The New Riddle of Induction: Neutral and Relative Perspectives on Color

Nel­son Good­man’s “New Rid­dle of In­duc­tion” has pre­vi­ously been dis­cussed on Less Wrong at http://​​less­wrong.com/​​lw/​​8fq/​​bayes and http://​​less­wrong.com/​​lw/​​mbr/​​grue . Briefly, this rid­dle shows that any at­tempt to make fu­ture pre­dic­tions by gen­er­al­iz­ing from past ob­ser­va­tions, may de­pend on ar­bi­trary as­pects of the rea­soner’s lan­guage. This is illus­trated in the con­text of the pro­posed time-de­pen­dent color pred­i­cates “grue” and “bleen”.

In this ar­ti­cle, I pro­pose that the re­s­olu­tion to this ap­par­ent para­dox lies in the recog­ni­tion that a neu­tral per­spec­tive ex­ists, and that while an agent can­not know with cer­tainty whether their per­spec­tive is neu­tral, they can as­sign sig­nifi­cantly higher cre­dence to their per­spec­tive be­ing neu­tral be­cause evolu­tion had no rea­son to in­tro­duce an ar­bi­trary time-de­pen­dent term in their color de­tec­tion al­gorithm. I am un­sure whether the ar­gu­ment is novel, but as far as I can tell, this par­tic­u­lar solu­tion is not dis­cussed in any of the pre­vi­ous liter­a­ture that I have con­sulted.

(Note: This ar­ti­cle was origi­nally writ­ten as course­work, and there­fore con­tains an ex­tended sum­mary of Good­man’s origi­nal pa­per and of pre­vi­ous dis­cus­sion in the aca­demic liter­a­ture. Read­ers who are fa­mil­iar with the pre­vi­ous liter­a­ture are en­couraged to skip to the sec­tion “The Grue Sleep­ing Beauty Prob­lem”)

The Prob­lem of In­duc­tion and Its Dissolution

In a se­ries of lec­tures pub­lished as the 1954 book “Fact, Fic­tion and Fore­cast”, Nel­son Good­man ar­gued that Hume’s tra­di­tional prob­lem of in­duc­tion has been “dis­solved”, and in­stead de­scribed a differ­ent prob­lem that con­founds our effort to in­fer gen­eral facts from limited ob­ser­va­tions. Be­fore I dis­cuss this new prob­lem—which Good­man termed “The New Rid­dle of In­duc­tion”—I briefly dis­cuss what Good­man means when he as­serts that the origi­nal prob­lem of in­duc­tion has been “dis­solved”.

In Good­man’s view, it is true that there can be no nec­es­sary re­la­tion­ship be­tween past and fu­ture ob­ser­va­tions, and it is there­fore fu­tile to ex­pect log­i­cal cer­tainty about any pre­dic­tion. In his view, this is all there is to the prob­lem of in­duc­tion: If what you want from an in­duc­tive pro­ce­dure is a log­i­cal guaran­tee about your pre­dic­tion, then the prob­lem of in­duc­tion illus­trates why you can­not have it, and it is there­fore fu­tile to spend philo­soph­i­cal en­ergy wor­ry­ing about knowl­edge or cer­tainty that we know we can never have.

Good­man there­fore ar­gues that the real prob­lem of in­duc­tion is rather how to dis­t­in­guish strong in­fer­ences from weak ones, in the sense that strong in­fer­ences are those which a rea­son­able per­son would ac­cept, even in the ab­sence of log­i­cal guaran­tees. In other words, he is in­ter­ested in de­scribing rules for a sys­tem of in­fer­ence, in or­der to for­mal­ize the in­tu­itions that de­ter­mine whether we con­sider any par­tic­u­lar pre­dic­tion to be a rea­son­able in­fer­ence from the ob­served data.

