Is this an example of extrapolation?

I saw a joke on facebook today where professor cat states:

There are two types of people in this world: Those who can extrapolate from incomplete data

My brother thinks that this really isn't extrapolation. M-W defines extrapolate as:

to infer (values of a variable in an unobserved interval) from values in an already observed interval.

The joke seems to fit this definition, but I'm not convinced one way or the other. What do you think? Is there a better word for this kind of reasoning?

The straightforward answer to your question is that the joke is an example of extrapolation.

A discussion of the mathematics of extrapolation misses the point. The word extrapolate is not limited in meaning to a mathematical sense. The word has a more general primary sense¹ which is the sense of the word used in the joke:

to say what is likely to happen or be true by using information that you already have (Macmillan Dictionary)

When you work from information that you already have (the text of the joke) to discover what is likely to be (the implied punchline: “and those who can’t”), you are extrapolating.

The general sense of the word is about as long-lived as the mathematical one. Online Etymology Dictionary says examples of the word extrapolation date back only to 1867. The original meaning of extrapolation was mathematical, but the “[t]ransferred sense of ‘drawing a conclusion about the future based on present tendencies’” was already in use by 1889. The verb (extrapolate) began to be seen in 1874.

(I recommend Macmillan and Oxford over Merriam-Webster.)

Notes

1. To check this fact, I consulted all online dictionaries on the first page of the Google search results for [ define extrapolate ]. Except specialized dictionaries, most report the general sense of extrapolate as the primary sense. All general-purpose dictionaries but Merriam-Webster qualify the mathematical sense as only applicable to mathematical, scientific, or statistical contexts:
   – American Heritage Dictionary of the English Language, Fourth Edition
   – Bing Dictionary
   – Collins English Dictionary – Complete and Unabridged – technical sense first
   – Dictionary.com
   – Google Dictionary
   – Macmillan Dictionary
   – Merriam-Webster – technical sense first
   – Oxford Dictionaries
   – Random House Word Menu
   – V2 Vocabulary Building Dictionary
Specialized dictionaries (technical sense only):
   – American Heritage Science Dictionary (science dictionary)
   – Business Dictionary by WebFinance (business dictionary)
   – Mosby’s Dental Dictionary (medical dictionary)
   – Oxford Dictionary of Biochemistry (science dictionary)
   – Saunders Veterinary Dictionary (medical dictionary)

I'd be hesitant to claim myself to be a notable expert but, yes, that does infer extrapolation.

A more common example of extrapolation would be if a company's sales figures were:

• Year 1 - £10,000
• Year 2 - £15,000
• Year 3 - £20,000

Extrapolating that set of data, one would suggest that Year 4 would yield £25,000 of sales and Year 5 would see £30,000.

In your example, there are two items in the data set; two types of people.

Having been given the first type, those who can extrapolate (which is the observed interval), one will assume that the second type (the unobserved interval) is those who can't extrapolate.

Here, we know that there are two items of data and logic infers that the second will be the antagonist of the first.

It's essentially extrapolation on the smallest of intervals. It works because the range is defined as 2. If the range was defined as larger, as in "there are twenty types of people", one couldn't extrapolate the data.

Speaking as someone who has used statistical analysis a bit, I agree with Ste's example of extrapolation, but I think that the reasoning involved in the joke is not quite as described. I also think that the definition is incomplete in that it misses¹ a crucial component of what makes something an extrapolation: the function used.

If you're not quite familiar with the function concept, the pictures in the linked article are I think a good start. You can think of a function as a way of associating some input to some output. For example, if you turn on the burner on your stove, the amount of heat that it gives off is a function of the amount that you turn the control handle. If you drop a rock in a glass of water, the water level will rise. The distance that it rises is a function of the size (volume) of the rock. When you fill out a form, the box that you check for sex is a function of you. Different people will check different boxes, but a person has (normally) only one sex. These examples all use what we'd recognize as a rule, but you can also have arbitrary functions, where the value that comes out is not related in an obvious way to the value that went in; you might say the assignment is random (but I won't pursue the precise meaning of this).

In Ste's example, sales are a function of time (in yearly intervals).

Here's how I think extrapolation works at its simplest. You have two sets, an input set and an output set. In the terminology of the OP's definition, the observed interval is the input set, the variable ranges over the input set, and the value of the variable is the output set. The thing that relates a variable to its value is a function. An extrapolation assumes a definition for the function and then just plugs values into it. In Ste's example, the input set is the years {1,2,3}, and the output set is the sales {10000,15000,20000}. The function is:

5000*y + 5000 = 5000(y+1)

where y is the year. Thus, when you extrapolate and plug in a value to the function that is not in the input set, e.g., 4, you get:

5000*4 + 5000 = 5000(4+1) = 25000

Assuming that the input set and output set are defined by some particular function is the core power and risk of extrapolating. Ste and I both appear to have thought of the same function, but we might have picked different ones. Two functions might agree on the outputs for some inputs but disagree on the outputs for other inputs. This is why choosing the right function is tricky and important.

To see the importance of the function itself, consider whether making a prediction counts as extrapolating if you merely flip a coin to get the answer. I would say no, that it rather counts as something like guessing or leaving things to chance. Interestingly, mathematics also recognizes this possibly vague notion of what counts as a legitimate prediction function. According to the modern definition of a function, even something that arbitrarily assigns output to input counts. But this was not historically the case, and the difference between the old and new concepts is not trivial. Much mathematical research centers around the difference between arbitrary functions and those that are definable or concrete in some technical sense. This distinction probably captures what matters in the everyday case: rule-based vs. arbitrary.

So what is the function for the joke? How do you know how to complete the sentence? The joke depends on your being familiar with the cliche or snowclone:

There are two types of people in the world: those who __ and those who don't/can't.

There are some other variants too, which change the number of people or such. I like this one:

There are 10 types of people in the world: those who understand binary and those who don't.

Another variant would produce something like this:

There are two types of people in the world: those who don't understand quantum mechanics and those who only think they understand quantum mechanics.

(Um, this is funny (to me; I just made it up) because it implies that no one in the world does understand quantum mechanics, which was a favorite claim of one of its greatest teachers and practitioners, Richard Feynman.)

The OP's joke works by your knowing the pattern that these sayings follow; the pattern is the function. When you apply the function to the beginning of this sentence (the input):

There are two types of people in the world: those who can extrapolate from incomplete data

you have to extrapolate, using the function, to get the output:

and those who can't.

Or possibly, you could consider the whole sentence as the output. This is immaterial.

This is a little silly as an extrapolation since the output is only marginally a function of the input. You just have to know whether you should use don't or can't or whatever will make the sentence work grammatically. It is almost a constant function, which a rather trivial function. But I suppose it counts as extrapolation. The suggestion that it's induction is also interesting. I'm not sure what the difference between induction (as opposed to deduction) and extrapolation is off the top of my head. Maybe this would be a good other question.

Also, note that incomplete is really a pun here because the input datum is an incomplete sentence. :^)

Incidentally, there is another function related to the joke. It takes a person and assigns to them the value can extrapolate or can't extrapolate. The joke actually serves as an implementation of this function, which is kind of cute, methinks.

1. On second thought, perhaps they are trying to capture the role of the function with infer. If so, I think this is asking a lot of that word (and their readers' attention) even if you could technically argue that it implies some kind of rule-based process or such.