description uses the same formulae or sentences with the coordinates entering them in exactly the same way. We keep track of which sentence or formulae belongs to which coordinate system by the absence or presence of the primes. That indexing leads to the common labeling of the descriptions as "notational variants" of one another. In the strongest cases, the very same sentences can be used not just for the descriptions, but also for the transformations that we use to translate between them. Hence:
C. For symmetrical descriptions, a failure of physical equivalence must be grounded in external facts, that is, in facts not expressed within the descriptions themselves.
This claim derives directly from the symmetry of the descriptions. For every sentence or formulae in one, there is a corresponding sentence or formula in the other; and conversely. Thus if we are to be assured that the corresponding sentences or formulae do not represent the same facts, we must call upon further facts that are not expressed within the symmetrical descriptions. As the scope of the symmetric descriptions becomes larger, there are fewer facts that can be called upon. The limiting cases are:
Da. Total science: If symmetrical descriptions cover all of science, then facts that establish the failure of physical equivalence must lie outside science.
Db. Total knowledge: If symmetrical descriptions cover all of our formally expressed knowledge, then facts that establish the failure of physical equivalence must lie outside our expressed knowledge, that is, in ineffable facts.
Neither Da nor Db necessitates the physical equivalence of the symmetrical descriptions. Both, however, make the presumption of physical equivalence very strong. If, for example, we have symmetrical descriptions of our total science, we might decide that only one properly describes the total science and that the other somehow fails. But a warrant for that decision must come from facts that lie outside of science; or perhaps from an extrascientific decision to give non-standard meanings to terms in one of the descriptions.
Illustration: the principle of relativity
	The best known example of symmetrical descriptions are afforded by relativity principles, such as the principle of relativity of inertial motion of special relativity. In that case, the description of the totality of all admissible physical processes will be identical no matter which inertial frame of reference is used for description. For example, limiting ourselves to one dimension of space, the totality of possible light signals will be described in one inertial frame (x,t) by the set of world lines {x=ct+K, x=–ct+M: for all real K, M}, where x and t are the space and time coordinates and c the speed light. In another frame, the totality of possible light signals will be described by {x'=ct'+K, x'=–ct'+M: for all real K, M}, an identical description, but where primes have been added to the x and t coordinates to remind us that they relate to a different inertial frame.
	One might be inclined nonetheless to insist that one of the frames of reference can be designated as factually preferred—the true rest frame. However all such efforts face formidable obstacles. It is not just that no observable fact distinguishes between the frames. It is worse. No fact expressible in the theory, observable or otherwise, allows any distinction between the frames. For example, we might note that some frame F is distinguished by its admitting a very massive sun at rest with planets orbiting around it. Might this be the distinguishing mark of the preferred frame? No. The symmetry of descriptions derived from the principle of relativity assures us that precisely the same process will be licit for a very massive sun at rest in another inertial frame G. The same fate awaits any factual result that we may seek to use to single out frame F over frame G.
	In short, when one has a perfect symmetry among the descriptions, one can insist that they describe physically distinct systems; however, as Result C asserts, it is at the cost of insisting that there are physical differences that literally cannot be expressed in the descriptions themselves.
The nocturnal expansion…
	It will be convenient to consider one example, already much discussed in the philosophy of space and time literature. (e.g. Grünbaum, 1973, pp. 42-43) What if everything in the universe doubled in size overnight? Would we know? If nothing else changed, we certainly might be able to figure it out. If everything doubled in size overnight, then light would take twice as long to traverse the distance from the sun to here. These and a host of other observable effects would tip us off to the expansion. But what if, at the same time, there were other compensating changes? What if temporal processes slowed to a half so that our clocks take twice as long to complete a cycle and we no longer notice that light is taking twice the time in its journey? It turns out that there is a simple recipe for altering quantities so that all the non-gravitational laws of physics remain unaffected. These laws are built on three basic units: mass M, length L and time T. The rule is simply to double all lengths and times and to halve all masses:
LL'=2L     TT'=2T     M  M'=M/2
This transformation will leave all the non-gravitational laws unchanged. Most important, the two fundamental constants, the speed of light c and Planck's constant h will remain numerically unchanged. One quickly sees that no non-gravitational law would allow us to discern whether this doubling had happened last night or not. For example, quantum theory tells us that the energy E of a photon is related to its frequency  by the law E = h. If the expansion happened last night, the photon's frequency  with units T-1 would be halved. We would not notice the halving directly since all our clocks would be correspondingly slowed. And we would not notice it through the law E=h, which entails that the photon's energy E would be halved. Since energy has units ML2T-2, all energies would be halved and we would not discern the halved energy of the photon against the background of halving of all energies.