He mod­els his ap­proach to in­duc­tion on a nar­ra­tive about how the rules of de­duc­tion were (or con­tinue to be?) de­vel­oped. In his view, this oc­curs through a feed­back loop or “vir­tu­ous cy­cle” be­tween the in­fer­en­tial rules and their con­clu­sions, such that the rules are re­vised if they lead to in­fer­ences we are un­will­ing to ac­cept as log­i­cally valid, and such that our be­liefs about the val­idity of an in­fer­ence are re­vised if we are un­will­ing to re­vise the rules. Any at­tempt at de­duc­tive in­fer­ence can then be judged by its ad­her­ence to the rules; and if a situ­a­tion arises where our in­tu­ition about any par­tic­u­lar in­fer­en­tial val­idity con­flicts with the rules, we would have to ad­just one of the two. In Good­man’s view, a similar ap­proach should, and is, used to con­tin­u­ously im­prove on an set of rules that cap­ture hu­man be­liefs about what gen­er­al­iza­tions tend to pro­duce good pre­dic­tions, and what gen­er­al­iza­tions tend to fail.

How­ever, as we will see, even if we do not seek the kind of cer­tainty that is ruled out by the lack of log­i­cal guaran­tees and nec­es­sary con­nec­tions, and fol­low Good­man to fo­cus our at­ten­tion on dis­t­in­guish­ing “good” gen­er­al­iza­tions from bad ones, there are im­por­tant ob­sta­cles that arise from the fact that our in­fer­en­tial mechanisms are un­avoid­ably shaped by our lan­guage, in the sense that the pre­dic­tions de­pend on seem­ingly ar­bi­trary fea­tures of how the ba­sic pred­i­cates are con­structed.

The New Rid­dle of In­duc­tion: Good­man’s Argument

Good­man’s goal is to de­scribe the rules that de­ter­mine whether in­fer­ring s2 from s1 is in­duc­tively “valid”, in the sense that a rea­son­able per­son would con­sider the truth of s2 to be “con­firmed” by the truth of s1 even with­out the sup­port of an ar­gu­ment that es­tab­lishes s1-->s2 by de­duc­tive logic. In par­tic­u­lar, we are in­ter­ested in situ­a­tions where s1 and s2 are very similar state­ments, where each state­ment makes some claim about a prop­erty that ap­plies to some set of el­e­ments/​ob­jects, such that the prop­erty is equal be­tween s1 and s2, but such that s2 refers to a larger, more gen­eral set of ob­jects (i.e. where the set of el­e­ments referred to by s2 is de­ter­mined by re­lax­ing/​ex­pand­ing the set of el­e­ments that are referred to by s1). Good­man adopts Hem­pel’s view that in­duc­tion is de­scribed by a re­la­tion R over these state­ments, such that s1Rs2 holds if and only if s2 is “con­firmed” by s1. The goal is then to de­scribe the con­di­tions that R must meet in or­der for a rea­son­able per­son to con­clude that s1Rs2

In or­der to demon­strate that this is more difficult than it sounds, Good­man pre­sents the “New Rid­dle Of In­duc­tion”. In this thought ex­per­i­ment, he posits that we have ob­served sev­eral fine stones, and have noted that all the emer­alds we have seen so far have been green. Our goal is to de­ter­mine whether we would be jus­tified in be­liev­ing that all emer­alds are green, or at least in pre­dict­ing that the next emer­ald we see will be green. Let us sup­pose we have seen enough green emer­alds to be con­fi­dent that the next emer­ald will also be green; and we have checked that the pro­ce­dure we fol­lowed to reach this con­clu­sion meets all the rules of our in­duc­tive frame­work.

How­ever, we have a friend who speaks a differ­ent lan­guage, which does not have a word for “green”. In­stead, they have a word for “Grue”, which trans­lates as “green be­fore time t, blue af­ter time t”. This lan­guage also has a word “Bleen”, which trans­lates as “blue be­fore time t, green af­ter time t”.

Since our friend saw the same emer­alds as us, and since they were all ob­served be­fore time t, he has cor­rectly ob­served that all the emer­alds he has seen so far have been grue. More­over, since he is fol­low­ing the same in­duc­tive rules as us, he is pre­dict­ing that the next emer­ald, to be ob­served af­ter time t, will also be grue. Since we are fol­low­ing the same in­duc­tive rules, his pre­dic­tion ap­pears to be based on equally strong in­fer­ence as our pre­dic­tion. How­ever, at most one of us can be cor­rect. This raises an im­por­tant challenge to the very idea of rea­son­ing about whether an at­tempted in­fer­ence is valid: Any set of rules, even when ad­hered to strictly, can lead to differ­ent pre­dic­tions de­pend­ing on how the ba­sic pred­i­cates are defined. In fact, this ar­gu­ment can be gen­er­al­ized to show that for any pos­si­ble fu­ture pre­dic­tions, there ex­ists a “lan­guage” such that in­duc­tive rea­son­ing based on that lan­guage will re­sult in the de­sired pre­dic­tion.

