# Limits of a Language



## Meyer Wolfsheim

Hello everyone,
                    How does one calculate the total possible logical phrases constructible in any given language?  I know this is not possible for a living language because it is constantly gaining new words, inflections, etc., but say a time period is chosen, like the English language in the year 1900.  Once the interval is set, there is a limit to how many words/inflections exist at a given time, because the English of 1900 must have had less possible meanings (i.e. the English of 2000 is capable of X more meanings than that of 1900).  The English language has no recursive features; only words and a handful set of inflections (and even if it did these are all blocked out as it only includes those words/inflections available at that given interval) so there must be a finite number of meanings possible.  

To explain my question better:  Assume there is a language which is completely analytical; there exist absolutely no inflections (or all words carry the same exact inflection, a null bound); a word is completely invariable.  This language has a finite number of words and because word order is the only method of creating meaning, there is ultimately a finite number of possible combinations, however large they be (this is not even including constructions which are not grammatically correct).  There are no recursive features (i.e. compounding, repetition of words, words which take multiple meanings); thus it can be said that this language has some limit to the total possible meanings that it can output.  

The likewise must be true for an exact fossilized form of English; there has to be a limited number of words and only a handful of words can take on several possible inflections; how would one go about this to find all the grammatically correct meanings possible?  

I have a feeling this question might be out of the scope of these forums, so if any of you have any works you could recommend me to I would greatly appreciate it.


----------



## trance0

Meyer Wolfsheim said:


> ... but say a time period is chosen, like the English language in the year 1900.  Once the interval is set, there is a limit to how many words/inflections exist at a given time, because the English of 1900 must have had less possible meanings (i.e. the English of 2000 is capable of X more meanings than that of 1900).  *The English language has no recursive features*(This does not hold water, see my comments below); only words and a handful set of inflections (and even if it did these are all blocked out as it only includes those words/inflections available at that given interval) so there must be a finite number of meanings possible.
> 
> To explain my question better:  Assume there is a language which is completely analytical; there exist absolutely no inflections (or all words carry the same exact inflection, a null bound); a word is completely invariable.  This language has a finite number of words and because word order is the only method of creating meaning, there is ultimately a finite number of possible combinations, however large they be (this is not even including constructions which are not grammatically correct).  *There are no recursive features (i.e. compounding, repetition of words, words which take multiple meanings)* (I don`t believe there has ever existed such a language. Every living language at any given fixed period of time in history had synonyms and homonyms, some more than others, but I doubt there have ever been any languages out there that haven`t had this); thus it can be said that this language has some limit to the total possible meanings that it can output (It probably has, but it is impossible to define what a meaning would be, especially counting all word puns, literature etc. There is no way of setting the parameters for this 'calculation').
> 
> *The likewise must be true for an exact fossilized form of English*(This is not true, see my comment above); there has to be a limited number of words and only a handful of words can take on several possible inflections; how would one go about this to find all the grammatically correct meanings possible?  (Simple answer, one would not go about this, because it wouldn`t have any sense. English or any other language for that matter is not mathemathics)
> 
> I have a feeling this question might be out of the scope of these forums (It most certainly would), so if any of you have any works you could recommend me to I would greatly appreciate it.


----------



## entangledbank

Mary ate a sandwich, then she ate a biscuit, then she ate another sandwich, then she ate another biscuit, then . . .

Lather, rinse, repeat; infinite sentence generation using 1900 English and no recursion, just linear right-hand branching.


----------



## xmarabout

If I understand well, your question can be reduced to that one: with exactly the same vocabulary, what is the probability somebody not knowing Shakespeare, rewrite Macbeth... Very difficult to calculate, isn't it ! And then you will have to do the same for all the documents (written and spoken) of the same period... Considering people of that period wrote everything that can be written...
Impossible even with the best computers !


----------



## brian

Meyer Wolfsheim said:
			
		

> There are no recursive features (i.e. compounding, repetition of words, words which take multiple meanings)



I don't think you can include such a statement. As far as I know, all known languages are recursive in some way. It's a basic principle of natural language semantics/syntax. Example: _I saw the man who saw the man who saw the man who..._

While the number of words in a given natural language (or in any one speaker's idiolect) may be finite, the number of possible constructions is infinite.

Edit: Sorry, this is basically exactly what entangledbank already said.


----------



## Meyer Wolfsheim

But what would be the point of repeating a construction over again over than to prove that recursiveness exists?  If Mary ate a biscuit, then a sandwich, then another biscuit, you could simply express this as Mary ate (sum of total biscuits) and (sum of total sandwiches).  Do not be recursive for the sake of being recursive.  

If we have an artificial language which has a finite number of words (any element which can stand on its own in a string of elements and carries individual meaning by itself) and it bases all of its meaning off of the order of elements, there is a finite order of elements and thus a limit to the possible combinations possible, ruling out the repetition of those elements already in the set and the compunding of words (thus the Mary case given would be illegal).  