…and its symmetry
	The descriptions of the non-gravitational world with and without nocturnal expansion are symmetrical in the sense given above. That is, the totality of admissible physical processes will be described by the same regularities in each case, down to the detail of the numerical constants. We might try to find a way of extending the transformation so that it includes gravitational processes in the symmetrical descriptions, but I am not sure how that could be done. The simplest expedient is this. For the purposes of this discussion and the remainder of the paper, let us set aside gravitational processes, imagining that there is no such thing. That small fiction gives us a simple illustration of what it is like for the expansion symmetry to embrace all physically possible processes.
	In this case, one might want to insist that there is a fact of the matter as to whether the type of nocturnal expansion described actually happened last night. But we also know that no factual statement in the expanded world will reveal the expansion; every such statement has an exact, numerically identical counterpart in the unexpanded world. For this reason I regard the two descriptions as merely different descriptions of the same thing. Alternatively, you could insist that there truly is a factual difference between the expanded and unexpanded systems since that is a logical possibility. But you must accept that the difference resides in facts that outstrip the expressive power of either system, a most dubious result for anyone who thinks that a total science exhausts physical facts.
	There is another way to see that the two descriptions are almost irresistibly of the same physical system. In both unexpanded and expanded systems, corresponding processes will be described by numerically identical laws. One way to account for that is to say that, in the expanded world, all spaces and times have doubled and we measure all processes with correspondingly doubled units. That treats the doubling as a real effect. Another way to account for it is just to say that we have left the world unchanged, but have merely redescribed it in units of space and time that are halved. Using those small units we mistakenly judge the regular units to be doubled and thus we mistake unexpanded processes  for expanded processes. That is, the entire apparatus of the nocturnal expansion can be simulated by merely rescaling units without any physical change in the world.
3. The Symmetry of Grue/Bleen and Green/Blue
In the restricted emerald-world
	If we return to grue/bleen and green/blue, we immediately see that we have just the sort of symmetrical description developed above. As long as we restrict ourselves to the symmetrical descriptions, we have good reason to think of grue and green as mere notational variants of one another, so that no account of induction should pick between them. Of course restricting ourselves to the symmetric descriptions is a very strong, artificial constraint and quite hard to visualize. One way to get a sense of what it amounts to is to imagine an extremely impoverished, fictional world: emerald-world. All it contains is a set of individuals—let's call them "emeralds" just to give them a name—whose sole function is to bear one of two mutually exclusive properties. They will correspond to the properties of being green or grue, etc., but let us just call them "property one" and "property two" for now, to remind ourselves that the ordinary connotations of green and blue cannot be presumed here. The sole activity in the world is the time indexed affirmation that this or that emerald bears either property one or property two. Each emerald may bear the affirmation at most once. To secure this, imagine, if you like, that the affirmation destroys the emerald, akin to the destructive testing of materials. So we might affirm at time tearly<t that one emerald has property one; and affirm at time tlate>t that another emerald also has property one.
	Now comes the odd part: the emerald-world supplies us with no notion of whether the first property one is the same as the second property one. The fact of sameness or its failure just is not something that obtains in this world. A sequence of affirmations at times 1, 2, 3, 4, 
"property one, property one, property one, property one"
could be stated with equal justice to the facts as
"property one, property one, property two, property two."
If we decide to call property one "green" and property two "blue," that amounts to saying that the affirmations
"green, green, green, green"
are factually the same as
"green, green, blue, blue"
If the time t of the definition of grue arises between times 2 and 3, this amounts to saying that the affirmations are factually the same as
"grue, grue, grue, grue"
So, it is just as true to say that "All emeralds are green." as "All emeralds are grue." exactly because neither says very much. With each affirmation, we conventionally decide whether to call the outcome "green" or "blue" (if we use one notational scheme) or "grue" or "bleen" (if we use the other). So a history of affirmations can be equally described as "Some emeralds were affirmed green." and "Some emeralds were affirmed grue." And each can confirm "All emeralds are green." and "All emeralds are grue." respectively without there being any problem, since the two universal propositions are not contradictory.
