Naked Science Forum
On the Lighter Side => New Theories => Topic started by: mg1 on 05/09/2012 22:13:45
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So I've seen this discussed before on other websites but I've never been too satisfied with the answers I've seen. The question is: Is it possible to create a list of all possible grammaticaly correct sentences?
Alot of people say it is an impossible task due to the amout of storage space required or even that based on certain rules of grammar such a list is impossible or infinite. My problem is every time I see this discussed it's generally in reference to English being the language used. Knowning what I do about English, mainly that most rules have exceptions and many words have multiple meanings, I can't help but feel it is entirely wrong for such a project in the first place. Creating an entirely new, simple language meant for the project would, to me, eliminate many of the issues. Perhaps making it possible?
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If you made a language with 10 verbs, 10 nouns, a few conjunctions, adjectives, and etc, you might have a chance.
Certainly there would be a problem with run-on sentences, and various clauses. You would have to have a rule preventing redundancy (which can, in fact, be used for emphasis in English).
John went to school to school to school again.
You still would run into problems with somewhat valid recursive statements.
I meant the green ball inside of the circle inside of the square inside of the circle inside of the triangle inside of the circle inside of the square inside of the circle inside of the triangle inside of the circle inside of the square...
(meaning you have a large number of layers of boxes).
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The number of possible sentences in any language with a reference scope that is universal is infinite, and therefore there cannot be a catalogue of "all possible sentences".
Proof: Consider the sentence
"The number three hundred and sixty-five thousand seven hundred and thirty-eight is one larger than three hundred and sixty-five thousand seven hundred and thirty-seven."
There are an infinite number of sentences like this, because you can substitute any other whole number into the sentence, and there are an infinite number of whole numbers.
It is possible to devise an artificial language with a finite strictly defined vocabulary. It is difficult to do it in a way that allows universal reference. And that is not enough restriction to restrict possible sentences to a finite number -- it also takes some unusual and rather weird rules of grammar.
A possible way, perhaps the easiest way of going about it is to restrict valid words of an artificial language to those in a particular edition of a standard English dictionary, and to have a grammatical rule that a sentence must consist of, say, ten words or fewer. This would create a need for roundabout ways of naming large numbers, so that a sentence like the example given above might require several sentences to express the meaning that it embodies.
An artificial language with n available words and a grammatical restriction that a sentence has a maximum length of m words has a finite number of possible sentences. I will leave it as an exercise for the reader to demonstrate that this number is
(n(m+1) – 1) / (n – 1)
(or find a Maths student who can do so).
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In the artificial language of my last post, I neglected to point out that the rule of "m words maximum" per sentence is the only rule of grammar in the language, which, as a result, would include sentences like
The the the the in in the the of of.
For a normal and comprehensible artificial language of this sort, the number provided by my formula will only be an upper limit on the number of possible sentences, and a hugely overstated one at that!
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"The number three hundred and sixty-five thousand seven hundred and thirty-eight is one larger than three hundred and sixty-five thousand seven hundred and thirty-seven."
There are an infinite number of sentences like this, because you can substitute any other whole number into the sentence, and there are an infinite number of whole numbers.
There are tribes which have only two numbers in their language: "one" and "many", so that would prevent an infinite number of numbers from appearing. Rather than having a specific word limit, which is too artificial a restriction to be practical, it would be more natural to say that a sentence can only contain up to three items, those being a subject, verb and object:-
I locked the door by turning the key --> I turned key. Previous caused following. Key locked door.
The blue car was fast --> Car was blue. Car was fast.
[Many languages don't use articles like "a" and "the", so they can be dropped.]
You can communicate everything important with a few hundred words, though a lot of cheating would be involved by combining words to cover other meanings (e.g. white horses), so a thousand would be better, and 3000 is generally considered to be a practical working vocabulary. Let's go for 3000 and make a third of them verbs/prepositions/conjunctions (prepositions and conjunctions are disguised verbs: the book in the library = the book that has location that is the library's interior's part; he died before the war = his death preceded the war), though this only restricts the verb position in the sentence - verbs can serve as nouns and be the subject or object of other verbs, so the subject and object positions in the sentence are not restricted. We now have 3000 x 1000 x 3000 possible sentences, or 9 billion, though of course most of them would be silly.
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I really think that you have missed my point David. It was about proofs, not practical constructions.
The first part of my post was a proof that the number of possible sentences in any actual modern language was infinite. It is not the only construction that provides a proof. I suppose that there might be some languages still extant that only have words for the first few integers; most of them allow for stacked conjunctions, so we could obtain a large integer from "... four and four and four and ..." , or build an infinite number of sentences that way.
The second part was not an attempt to produce the most practical or most efficient language with a finite number of sentences -- it was simply a proof that a language with a finite number of sentences was conceivable, and that it was possible to construct one in such a way that it was a genuine language -- that is, that could express any ideas we might want to communicate as well as any modern language.
A sentence in a modern language is supposed to express a single idea. Did you know that there is a single and very simple idea that can be expressed in an infinite number of ways, but only in infinitely long sentences?
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I didn't miss the point - I was answering the OP and simply used a quote from you to lead into an explanation of how the issue of a range of infinite numbers can be avoided, so it wasn't intended as a rejection of anything you had said. The OP was interested in the idea that the number of possible distinct sentences in a language could be finite if the right language was chosen or created, and I wanted to show how that could indeed be the case while also ensuring that the language could still be fully practical as a system of communication.
Did you know that there is a single and very simple idea that can be expressed in an infinite number of ways, but only in infinitely long sentences?
Don't think I did, but I've certainly never read such a sentence. Would be interested to hear more.
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I didn't miss the point - I was answering the OP and simply used a quote from you to lead into an explanation of how the issue of a range of infinite numbers can be avoided, so it wasn't intended as a rejection of anything you had said.
My apologies for being over-sensitive.
Did you know that there is a single and very simple idea that can be expressed in an infinite number of ways, but only in infinitely long sentences?
Don't think I did, but I've certainly never read such a sentence. Would be interested to hear more.
Consider the sentence "The moons of Mars are Phobos and Deimos" and the sentence "The moons of Mars are Deimos and Phobos". Two different ways of saying the same thing, both legitimate. (And that last "sentence" of mine is not grammatically legitimate, even though it completely expresses an idea! Even standard grammars have artificial aspects.)
Now consider the sentence "The natural numbers are 5 and 7 and 58 and 2 and ...". I will not write the rest of it because it would waste too much of TNS bandwidth [;D], but you can see where it is going!
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Consider the sentence "The moons of Mars are Phobos and Deimos" and the sentence "The moons of Mars are Deimos and Phobos". Two different ways of saying the same thing, both legitimate. (And that last "sentence" of mine is not grammatically legitimate, even though it completely expresses an idea! Even standard grammars have artificial aspects.)
What that actually tells you is that standard grammars are wrong if they declare your sentence to be illegitimate, because it's perfectly standard - when it comes to languages, what is normal is right, even if it's logically wrong (though that doesn't apply in this instance).
Now consider the sentence "The natural numbers are 5 and 7 and 58 and 2 and ...". I will not write the rest of it because it would waste too much of TNS bandwidth [;D], but you can see where it is going!
I was hoping it might be something more exciting than an infinite set of numbers again, but it's still interesting to find out that is isn't.