An infinite number of possible sentences

Discussion in 'Linguistics' started by Magical Realist, Oct 16, 2014.

  1. Magical Realist Valued Senior Member

    Do you think there are? Could a sentence be infinite in length? Given there are about 500, 000 words in the english language, how many possible sentences ARE there? Could we ever reach the point of having said all there is possible to say in our language? What kind of language would we have to invent that would allow for infinite combinations? While you're pondering that, here's some thoughts on this conundrum:

    Robert Mannell (1999)

    "The claim, often repeated by linguists, that we can potentially produce an infinite number of sentences in a language, is not strictly true. For such a claim to be true there must either be the possibility of sentences of infinite length or there must be an infinite number of words in the language. Of course, neither of these requirements are possible for any human language (or for any imaginable communication system made up of discrete word-like units of meaning). We will examine the "infinite sentences" claim for English. If we assume:-

    • that English has about 500,000 words (there are about 450,000 in the 20 volume Oxford English Dictionary, but this excludes many colloquial forms - although it does include many obsolete forms),
    • that English sentences can be up to 100 words in length (a fairly reasonable working assumption)
    • that any individual word can occur 0 to 100 times in a single sentence (an unrealistic assumption)
    • that words can be combined in any order (a false assumption)
    then we can determine that there could be as many as about 10570 possible sentences (very much greater than estimates of the number of atoms in the observable universe). Grammatical rules would greatly reduce this number of sentences, as would the requirement that all sentences be meaningful, but the resulting number of possibilities would still be extremely large (more than could ever be spoken in the entire history of human languages let alone during the much shorter life span of an individual language). So for all practical, non-mathematical, purposes we can say that the English language, or any other living language(1), is an open system. It's actually quite easy to come up with a unique, never before produced, sentence. To do so, for example, combine an unlikely (or impossible, or meaningless) event with a particular named person on a particular date. For example: "On 31st October 1999, whilst writing a lecture on animal communication, Robert had a colourless green idea." (2) Once this sentence has been written or spoken, subsequent productions of this same sentence are not unique, but unique sentences may potentially be generated from it by making slight changes to it (eg. change "green" to "red").

    If we consider spoken language, then we would come to a similar conclusion if we examine only the word content of spoken sentences. We might additionally consider the manipulation of vocal resonance (frequencies of spectral peaks), vocal pitch (fundamental frequency), vocal loudness (intensity), rate of utterance and the placement and timing of pauses that occur as a consequence of the combined effects of prosody, vocal emotion, and size, age and gender differences. There are potentially an infinite number of infinitesimally different productions of any sentence (infinitesimal differences of frequency, intensity and timing). It is well known, however, that the human brain is only able to discriminate discrete (step-wise) changes in each of these dimensions. Across the possible human vocal range of these acoustic dimensions there is only a finite number of discriminable (just noticeable) steps. Additionally, it is also well established that meaningful changes in each of these dimensions tend to involve significantly larger changes than those changes that are just noticeable perceptually. This means that all of the meaningful vocal nuances of all of the possible sentences in English would be a large, but finite, number.

    I've recently (2011) been alterted to the following web page by Ken Wais that I found quite interesting:-

    Linguistic Combinatorics: Infinity and Human Language

    Also have a look at the following page that was suggested to me by John Fry. Especially look at the section entitled "4.4 Infinite vs unbounded". In that section it is argued that the length of English sentences (or sentences in any other human language) is not potentially infinite, but is unbounded in that we cannot define an upper limit to sentence length. Unbounded does not directly prohibit the possibility of an infinite set of sentences, but it does suggest that sentence length is always finite. In order for there to be an infinite number of sentences in a language there must either be an infinite number of words in the language (clearly not true) or there must be the possibility of infinite length sentences. The product of two finite numbers is always a finite number."===
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  3. mathman Valued Senior Member

    Not true! As long as there is no upper bound on the number of words per sentence, all sentences may be finite. This is elementary mathematics!
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  5. Captain Kremmen All aboard, me Hearties! Valued Senior Member

    If a series of random numbers was infinite in length then somewhere in it would be every other sequence of non infinite length.
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  7. Aqueous Id flat Earth skeptic Valued Senior Member

    Actually, the source that comes to mind is Argentinian philosopher Luis Jorge Borges, who took up this question in his short story The Library of Babel:

    Borges' narrator describes how his universe consists of an enormous expanse of adjacent hexagonal rooms, each of which contains the bare necessities for human survival—and four walls of bookshelves. Though the order and content of the books is random and apparently completely meaningless, the inhabitants believe that the books contain every possible ordering of just 25 basic characters (22 letters, the period, the comma, and the space). Though the vast majority of the books in this universe are pure gibberish, the library also must contain, somewhere, every coherent book ever written, or that might ever be written, and every possible permutation or slightly erroneous version of every one of those books. The narrator notes that the library must contain all useful information, including predictions of the future, biographies of any person, and translations of every book in all languages. Conversely, for many of the texts some language could be devised that would make it readable with any of a vast number of different contents.
    From which Borges conceives of every possible permutation of hybrid languages in which all possible letters are arranged onto 401 pages (per volume), most notably, this memorable variant which Fraggle Rocker should appreciate:

    a Samoyedic Lithuanian dialect of Guarani, with classical Arabian inflections.

    "The Library of Babel"

  8. Captain Kremmen All aboard, me Hearties! Valued Senior Member

    Fraggle owned one dog.
    Fraggle owned two dogs
  9. kx000 Valued Senior Member

    You can only count so high, Cpt.
  10. Captain Kremmen All aboard, me Hearties! Valued Senior Member

    It isn't me counting, it's Fraggle.
  11. rpenner Fully Wired Staff Member

    Obviously \(10^{570}\) was meant, and indeed appears that way in the original.

    Indeed \(\sum_{n=1}^{100} { \left( 5 \times 10^{5} \right) } ^ n = \frac{ { \left( 5 \times 10^{5} \right) } ^{101} - 5 \times 10^{5} }{5 \times 10^{5} - 1} \approx { \left( 5 \times 10^{5} \right) } ^ {100} \approx 0.78886 \times 10^{570}\) so the claim is true.

    However, since the claim is predicated on assuming that English sentences can't be longer than 100 words, it is doubtful the calculation is ultimately meaningful.

    This calculation can also proceed approximately via logarithms.
    \(\log_{10} 2 \approx 0.3\) so \(\log_{10} \left(5 \times 10^5 \right) = \log_{10} \frac{10^6}{2} \approx 6 - 0.3 = 5.7\) so \(\log_{10} { \left(5 \times 10^5 \right) }^{100} = 100 \times \log_{10} \frac{10^6}{2} \approx 570\), thus \({ \left(5 \times 10^5 \right) }^{100} \approx 10^{570}\).

    And since \(\log_{10} 2 \approx \frac{28}{93}\) it follows that \({ \left(5 \times 10^5 \right) }^{100} \approx 10^{- \frac{10}{93} } \times 10^{570} \approx 0.78 \times 10^{570}\).
    Last edited: Nov 1, 2014
  12. Captain Kremmen All aboard, me Hearties! Valued Senior Member

    If the number of words is limited to 100, then even using random words you would not get an infinite number of combinations.
    Unless you allow numbers, as in my example. Then there is no limit.
  13. mathman Valued Senior Member

    What you say requires that there be an upper limit on the number of words per sentence as well.
  14. Captain Kremmen All aboard, me Hearties! Valued Senior Member

    Here are the rules of grammar for numbers:

    So, any long number can be written as a word in a sentence using digits and commas.
    As there are infinite numbers, there are infinite possible sentences of the form
    x owned y dogs.

    There would be a finite number of the form
    y dogs were owned by x
    because the number would need to be spelled out.
  15. Fraggle Rocker Staff Member

    I would not allow "twenty-three hundred sixty-one." That's colloquial English, suitable only for casual speech. It should be written, "two thousand three hundred sixty-one."
  16. Captain Kremmen All aboard, me Hearties! Valued Senior Member

    According to the rules on the site I linked to earlier, once the number becomes easier to read in digits than it is in words,
    then you write it as digits.
    So, 1,001,018 would not be written as "One Million........etc",
    and "The Two Gentlemen of Verona", would not be written as "The 2 Gentlemen of Verona".

    The key question is this.
    Are "1,001,018" and its like, words?
    I would say yes.
    If so, then there are infinite potential sentences.
  17. Captain Kremmen All aboard, me Hearties! Valued Senior Member

    On further reflection, 1,001,018 is not a word.
    It is a series of symbols representing a number of words.

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