	This failure of a factual sameness to obtain is essential if the symmetry of grue/bleen and green/blue descriptions is to be maintained. With a notion of sameness added, whether property one measured at tearly is the same as property one measured at tlate becomes a factual matter. Imagine it holds and let us call property one "green" and the exclusive property two "blue." So the emerald measured at tearly is green; the emerald measured at tlate is green; and these are factually the same (and thus not blue). Since tearly and tlate straddle the time t invoked in the definition of grue, it is now factually false that the emerald of tearly and the emerald of tlate are both grue. That is, the addition of the notion of sameness has violated the symmetry of the grue/bleen and green/blue descriptions. More generally, with a notion of sameness added to the world, at most one of "All emeralds are green." and "All emeralds are grue." can be true, a violation of the symmetry.
	One might be tempted to add the notion of sameness to the world exactly because it engenders a factual difference between green and grue and thus seems to restore the new riddle of induction. The outcome, however, would be to defeat the riddle, in so far as it does anything other than to raise a problem for an unqualified enumerative induction. For with sameness added, one quite naturally augments the scheme of enumerative induction to read:
Augmented Enumerative induction: Some As are B confirms all As are B,
where all the Bs must be the same.
This augmented scheme would allow observed green emeralds to confirm all emeralds are green; but it would preclude observed grue emeralds confirming all emeralds are grue since grue violates the requirement of factual sameness.
	In short, if we are to retain the essential symmetry of grue/bleen and green/blue we must restrict ourselves to such an impoverished physical world that there is no physical difference between "All emeralds are green." and "All emeralds are grue." Or, if there is a difference, it transcends the facts prevailing in the world. We should not expect any account of inductive inference to pick between them.
The natural escape
	The literature on grue is enormous. Stalker (1994) includes a 177 page annotated bibliography, so I cannot hope to canvas the many resolutions offered. The discussion of symmetry, however, suggests that the natural escape is simply to go outside the domain in which grue/bleen and green/blue afford symmetrical descriptions and use an account of induction rich enough to exploit the resulting asymmetry. That is the sort of escape I favor. Quine (1970), for example, notes that green but not grue is a natural kind term in our broader science, so that we can rescue induction by restricting schemes like enumerative induction to natural kind terms. Alternatively, in a Bayesian analysis, our prejudice against grue-ified predicates could be encoded in our assigning "All emeralds are grue." a much lower (or even zero) prior probability, compared with "All emeralds are green." Conditioning on the observation of a green=grue emerald prior to t, would boost the probability of both, but the probability of "All emeralds are grue." would remain small (or zero). More generally, we escape the grue like problems by tracing back the warrant of an inductive inference to facts rather than inductive inference schemes; this is the basic idea of what I call "material theories of induction" in Norton (forthcoming). In so far as these facts can draw upon the asymmetries, they can protect induction from grue-like problems.
The total science objection fails
	There is a powerful objection that can be raised to the natural escape just sketched. The escape presumes that we can expand our scope into a larger context in which symmetry breaking facts prevail. But what if we provide an alternative, grue-ified description covering that larger scope as well? Then the symmetry breaking facts would now lose their power since there would be corresponding grue-ified facts in the alternative description. That still does not preclude the possibility of further symmetry breaking facts outside that scope. They can be precluded once and for all by extending the scope of the grue-ified descriptions to cover our total science. We would now have two accounts of our total science, the regular and the grue-ified. No symmetry breaking scientific facts would be available and there is no larger context left in which to seek them. Any candidate symmetry breaking fact in one description (e.g., that green is a natural kind term) would be matched by a corresponding symmetry breaking fact in the other (e.g., that grue is a natural kind term). The escape is blocked.
	Claim Da of the discussion above already suggests the answer to the objection. If we grue-ify our total science in this way, then we have simply generated a symmetric description of the very same facts; or at least we have a symmetric description that can only fail to describe the same facts if we presume that there are further distinguishing facts that somehow elude expression in our total science—not a position anyone should be eager to defend. So the total science objection fails exactly because it presumes eradication of any difference between the original and grue-ified descriptions.
	Claim Db indicates that a similar failure awaits efforts to salvage grue by extending the grue-ified descriptions not just to embrace all science but also to cover all formally expressed knowledge of any kind.
