In Hofstadter and Sander’s cognitive science book Surfaces and Essences an interesting thought experiment is presented. Called the “copycat domain,” we are invited to consider a few situations in a world where strings of letters can talk about their transformation. The book itself is quite verbose, so I’ll focus on the essence of this particular thought experiment.
Imagine for a moment a domain where we deal with strings of letters. Our first character is the string abc. It says: “I recall the time when I was changed to abd!”
- The second character is the string
efg. It says: “Exactly the same thing happened to me!” What do you think happened toefg? Write down your answer to this, and subsequent scenarios. - The next string is
pqrs. It, too, says: “Exactly the same thing happened to me!” What happened? - Another string, this time
acesays: “Exactly the same thing happened to me!” What is it talking about? - The next string is
iijjkk. It again says: “Exactly the same thing happened to me!” What do you think happened to it? - The next string is
mrrjjj. It also says: “Exactly the same thing happened to me!” What happened this time? - Finally, we have the string
xyz. It also says: “Exactly the same thing happened to me!” What happened now?
Consider your answers. Do you think there is only one possible answer for each of the questions? What do you think the word “exactly” means in each of these sentences?
Intelligence
I would hazard a guess that you didn’t say that, in each of the cases, the strings were all transformed to abd, yet, by the dictionary definition of the word “exactly” that’s precisely what should have happened. We all have some kind of innate instinct that precludes this answer: we expect that whoever set the task of guessing the resulting sequence had put some thought into it and there is some kind of interesting, aesthetic pattern that we have to follow when coming up with the solution.
I will relate the “correct,” or, I should say, “intended” answers to each scenario in a moment. This is the kind of task that is often presented in so-called intelligence tests, which purport to measure a value called “intelligence quotient,” or, for short, “IQ.” Even if one has never actually taken an “IQ” test, one recognises them as such, as they are not based on general knowledge, but on the ability to make analogies, categorise objects, guess the next logical thing or simply intuit the train of thought of the question-setter. These kinds of questions even happened to merit their own game trivia shows (e.g. The 1% Club).
Of course the problem with these kinds of questions is that they are very nebulous and often may have multiple defensible answers. In ordinary trivia shows and quizzes, general knowledge questions typically have one accepted answer. The question-setters strive to make the form of the question as unambiguous as possible, so that no answer other than the intended one can be considered correct (the technical term for this kind of refinement is pinning). To the contrary, the “IQ” test questions might have multiple defensible answers, even if only one was intended.
The answers
The intended answers are as follows.
In the case of efg, the intended answer is efh: by “exactly the same thing” it meant “the last letter in the string was changed to its alphabetical successor.” There are, of course, other options, such as efd (“the last letter in the string was changed to d”), efg (“replace c with d,” since there is no c in the string, it remains unchanged) and of course the “dictionary” option: abd.
With pqrs we have essentially the same thing happening with the same explanation, the intended answer is pqrt. The scope of possible “wrong” answers is analogous to the previous one, with the addition of pqss (“the third letter in the string was changed to its successor”) and maybe also pqds (“the third letter in the string was changed to d”).
As for ace, the intended answer is acg: since the internal structure of the string is analogous to abc, but “doubled,” the transformation is doubled as well and the final letter is the “double successor” to the original.
Next, with iijjkk the intended answer is iijjll: we once again have a group of three letters, but since the letters are doubled, by “exactly the same thing” it meant “the last group in the string was changed to its alphabetical successor.” The set of other possible answers expands considerably: in addition to the previous explanations we might also have answers such as iijjkd, iijjd, iijjdd, iikjkk and others. What is interesting here is that in the intended answer we are now completely removed from the concrete elements of the original situation, namely the third position and the letter d, everything is, in a way, abstracted (one familiar with category theory might even say, functorial). In each of the “wrong” answers the beautifully abstract world is contaminated by some concretion from the predicate transformation.
The intended answer to the penultimate transformation is by mapping letters to numbers: mrrjjj by “exactly the same thing” meant being mapped to mrrjjjj. (abc is the first, second, and third letter, and we have groups of one, two, and three letters respectively; the two conceptual jumps are from d to 4, and then to jjjj). I won’t list all other possible answers here. I’m certain if you came up with a different one, you have a good explanation for it.
Finally, the last of the dramatis personae, xyz inverts the entire thing. Whereas in the case of abc the final letter transformed to its successor, here, the first letter transformed to its predecessor, hence, the intended answer is wyz.
Quality
What I think makes this experiment interesting is the “kinds” of answers one might give. The first is the “objective” kind, which is answering abd to each of the scenarios since that is the only literal way of interpreting the word “exactly.” The second is the “subjective,” which is whatever you chose to answer.
But there is a secret third thing: the answers intended by the question-setter and, in a way, deducible by the one answering. Even if you answered something other than the “intended” answer, I think the explanations for what the word “exactly” meant in each of the examples is convincing enough to make you reconsider your subjective choice and altogether discard the objective choice.
In Zen and the Art of Motorcycle Maintenance, Pirsig wrote that Quality is the instinctive edge of experience, something neither subjective nor objective, but in a way uniting both and in another having nothing to do with either. This is precisely what happens when you encounter the answer wyz for the string xyz.
Notice the sequence of your own thinking: there’s often an immediate sense of “ah, right” before you can explain why. Only afterwards does your conscious mind catch up and articulate the reasoning. This pre-intellectual recognition, this feeling of rightness that precedes analysis, is Quality making itself known.
What makes certain answers feel more “right” than others? The intended answers share an aesthetic coherence - they preserve structural relationships, they find the deeper pattern rather than the surface one. When iijjkk becomes iijjll, we instinctively appreciate how the transformation respects the doubled structure. When xyz becomes wyz, we sense the elegant inversion. These answers might not be objectively correct, but they’re beautifully correct.
This same instinct operates when we recognise elegant code, a well-crafted function, or a clean architectural pattern. We know quality when we encounter it, even before we can enumerate the principles that make it so. And if you managed to intuit all the intended answers from the start, congratulations! It’s clear that your instinct for Quality is very good. We might even call it “IQ” for short.
