follows is a fairly technical proof of the idea that parts of our linguistic system are at least plausibly construed as an innate, in-built system. If you aren’t interested in this proof (and the problems with it), then you can reasonably skip ahead to section 6.4.
The argument in this section is that a productive system like the rules of Language probably could not be learned or acquired. Infinite systems are in principle, given certain assumptions, both unlearnable and unacquirable. Since we’ll show that syntax is an infinite system, we shouldn’t have been able to acquire it. So it follows that it is built in. The argument presented here is based on an unpublished paper by Alec Marantz, but is based on an argument dating back to at least Chomsky (1965).
1 First here’s a sketch of the proof, which takes the classical form of an argument by modus ponens:
2 Premise (i): Syntax is a productive, recursive and infinite system.
3 Premise (ii):Rule-governed infinite systems are unacquirable.
4 Conclusion: Therefore syntax is an unacquirable system. Since we have such a system, it follows that at least parts of syntax are innate.
There are parts of this argument that are very controversial. In the challenge problem sets at the end of this chapter you are invited to think very critically about the form of this proof. Challenge Problem Set CPS8 considers the possibility that premise (i) is false (but hopefully you will conclude that, despite the argument given in the problem set, the idea that Language is productive and infinite is correct). Premise (ii) is more dubious, and is the topic of Challenge Problem Set CPS9. Here, in the main body of the text, I will give you the classic versions of the support for these premises, without criticizing them. You are invited to be skeptical and critical of them when you do the Challenge Problem sets.
Let’s start with premise (i): i-language is a productive system. That is, you can produce and understand sentences you have never heard before. For example, I can practically guarantee that you have never heard or read the following sentence:
18) The dancing chorus-line of elephants broke my television set.
The magic of syntax is that it can generate forms that have never been produced before. Another example of this productive quality lies in what is called recursion. It is possible to utter a sentence like (19):
19) Rosie reads magazine articles.
It is also possible to put this sentence inside another sentence, like (20):
20) I think [Rosie reads magazine articles].
Similarly, you can put this larger sentence inside of another one:
21) Drew believes [I think [Rosie reads magazine articles]]
And, of course, you can put this bigger sentence inside of another one:
22) Dana doubts that [Drew believes [I think [Rosie reads magazine articles]]].
and so on, and so on ad infinitum. It is always possible to embed a sentence inside of a larger one. This means that i-language is a productive (probably infinite) system. There are no limits on what we can talk about. Other examples of the productivity of syntax can be seen in the fact that you can infinitely repeat adverbs (23) and you can infinitely add coordinated nouns to a noun phrase (24):
23)
1 a very big peanut
2 a very very big peanut
3 a very very very big peanut
4 a very very very very big peanut etc.
24)
1 Dave left
2 Dave and Alina left
3 Dave, Dan, and Alina left
4 Dave, Dan, Erin, and Alina left
5 Dave, Dan, Erin, Jaime, and Alina left etc.
It follows that for every grammatical sentence of English, you can find a longer one (based on one of the rules of recursion, adverb repetition, or coordination). This means that the production of syntax is at least countably infinite. This premise is relatively uncontroversial (however, see the discussion in Challenge Problem Set CPS8).
Let’s now turn to premise (ii), the idea that infinite systems are unlearnable. In order to make this more concrete, let us consider an algebraic treatment of a linguistic example. Imagine that the task of a child is to determine the rules by which her language is constructed. Further, let’s simplify the task, and say a child simply has to match up situations in the real world with utterances she hears.9 So upon hearing the utterance the cat spots the kissing fishes, she identifies it with an appropriate situation in the context around her (as represented by the picture).
25) “the cat spots the kissing fishes”
Her job, then, is to correctly match up the sentence with the situation.10 More crucially she has to make sure that she does not match it up with all the other possible alternatives, such as the things going on around her (like her older brother kicking the furniture or her father making breakfast for her, etc.). This matching of situations with expressions is a kind of mathematical relation (or function) that maps sentences onto particular situations. Another way of putting it is that she has to figure out the rule(s) that decode(s) the meaning of the sentences. It turns out that this task is at least very difficult, if not impossible.
Let’s make this even more abstract to get at the mathematics of the situation. Assign each sentence some number. This number will represent the input to the rule. Similarly, we will assign each situation a number. The function (or rule) modeling language acquisition maps from the set of sentence numbers to the set of situation numbers. Now let’s assume that the child has the following set of inputs and correctly matched situations (perhaps explicitly pointed out to her by her parents). The x value represents the sentence she hears. The y is the number correctly associated with the situation.
| 26) Sentence (input) | Situation (output) |
| x | y |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
Given this input, what do you suppose that the output where x = 6 will be?
| 6 | ? |
Most people will jump to the conclusion that the output will be 6 as well. That is, they assume that the function (the rule) mapping between inputs and outputs is x = y. But what if I were to tell you that in the hypothetical situation I envision here, the correct answer