An illustration
	The principle of this answer is clear. However it would be nice to see some of the details of the grue-ified total science and how it ends up describing the same facts as the original science. I have not been able to see a simple way of implementing universally the original transformation of green and grue. However it turns out that we have already seen an essentially similar transformation in the example above of nocturnal expansion. The effect of the expansion is to double the time of all processes. That means that the expansion will halve the frequency of all light waves. This is not a simple exchange of green and blue at time t  as in Goodman's example. Instead, at time t, all colors are shifted away from the violet towards the red end of the spectrum. Green light of frequency 550 THz would be shifted to infrared light of 275 THz; and ultraviolet light of 1100 THz would be shifted to green light of 550 THz. And so on.

…	1100THz
ultraviolet	550 THz
green	275 THz
infrared	
				
1100 THz
ultraviolet	550 THz
green	275 THz
infrared	…	

	Let us consider the fate of an emerald. Prior to t, it reflects green light of frequency 550 THz. In the ordinary description, after t, it still reflects green light of frequency 550 THz. In the description corresponding to the expanded world, however, after t, it now reflects light of 275 THz. Emeralds observed before t are green; emeralds observed after t are infrared. Let us call this compound property "green#", the expanded world's analog of grue.
	Prior to t, we observe emeralds to be green, or, equivalently, green#. Does this confirm that all emeralds are green; or that all emeralds are green#? The answer is both and that this engenders no problem. To begin, nothing observable depends on whether we decide that all emeralds are green or green#. If the first is true, emeralds observed after t will reflect green light of frequency 550 THz. If the second is true, we now shift to what is putatively a new world that expands at t. In it, after t, emeralds will reflect infrared light of frequency 275 THz. But no experiment in this expanded world will reveal the change. All temporal processes will be slowed correspondingly. Indeed any device capable of measuring the color of light—prisms, diffraction gratings, standard color swatches for comparison—will all be altered so as to make the shift undetectable. Since no fact expressible in the description of the expanded world, whether pertaining to observables or not, can distinguish it from the unexpanded world, I have urged above that we simply regard it as a variant description of the same world.
	When we grue-ify or, analogously, nocturnally expand, our total science, the resulting perfect symmetry of the descriptions defeats the problem of grue. That symmetry gives us no grounds to presume a factual difference between the assertions that all emeralds are green or grue (or, in the expanded worlds, green or green#). So each can properly be confirmed by our history of observation of green emeralds.
Can all emeralds really be both green and grue?
	I now want to describe and dissolve a mistaken objection to the presumed physical equivalence of symmetric descriptions. Let us assume, the objection runs, that we are in a world in which all emeralds really are green. Hence it is false that all emeralds not observed before t are blue. Hence it is false that all emeralds are grue. That is, the two sentences "all emeralds are green" and "all emeralds are grue" cannot both be true and thus cannot be physically equivalent descriptions.
	What the objection overlooks is that the two sentences pertain to different members of a pair of total descriptions. The total descriptions are symmetric and presumed to cover a scope sufficiently broad to preclude the obtaining of symmetry breaking facts outside that scope. In the normal total description, it will be true that "all emeralds are green" and, for the reasons the objection states, false that "all emeralds are grue." Matters will be exactly reversed if we shift to the grue-ified total description. There "all emeralds are grue" will be true and "all emeralds are green" will be false. Thus both "all emeralds are green" and "all emeralds are grue" can both be true as long as they are referred to the appropriate total description. Following the earlier analysis, they will turn out to be describing exactly the same physical properties of emeralds.
5. Conclusion
	As long as we restrict ourselves to worlds, impoverished or otherwise, in which grue/bleen and green/blue provide  fully symmetric descriptions, then grue raises no problems for any account of induction. The descriptions are strong candidates for descriptions of the same physical world; any physical difference between them must be inexpressible in the descriptive apparatus presumed. No account of induction ought to be expected to distinguish between such descriptions. Their failure to do so is their success at not distinguishing what should not be distinguished.
	If we can embed our account of emeralds and their colors in a broader context in which the symmetry fails, then an account of induction ought to distinguish "All emeralds are green." and "All emeralds are grue." The unaugmented enumerative induction described in Section 2 fails to do this since it does not avail itself of the symmetry breaking facts. For this reason, it and other accounts like it fail as accounts of induction. This failure is the full extent of the real problem grue brings. Grue does not show, as is widely supposed, that there is a deeper problem with the projection of regularities by any inductive means whatsoever. Other inductive means are free to exploit the symmetry breaking facts and I have pointed briefly to some of these in Section 3 above. Of course there may still be a problem in projecting regularities, but, if there is, analyses extending beyond grue will be needed to demonstrate it.
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