First, a note of trivia (well, trivia if what you are concerned with is syntax, rather than vision): This light box works on the same principle that a television does, with additive color mixing of (additive) primary colors. The reason it works has to do with the biology of the human eye, which has three types of cones in the retina each detecting one of (essentially) red, green, and blue. That is, a red light and a green light don’t really make yellow. But the human eye can’t tell the difference, because actual yellow has the same effect (stimulating red and green cones). Hence: Television, which capitalizes on this principle. Our light box is something like a pixel on a television. Cool, no? The light box described in the problem is in fact somewhat related to the color perception of a person with red-green colorblindness.

The reason, incidentally, that the light box is black has nothing to do with vision. It is actually more of a scientific metaphor, and perhaps not completely accurately applied in this case. When one studies a “black box“, one essentially presumes nothing about what goes on inside the box but looks only at what it is given as input (here, the switch setting) and what it returns as output (here, the color and the buzzing sound) and tries to model the relationship between them. In this problem, we do know that the light box has within it a red bulb, a green bulb, and a blue bulb, so we do presume a bit more about the inner workings of the box than we would if this were really a “black box” we were studying.

Great, now what does this have to do with humans putting sentences together? There are a couple of ways in which this problem relates to what we’re doing. One is this: Thought of in the right way, we’re creating a model of a different species-specific, abstract cognitive system: color perception. Like the grammatical system, the vision system is not open to introspection: we can’t just sit and try to “remember” the principles that govern the way we parse visual information into color representations.

The other, and probably a bit more concrete and accurate, way that this problem relates to what we’re doing is that it is a system that lends itself to being modeled by way of features and can give you some practice in theory construction and the concepts of overgeneration and undergeneration. We will be spending a lot of time with features this semester, and I wanted you to get comfortable with them right away. We won’t be spending any more time with the vision system, but the principles will be the same.

“Intriguing,” I hear you say. “But now how do I start?” Let me lay out the analogy to the singular/plural discussion in the book and in class a bit more explicitly. Imagine that we have a black box whose input is a noun from the lexicon (with whatever information we have stored in our memory about this noun), and whose output is the form that the verb to be would take if that noun were the subject. So, given the lexical entry for scissors, the black box would yield are. Given the lexical entry for book, the black box would yield is. No matter what noun we give this black box, it will give us either is or are. The black box therefore divides nouns into two classes, which we can call “plural” (are) and “singular” (is). We want to model this system, this black box. It is sensitive to some feature of the lexical item, a feature that determines whether the box returns is or are.

Roughly following the course of the discussion in the book/lecture, we could suppose that among the properties (features) a noun can have in the lexicon are [sg] and [pl]. Thus, if the black box gets a noun with [pl], it yields are, and if it gets a noun with [sg], it yields is. The problem with this is that nothing guarantees that a noun has exactly one of these features: Since both [sg] and [pl] are features a noun might have, this leaves open the possibility that a noun might have both or have neither. We could stipulate that all nouns in the lexicon have exactly one of these two features, but this is a complication to our model. If we can find a simpler model that captures the observed facts without needing this stipulation, the simpler model would be a better one (Ockham’s razor).

We could say that the box yields are if the noun has a [pl] feature, and yields is otherwise. That way, a noun with both [sg, pl] would result in are and a noun with neither would result in is. But at that point, the [sg] feature is irrelevant. It doesn’t matter whether it is there or not, it is the presence or absence of the [pl] feature that determines the output. Thus, the simpler model would be one where we only posit one feature, [pl], that nouns in the lexicon either have or lack. Coming at this another way, if we posit that the box is sensitive to both [sg] and [pl], we predict that it would in principle be possible for the box to yield four different outputs, but we see only two. The model predicts things that aren’t attested, it overgenerates.

The discussion above concerned privative features, specifically the feature [pl], which can either be a property of a noun or not. Another way to do this would be use a binary feature, saying that nouns either have a [+pl] feature or a [−pl] feature. The analysis with a binary feature would be essentially the same: For nouns with [+pl], the black box yields are, and for nouns with [−pl], it yields is. The overgeneration problem with assuming both a [+/−pl] and a [+/−sg] feature would also be the same.

Back to the problem at hand, with the light box. Your task is similar to that just outlined with the singular/plural distinction, except that instead of giving the black box a noun, you’re giving it a configuration of light bulb states (as set by the switch). When the switch is in its first position, the red light bulb is on, the green and blue light bulbs are off. The resulting color is red. You are asked to model this in terms of binary features, and it’s a pretty straightforward task to describe the state of a light bulb in terms of a binary feature. That is, if the blue light bulb is on, you can describe this as [+blue], and if the blue light bulb is off, you can describe this as [−blue]. As suggested in the problem, you can look at the model in Part 3 to see how you might write this out, although there is an issue with the model presented in Part 3 (so your answer to Part 1 will be different in content, although the same in form). You might in fact be better off starting with Part 3 to try to identify the problem with it, and then go back to Part 1 and provide a model that doesn’t have the problem.

Anyway, I’ll leave it there. I hope this helps get you started with the problem if you were having difficulty grasping what was being asked for. Email me or post comments here if questions occur to you.