Nelson Goodman introduced a problem with human intuition through induction in his book "Fact, Fiction and Forecast" published in 1965. The problem has to do with examining the color of an emerald at different times. First Goodman defines the word grue. The definition of grue has two parts. Objects that are assigned the property of grue are:
• Objects FIRST examined before January 1, 2003 that are green.
• Objects not examined before January 1, 2003 that are blue.

The time of January 1, 2003 is completely arbitrary. The date serves to distiguish between objects examined before a certain time and objects examined after that time.

This definition of grue can be difficult to understand initially. Assigning the property of grue to an object depends on the time it was FIRST seen and its color. A green object FIRST seen before the date is called grue. Likewise, a blue object FIRST seen after the date can be called grue as well. However, green objects FIRST seen after the date and blue objects FIRST seen before the date are not grue. Once an object is described as having the property of grue, the object can always be described as having the property of grue. For example, a green object seen before the date is given the property of grue. After the date, that object can still be described as being grue. This is because the object was first seen before the date.

With the definition of grue out of the way, Goodman continues the riddle with the examination of emeralds. One examines a set of emeralds before the date. The emeralds are seen to be green. As it is before the date that the emeralds are examined, they are also described as being grue. One will induce that all emeralds are green and grue.

After the date one picks up a new emerald, one that has not been previously examined. The expectation, based on induction, is that the emerald will be grue and green. For the emerald to be grue, it has to have the color of blue. Grue objects first examined after the date must be blue by definition. A blue emerald directly violates the other induction, that emeralds are green. This contradiction is Goodman's proof for showing that induction is totally useless.

Note: In response to Rose Thorn's write up: Goodman's New Riddle of Induction does not mention mathematical induction, in fact, it has nothing to do with mathematical induction. Goodman discusses philosophic induction, the generalization about the whole based on a part (See Webster's write up in Induction, specifically the part on philosophy). In other words, although we witness gravity everyday, there is no guarantee that gravity will be around forever.

• This is not a valid induction. Induction is defined on sequences. Sequences are typically interpreted in terms of set theory or functions (In fact, if functions and sequences are given a set-theoretic interpretation, sequences ARE functions int->alpha, where alpha is some type). Sets exist outside of time, so mathematical induction is not the right tool to deal with time-dependent entities. Set-theoretic arguments, and most arguments grounded in classical logic and classic mathematical logic are only useful for describing things that are eternal, or for describing an instant in time, unless we specifically take account of change in the world.

The above is also not a valid induction because it does not have a concept of 'successor': there is no order, not even partial order, defined over emeralds. As such, there is no "successor axiom" relating item n to item n+1.

I claim that the above example is in fact a deductive argument based on the following:

Axiom 1: The Grue Axiom
grue(A), for all A is true if:
• A was FIRST examined before January 1, 2003 AND is green. OR:
• A was not examined before January 1, 2003 AND is blue.
Anything that has ever been grue is always grue.
Axiom 2:Premise of Grue's time-independance
This is the axiom that grue(A) is a predicate that is time-independent.
Axiom 3:Premise of Universalisability of predicates over emeralds
This is the premise that all emeralds for are alike, so we may take a representative sample.
Clearly, the first and second axioms are incompatible. The final axiom is interesting, because it interacts with the definition of an emerald. If we require that all emeralds be grue, then nothing examined after 01/01/2003 is an emerald. (We may similarly require that emeralds be examined before 01/01/2003). If we do not require emeralds to be grue, we must resort to the third axiom to decide that all emeralds are grue.

This third axiom is what Mr. Frog would refer to as the inductive step. Clearly, it makes no sense in this context to apply it? Why? Because we consider emeralds to have a temporally-defined property (Time of examination), yet we also require all emeralds to be the same. Clearly, all emeralds in our sample will have the same time of examination, yet we do not conclude that all emeralds are examined at that time (Because, for there to be any problem with believing all emeralds grue, we admit emeralds not examined at this earlier time). If the logic is to be sound, we must refuse to believe in the existence of emeralds examined at another time. Using this argument to show that "induction" is worthless is like my using the fact that my car doesn't drive well on lakes to show that cars are useless.

See false mathematical proofs.

the rebuttal that doesn’t need a logic textbook

After extensive debate with Mr. Frog, I find the theorem absolutely useless.

To understand my first problem requires understanding the difference between states and visible colors. If an emerald is seen before the date and gives the visible characteristic of green light, Mr. Frog makes the conclusion that since both definitions enclose this occurrence that we then can induce that the emerald is both green AND grue. This is just not so. If the state of being green and the state of being grue both display the visible characteristic of being green, then it is impossible to know which one the emerald is, if it is either of them at all.

This does not change what the proof “proves”, only the method. Mr. Frog claims the true definition of grue is not based on an inner state, but objects become defined as grue automatically if they are seen before the date as displaying the color green.

In that case I define everyone I meet who has red hair to be named Jackass. If I meet someone who is really named Tony, I will tell them sorry, but I defined their name to be Jackass, so they have to be named Tony AND Jackass, because that is my definition. In reality the individuals name is Tony, but through an irrational definition I can now claim he is defined as a Jackass.

Now say there actually is an individual whose name is Jackass, and his hair actually is red. Put him in a room with another guy that has red hair, and then walk me in. I know that there does exist an individual or multiple individuals whose names are Jackass, and that they all have red hair. I look at these two red haired individuals, and since someone named Jackass has red hair, then these red heads are named Jackass.

This logic doesn’t work. If I am human, and Bill Gates is human, I can’t prove to anyone that I’m Bill Gates with just that information. Now if I happened to be a little squirrelly guy who was the head of the largest computer software manufacturer on earth and I was famous for unscrupulous business practices, then I might be able to convince someone I was Bill Gates. They would be convinced only if presented with information that they believed.

My final problem is Mr. Frogs conclusion. Apparently Goodman makes this conclusion too, and that disturbs me.

This contradiction is Goodman's proof for showing that induction is totally useless.

In the induction write-up by Webster 1913, it is defined partly as “The act or process of reasoning from a part to a whole, from particulars to generals.” Gleaning something large from something small.

You show up to my house knowing you’ll be meeting my two best friends for the first time. I tell you only about one of them, named Josh. He is 5’9”, wears a baseball cap, always a Steelers hooded sweatshirt, shorts, sandals and has his eyebrow pierced. He has glasses, and never smiles. If you show up and see someone who matches these characteristics, you don’t know that it’s Josh, but chances are it is. You might induce that since most people wouldn’t dress exactly alike on the same day, that the odds of two friends wearing matching outfits when they are hanging out are fairly low. This isn’t wrong, induction isn’t useless, but it’s not perfection.

But Goodman proves that induction is “useless”. I’m glad he did, because now I know that although it was helpful for me to wear shoes in the past that doesn’t mean it always will be. Who knows, shoes could stop being useful tomorrow, so not wearing shoes is just as viable an option as wearing them. Come to think of it, these clothes I have on can go. Just because people historically think nakedness is a taboo in America doesn’t mean it will be tomorrow! Because I know induction is useless then that means that the past has no bearing on the future.

As humans we have five senses, which constitute a limited set of data. If we cannot induce generalities from a limited set of data, then we are crippled. If induction is useless, than so are our senses. If our senses are useless, I’m going to go find a knife, because life then is pointless.

Induction is not perfect, but induction is not useless. Induction is by definition an imperfect tool, and Goodman’s proof says nothing of importance.

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