[As per comments below, I have edited this question for greater clarity.]

I want to know whether a language without recursion, which would generate only a finite number of well-formed sentences, could still generate a potentially infinite amount of discourse.

As I understand it, the term "recursive" describes formal operations (including a grammar's generation of well-formed sentences and constituents) that can apply to their own output. For example, the rules for forming relative clauses amount to a recursive operation, since relative clauses can be added to a noun phrase within some matrix relative clause. Thanks to recursion, we have noun phrases such as the cat that ate the rat," "the cat that ate the rat that ate the cheese," "the cat that ate the rat that ate the cheese that attracted the roaches," and so on, ad infinitum.

When I say that recursion allows a grammar to generate a "potentially infinite" number of sentences, I mean that a formal description of the language's syntax would entail that the number of well-formed sentences in the language is infinite in principle.

When I say that a language without recursion is finite, I mean that such a language's syntax would entail that the number of well-formed sentences in the language would be finite. This does NOT imply that a finite language couldn't generate enough sentences for its speakers to express themselves across contexts. Neither does it mean that a finite language would necessarily be a simple one. Dan Everett, a linguist who joined the Piraha-speaking community in South America to learn and study their language, has claimed that Piraha is finite and has also described Piraha's verb formation as extremely complex.

Now so far, we've been talking only about sentences. But we all know that people don't communicate in one-sentence telegrams. We talk at length for various purposes. In other words, we exhibit discourse. AFAIK, a lot of human discourse is "potentially infinite." In other words, in many types of discourse, an instance of such a type of discourse could continue indefinitely in principle. Hence expressions like "That conversation will go on forever."

All this information prompted me to ask whether a language without recursion, which would have a finite number of sentences, could in principle allow discourse (such as conversations) that could in principle continue indefinitely. I suspect that speakers of any language, finite or not, could go on talking forever in principle. But I want to know if this is more than my suspicion.

  • The question and the current answer seem to depend on not trying to describe what is meant by recursion yet asserting that something about it is not true. You need to clarify your understanding of linguistic recursion and what you mean when you talk about it and terms like "non-finite". You could be talking at cross purposes or even creating a straw man argument. – hippietrail Aug 2 '13 at 5:25
  • AFAIK, "recursive" characterizes any operation that can be performed on its own output. For example, relative clauses are recursive, since a relative clause can modify noun phrases in a matrix relative clause. So we can have noun phrases such as "the cat that ate the rat," "the cat that ate the rat that ate the cheese," "the cat that ate the rat that ate the cheese that attracted the roaches," and so on. AFAIK, a finite language is one whose grammar generates a finite number of well-formed sentences. [cont'd] – James Grossmann Aug 2 '13 at 6:16
  • [cont'd from last comment] AFAIK, a natural language that has recursion would generate a potentially non-finite number of well-formed sentences, owing to the fact that recursion in grammar could occur an arbitrarily large number of times. However, a language without recursion would generate a finite number of well-formed sentences. This does NOT mean that the language's grammar would not generate enough sentences for its speakers to express themselves. Nor would it mean that the language would have a simple grammar. Everett reports that Piraha's verbs are very complex. [con'td] – James Grossmann Aug 2 '13 at 6:21
  • If I don't have the concept of "recursive" or "finite" right, feel free to correct me. However, be advised that a large number of posts on this list assume acquaintance with the terms used in said posts. Witness the posts on computational linguistics, which I don't understand. Yes, my post assumes that the reader knows what "recursion" and "finite" mean in this context. I think that the post and its answer reflect this understanding. – James Grossmann Aug 2 '13 at 6:26
  • Thanks for your clarifications James! But I think it would benefit your question and the audience who come to read it if you could rework all of these thoughts into a longer version of your question - because it is a good question but it has to be solidly asked. When well stated it will be easier for answerers to answer and to spot if there are any misconceptions. – hippietrail Aug 2 '13 at 6:33
up vote 3 down vote accepted

I presume you have the recursion of syntactic dependencies in mind. Yes, I agree: a language can never be finite as long as you can produce it. Even saying the exact same thing twice or thrice or a hundred times can at least theoretically add or change meaning.

What those people you are probably thinking of mean is that you cannot have an infinite number of different sentences without some kind of sub-sentence recursion. That is to some extent true, but it seems to focus a bit too much on the sentence as a unit. Because I can say more or less the same thing... In shorter or different units. Another thing to consider is the following. Sentence boundaries are often a stylistic matter more tied to the following. Writing, rather than speech. They also are the following. Defined or chosen in somewhat arbitrary ways.

  • 1
    +1 for the self-referent answer. – Otavio Macedo May 9 '13 at 11:41
  • Plus, of course, since humans are finite, no infinite-length sentences are possible. Unless by sentence one means a string of letters, instead of an actual utterance. In which case it's not a linguistic question. – jlawler May 9 '13 at 14:27
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    @jlawler: If if it means a string of letters, humanity and its products will in all likelihood one day perish, thereby ending all sentences. So we can't have infinite sentences anyway. – Cerberus May 9 '13 at 16:16
  • And in that case there won't have been an infinite number of them, either. – jlawler May 9 '13 at 16:38
  • Dan Everett says exactly that Pirahã does use the method of several shorter sentences to express what would be expressed with one complex sentence in other languages, and that this is what it means to not have recursion. – hippietrail Aug 2 '13 at 5:19

Now that you've substantially fleshed out your question it's easy to answer.

I want to know whether a language without recursion, which would generate only a finite number of well-formed sentences, could still generate a potentially infinite amount of discourse.

Yes of course! The linguistic theory of recursion is about syntax, within the sentence. It places no limits on what can happen among many sentences - discourse.

Languages are defined as subset of Kleene closure of alphabet.So the subset may be recursive set or enumerably recursive set,or non-enumerably-recursive set,the last one is not recursive,and in fact,the last two are not recursive in a strict sense.So,there are infinite languages that are not recursive.Or,there are as many non recursive infinite language as real numbers.

  • 1
    What about the many languages that don't have an alphabet? – Gaston Ümlaut Aug 2 '13 at 0:34
  • @GastonÜmlaut,It is in the sense of formal language.You may can map a natural language without alphabet to a formal language,which is called encoding.Actually,for example,Chinese characters are a kind of alphabet that components of the language mapped. – XL _at_China Aug 2 '13 at 2:36
  • What's Kleene closure? – James Grossmann Aug 2 '13 at 6:27
  • @JamesGrossmann,en.wikipedia.org/wiki/Kleene_star,it is from formal language theory,or logic.You know,Noam Chomsky has used a lot of idea and technics from mathematical logic.By the way ,I do not know who downvote my answer,I think the downvoter must have not known what the idea of recursion is :) – XL _at_China Aug 2 '13 at 11:28
  • I downvoted it, for the same reason Gaston commented on. Linguistic recursion doesn't seem to have anything to do with writing systems at all. – hippietrail Aug 2 '13 at 12:46

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