Good­man goes on to ar­gue that the ba­sic prob­lem is one of dis­t­in­guish­ing “lawlike” state­ments from “ac­ci­den­tal” state­ments, where lawlike state­ments are those that re­late to classes of ob­jects such that ei­ther all mem­bers have the prop­erty in ques­tion, or such that none have the prop­erty. It can then be ex­pected that lawlike pred­i­cates are pro­jectable, in the sense that in­duc­tion based on lawlike pred­i­cates lead to valid ex­trap­o­la­tion. How­ever, this move does not re­ally re­solve the is­sue: Essen­tially, it just shifts the dis­cus­sion down one step, to whether state­ments are lawlike, which it is not pos­si­ble to know un­less one has in­for­ma­tion that would only be available af­ter suc­cess­ful in­duc­tive in­fer­ence.

Can The Rid­dle Be Ad­e­quately Solved?

An ad­e­quate solu­tion to Good­man’s rid­dle would re­quire a prin­ci­pled way to dis­t­in­guish “pro­jectable” pred­i­cates from non-pro­jectable ones—a way to “cut re­al­ity at the joints” and es­tab­lish nat­u­ral, lawlike pred­i­cates for use in in­duc­tive rea­son­ing.

Ru­dolf Car­nap sug­gests that one can dis­t­in­guish lawlike state­ments from ac­ci­den­tal state­ments, be­cause lawlike state­ments are “purely qual­i­ta­tive and gen­eral”, in the sense that they do not re­strict the spa­tial or tem­po­ral po­si­tion of the ob­jects that the pred­i­cates re­fer to. Good­man con­sid­ers and re­jects this ar­gu­ment, be­cause from the per­spec­tive of some­one who thinks in terms of pred­i­cates such as grue, grue is time-sta­ble and green is the time-de­pen­dent pred­i­cate. There­fore, tem­po­ral­ity it­self can only be defined rel­a­tive to a given per­spec­tive, and there seems to be no ob­vi­ous way to give pri­or­ity to one over the other.

One po­ten­tial line of ar­gu­ment in fa­vor of giv­ing pri­or­ity to green over grue has been sug­gested by sev­eral writ­ers, who make the ob­ser­va­tion that the time “t” is seem­ingly cho­sen ar­bi­trar­ily, and that there are in­finitely many ver­sions of “grue(t)”, one for each pos­si­ble time t, with no rea­son to pre­fer one over the other. This ar­gu­ment also suffers from the prob­lem that a “grueist” per­son can take the same ap­proach, and es­tab­lish that there are in­finitely many pos­si­ble ver­sions of “green(t)” at which time green tran­si­tions from be­ing grue to be­ing bleen.

In fact, this is a com­mon theme in most ar­gu­ments at­tempt­ing to re­solve the prob­lem: They can gen­er­ally be restated from the grueist per­spec­tive, and used to reach the op­po­site con­clu­sion. Good­man there­fore holds that it is im­pos­si­ble to dis­t­in­guish lawlike state­ments from ac­ci­den­tal state­ments on grounds of their gen­er­al­ity. The best we can do is ask whether the state­ments are based on pred­i­cates that are “en­trenched”, in the sense that have in the past proved suc­cess­ful at pro­duc­ing ac­cu­rate pre­dic­tions. Such a track record then pro­vides some sup­port for a ten­ta­tive be­lief that the pred­i­cate has caught onto some as­pect of re­al­ity that is gov­erned by lawlike phe­nom­ena.

The Grue Sleep­ing Beauty Problem

Swin­burne (1968) sees an asym­me­try be­tween the pred­i­cates green and grue, which arises from the fact that an in­di­vi­d­ual can judge whether an ob­ject has the prop­erty “green” even if they do not know what time it is, and ar­gues that this asym­me­try can be ex­ploited to give pri­or­ity to green over grue. I find this ar­gu­ment per­sua­sive but in­com­plete, and will there­fore dis­cuss it in de­tail, in a slightly al­tered form.

