[ Note to instructors : this question requires some background in either formal logic or mathematical proofs.]
In the text, it was claimed that because language is recursive, it follows that it is infinite. (This was premise (i) of the discussion in section 4.3.) The idea is straightforward and at least intuitively correct: if you have some well-formed sentence, and you have a rule that can embed it inside another structure, then you can also take this new structure and embed it inside another and so on and so on. Intuitively this leads to an infinitely large number of possible sentences. Pullum and Scholz (2005) have claimed that one formal version of this intuitive idea is either circular or a contradiction.
Here is the structure of the traditional argument (paraphrased and simplified from the version in Pullum and Scholz). This proof is cast in such a way that the way we count the number of sentences is by comparing the number of words in the sentence. If for any (extremely high) number of words, we can find a longer sentence, then we know the set is infinite. First some terminology:
Terminology: call the set of well-formed sentences S. If a sentence x is an element of this set we write S(x).
Terminology: let us refer to the length of a sentence by counting the number of words in it. The number of words in a sentence is expressed by the variable n. There is a special measurement operation (function) which counts the number of words. This is called μ. If the sentence called x has 4 words in it then we say μ(x) = 4.
Next the formal argument:
Premise 1: There is at least one well-formed sentence that has more than zero words in it.
∃x[S(x) & µ(x) > 0]
Premise 2: There is an operation in the PSRs such that any sentence may be embedded in another with more words in it. That means for any sentence in the language, there is another longer sentence. (If some expression has the length n, then some other well-formed sentence has a size greater than n).
∀n[∃x[S(x) & µ(x) = n]] → [∃y[S(y) & µ(y) > n]]
Conclusion: Therefore for every positive integer n, there are well-formed sentences with a length longer than n (i.e., the set of well-formed English expressions is at least countably infinite):
∵∀n[∃y[S(y) & µ(y) > n]]
Pullum and Scholz claim that the problem with this argument lies with the nature of the set S. Sets come of two kinds: there are finite sets which have a fixed number of elements (e.g. the set {a, b, c, d} has 4 and exactly 4 members). There are also infinite sets, which have an endless possible number of members (e.g., the set {a, b, c, …} has an infinite number of elements).
Question 1: Assume that S, the set of well-formed sentences, is finite. This is a contradiction of one of the two premises given above. Which one? Why is it a contradiction?
Question 2: Assume that S, the set of well-formed sentences, is infinite. This leads to a circularity in the argument. What is the circularity (i.e., why is the proof circular)?
Question 3: If the logical argument is either contradictory or circular what does that make of our claim that the number of sentences possible in a language is infinite? Is it totally wrong? What does the proof given immediately above really prove?
Question 4: Given that S can be neither a finite nor an infinite set, is there any way we might recast the premises, terminology, or conclusion in order not to have a circular argument and at the same time capture the intuitive insight of the claim? Explain how we might do this or why it’s impossible. Try to be creative. There is no “right” answer to this question. Hint: one might try a proof that proves that a subset of the sentences of English is infinite (and by definition the entire set of sentences in English is infinite) or one might try a proof by contradiction.
Important notes:
1 Your answers can be given in English prose; you do not need to give a formal mathematical answer.
2 Do not try to look up the answer in the papers cited above. That’s just cheating! Try to work out the answers for yourself.
CPS9. ARE INFINITE SYSTEMS REALLY UNLEARNABLE?
[Creative and Critical Thinking; Challenge]
In section 4.3, you saw the claim that if language is an infinite system then it must be unlearnable. In this problem set, you should aim a critical eye at the premise that infinite systems can’t be learned on the basis of the data you hear.
While the extreme view in section 4.3 is logically true, consider the following alternative possibilities:
1 We as humans have some kind of “cut-off mechanism” that stops considering new data after we’ve heard some threshold number of examples. If we don’t hear the crucial example after some period of time we simply assume it doesn’t exist. Rules simply can’t exist that require access to sentence types so rare that you don’t hear them before the cut-off point.
2 We are purely statistical engines. Rare sentence types are simply ignored as “statistical noise”. We consider only those sentences that are frequent in the input when constructing our rules.
3 Child-directed speech (motherese) is specially designed to give you precisely the kinds of data you need to construct your rule system. The child listens for very specific “triggers” or “cues” in the parental input in order to determine the rules.
Question 1: To what extent are (a), (b), or (c) compatible with the hypothesis of Universal Grammar? If (a), (b) or (c) turned out to be true, would this mean that there was no innate grammar? Explain your answer.
Question 2: How might you experimentally or observationally distinguish between (a), (b), (c) and the infinite input hypothesis of 4.3? What kinds of evidence would you need to tell them apart?
Question 3: When people speak, they make errors. (They switch words around, they mispronounce things, they use the wrong word, they stop mid-sentence without completing what they are saying, etc.) Nevertheless children seem to be able to ignore these errors and still come up with the right set of rules. Is this fact compatible with any of the alternative hypotheses: (a), (b), or (c)?
CPS10. INNATENESS AND PRESCRIPTIVISM?
[Creative and Critical Thinking; Challenge]
Start with the assumption that i-language is an instinct. How is this an argument against using prescriptive rules?
CPS11. LEARNING PARAMETERS: PRO-DROP
[Critical Thinking, Data Analysis; Challenge]
Background: Among the Indo-European languages there are two large groups of languages that pattern differently with respect to whether they require a pronoun (like he, she, it) in the subject position, or whether such pronouns can be “dropped”. For example, in both English and French, pronouns are required. Sentences