Con­sider the fol­low­ing thought ex­per­i­ment, which is not due to Swin­burne but which I be­lieve illus­trates his ar­gu­ment: An evil tur­tle has ab­ducted Grue Sleep­ing Beauty, the princess of the King­dom of Grue, in or­der to perform ex­per­i­ments on her. Speci­fi­cally, he in­tends to give her sleep­ing pill, and then flip a coin to ran­domly de­cide whether to wake her be­fore or af­ter time t. In the room, there will be a green emer­ald. Bowser plans to ask what color the emer­ald is, with­out in­form­ing her about what time it is.

One pos­si­bil­ity is that Grue Sleep­ing Beauty gets it right: She will ex­pe­rience the emer­ald as be­ing grue if she is wo­ken be­fore time t, and as bleen if she is wo­ken af­ter time t. How­ever, this seems un­likely: It re­quires that the un­speci­fied psy­chi­cal phe­nomenon that pro­duces color, in­ter­acts with her qualia in a time de­pen­dent man­ner even when she can­not know what time it is. The other op­tions are that she gets it wrong—which seems like a big hit against the idea of “grue”—or that she ex­pe­riences no qualia at all, which seems un­likely, since all our ex­pe­rience tells us that non-col­or­blind hu­mans can in gen­eral iden­tify the col­ors of ob­jects.

Good­man might ar­gue that this parable begs the ques­tion, by im­plic­itly as­sum­ing that the di­a­mond re­mains green from the ex­per­i­menter’s per­spec­tive. One imag­ines his re­sponse might be to re­verse the thought ex­per­i­ment, and in­stead tell a story about a grueist stu­dent of philos­o­phy who makes up an elab­o­rate tale about ab­duct­ing Green Sleep­ing Beauty. There­fore, this line of rea­son­ing is in­com­plete, for the same rea­son as most other at­tempted solu­tions to the New Rid­dle of In­duc­tion. How­ever, I be­lieve the two ver­sions of the parable have differ­ent im­pli­ca­tions, such that a rea­son­able per­son would as­sign much higher cre­dence to the im­pli­ca­tions of the first ver­sion. To ex­plain this, in the next sec­tion I provide a “patch” to Swin­burne’s argument

Time In­de­pen­dence: Neu­tral and Rel­a­tive Perspectives

Let us as­sume there is an un­der­ly­ing reg­u­lar, lawlike phe­nomenon (in this case the wave­length of light re­flected by emer­alds), and that agents im­ple­ment an al­gorithm which takes this phe­nomenon as in­put, and out­puts a sub­jec­tively ex­pe­rienced color. A clas­sifier al­gorithm is said to be time-in­de­pen­dent if, for any par­tic­u­lar wave­length, it out­puts the same sub­jec­tively per­ceived color, re­gard­less of time. From the per­spec­tive of an agent that im­ple­ments any par­tic­u­lar clas­sifier al­gorithm, other clas­sifier al­gorithms will ap­pear rel­a­tively time-de­pen­dent if refer­ence to time is needed to trans­late be­tween their out­puts.

I ar­gue that there ex­ists a clas­sifier al­gorithm that is time-in­de­pen­dent even from a neu­tral per­spec­tive: It is the one that sim­ply ig­nores in­put on time. There­fore, a color clas­sifier al­gorithm is time-in­de­pen­dent by de­fault, un­less it ac­tively takes time into ac­count. More­over, if two color clas­sifier al­gorithms re­sult in pred­i­cates that are time-de­pen­dent rel­a­tive to each other, then at least one of the al­gorithms must con­tain a term that takes time into ac­count, ei­ther di­rectly or through con­tex­tual clues. Of course, this may be sub­con­scious and hid­den from the meta-level cog­ni­tion of the agent im­ple­ment­ing it, and the agent there­fore has no di­rect way of know­ing whether he is im­ple­ment­ing an al­gorithm that is time-in­de­pen­dent from a neu­tral per­spec­tive.

Both ver­sions of the Grue/​Green Sleep­ing Beauty thought ex­per­i­ment im­plic­itly as­sume that the in­ves­ti­ga­tor is im­ple­ment­ing a clas­sifi­ca­tion al­gorithm that is time-in­de­pen­dent even from a neu­tral per­spec­tive. Since the neu­tral per­spec­tive is unique, at least one of them must be wrong. Find­ing out which one (if any) is right, is an em­piri­cal ques­tion, but unan­swer­able be­fore time t, so any at­tempt to grant the nec­es­sary em­piri­cal ob­ser­va­tions would amount to beg­ging the ques­tion.

But now con­sider an evolu­tion­ary en­vi­ron­ment where agents must choose whether to eat deli­cious blue berries, or poi­sonous green berries. There is sig­nifi­cant se­lec­tive pres­sure to find find an al­gorithm that out­puts the same sub­jec­tive qualia, for any given wave­length. There is no rea­son at all that na­ture should put a term for time in this al­gorithm: To do so, would be to add need­less com­plex­ity and add an ar­bi­trary term that makes refer­ence to the time t, which has no con­tex­tual mean­ing in the set­ting in which the al­gorithm is op­er­at­ing and in which it is be­ing op­ti­mized.

This points us to the cen­tral differ­ence be­tween the grue and green pred­i­cates: The fact that we, as real hu­mans shaped by evolu­tion, are im­ple­ment­ing a par­tic­u­lar clas­sifi­ca­tion al­gorithm is highly rele­vant ev­i­dence for it not con­tain­ing ar­bi­trary com­pli­ca­tions, rel­a­tive to a hy­po­thet­i­cal al­gorithm that, as far as we know, does not ex­ist in any agent found in na­ture. The ac­tual hu­man al­gorithm is there­fore more likely to ig­nore in­put on time, and more likely to be time-in­de­pen­dent from the neu­tral per­spec­tive.

Note that even in the ab­sence of a the­ory of op­tics, it is rea­son­able to ar­gue that sub­jec­tive ex­pe­rience of col­ors arises from some phys­i­cal reg­u­lar­ity in­ter­act­ing with our sen­sory sys­tem, and that this sen­sory sys­tem would be much sim­pler if it ig­nores time. Thus, the ar­gu­ment still holds even if the un­der­ly­ing phys­i­cal phe­nomenon is poorly un­der­stood.

Conclusions

Nel­son Good­man pre­sents a rid­dle which illus­trates how our pre­dic­tions for fu­ture ob­ser­va­tions are, to some ex­tent, func­tions of the seem­ingly ar­bi­trary cat­e­go­riza­tion scheme used in our men­tal rep­re­sen­ta­tion of re­al­ity. A com­mon theme be­tween many of the sug­gested solu­tions to this rid­dle, is that they at­tempt to find an asym­me­try be­tween lan­guages that use the pred­i­cate grue, and lan­guages that use the pred­i­cate green. Such an asym­me­try would have to be im­mune to pre­dic­tive changes that arise when rephras­ing the prob­lem in the other lan­guage; it seems likely that no such asym­me­try can be found on purely log­i­cal or se­man­tic grounds.

In­stead, I ar­gue that one can bring in ad­di­tional back­ground be­liefs in sup­port of the con­clu­sion that the refer­ence frame im­ple­mented by hu­mans is “neu­tral”. In par­tic­u­lar, the hu­man color clas­sifier al­gorithm was cho­sen by evolu­tion, which had no rea­son to in­clude a term for time. This li­censes me to give sig­nifi­cant higher cre­dence to the be­lief that my rep­re­sen­ta­tion scheme is neu­tral, and that hy­po­thet­i­cal other clas­sifi­ca­tion al­gorithms that re­sult in con­structs such as grue are time-de­pen­dent even from a neu­tral per­spec­tive. In some sense, this line of rea­son­ing could be in­ter­preted as an ex­ten­sion of Good­man’s origi­nal ar­gu­ment about en­trench­ment, but al­lows the en­trench­ment to have oc­curred in evolu­tion­ary his­tory.

De­spite the fact that a solu­tion seems pos­si­ble, the rid­dle is still im­por­tant, since an agent can only com­pen­sate for the un­cer­tainty in his pre­dic­tions that re­sults from the refer­ence frame of his pred­i­cates, if he has a clear un­der­stand­ing of the prob­lems high­lighted by Good­man.