But, if a language can never have a true limit to its output, then it thus proves that human language has no limits to the information which it can express and if language is the mode of human thought, then human thought is not human in the sense that it has no limits.


----------



## brian

Meyer Wolfsheim said:


> But what would be the point of repeating a construction over again over than to prove that recursiveness exists?



Because the existence of recursiveness is crucial to the matter at hand. My example (_I saw the man who saw the man who saw the man who..._) only repeats the same information in a superficial way - insofar as the exact same words are used; but the underlying idea is not superficial at all: *constructions can be infinitely nested without repetition of the same information*. Think of it mathematically: just like variables, each relative clause in my example can be switched out with some other kind of clause, using different nouns and verbs, to create an infinite amount of uniquely meaningful sentences.

Or think of it grammatically: all the _man's_ and all the _who's_ are all unique individuals. The words are the same, but removing any part of that sentence changes the meaning of the entire sentence!* This is not a triviality.*



Meyer Wolfsheim said:


> If Mary ate a biscuit, then a sandwich, then another biscuit, you could simply express this as Mary ate (sum of total biscuits) and (sum of total sandwiches).  Do not be recursive for the sake of being recursive.



True, but that's why I think my example is a bit better: each part of the (infinite) structure is crucial to the meaning of the sentence as a whole.



Meyer Wolfsheim said:


> If we have an artificial language which has a finite number of words (any element which can stand on its own in a string of elements and carries individual meaning by itself) and it bases all of its meaning off of the order of elements, there is a finite order of elements and thus a limit to the possible combinations possible, ruling out the repetition of those elements already in the set and the compunding of words (thus the Mary case given would be illegal).



What do you mean by "order" here? I don't follow.



Meyer Wolfsheim said:


> But, if a language can never have a true limit to its output, then it thus proves that human language has no limits to the information which it can express [...]



Mmm.. no, not true. Once again, think of it mathematically: a function _f_ that maps from the set of all negative integers over to the set of all positive integers, e.g. _f_(-3) = 3, _f_(-5) = 5, etc. Both the input and output (i.e. domain and range in math-speak) are infinite sets, but that does not mean that _f_ has no limits: for example, it is unable to produce the number 0.5. So the idea of "limits" in a language is a bit ambiguous - even if a language has an infinite amount of constructions (and so is in a way "limitless"), it still has limits on what it can produce.



Meyer Wolfsheim said:


> and if language is the mode of human thought, then human thought is not human in the sense that it has no limits.



I think this is getting a bit to philosophical.


----------



## Meyer Wolfsheim

brian said:


> Or think of it grammatically: all the _man's_ and all the _who's_ are all unique individuals. The words are the same, but removing any part of that sentence changes the meaning of the entire sentence!* This is not a triviality.*


 
*Could you give me a real life example in English where a string goes on for that many clauses?*



True, but that's why I think my example is a bit better: each part of the (infinite) structure is crucial to the meaning of the sentence as a whole.



What do you mean by "order" here? I don't follow.

I mean that you can only arrange all the elements in a finite amount of orders. 


Mmm.. no, not true. Once again, think of it mathematically: a function _f_ that maps from the set of all negative integers over to the set of all positive integers, e.g. _f_(-3) = 3, _f_(-5) = 5, etc. Both the input and output (i.e. domain and range in math-speak) are infinite sets, but that does not mean that _f_ has no limits: for example, it is unable to produce the number 0.5. So the idea of "limits" in a language is a bit ambiguous - even if a language has an infinite amount of constructions (and so is in a way "limitless"), it still has limits on what it can produce.

So, if a language can have a limit on what it can produce, how do you find these limits?


----------



## entangledbank

Since you have ruled out everything about human language that makes it human language, the answer to your original question is now some _n_, where _n_ is a positive integer less than or equal to some upper bound _m!_, where _m_ is the number of head-words in the _OED_. As required, this tells us nothing at all about human language, and produces a finite cardinal.


----------



## entangledbank

Mary, meanwhile, has become an aspiring gourmet, and has eaten a sandwich containing egg and mutton but not lobster; followed by a sandwich containing egg, mutton, mustard, and Camembert but not spinach or sardines; followed by a sandwich containing layers of sardines over plums over sardines but not guacamole on its own or egg mixed with guacamole or any layers of Gorgonzola over rusks over Gorgonzola over guacamole, . . .


----------



## Meyer Wolfsheim

Yes, I have to agree with you entangled.  But another poster mentioned that every language has some limits to what it can express.  How do you find these limits or in truth can any language communicate over any information?


----------



## Hulalessar

Meyer Wolfsheim said:


> But another poster mentioned that every language has some limits to what it can express. How do you find these limits or in truth can any language communicate over any information?


 
The question of whether any language has limits to what it can express is a different one from whether the number of possible utterances is finite. The question has two aspects. The first is whether any language can completely describe the world and human experience. The second, not completely unrelated to the first, is whether language limits thought and/or the way we perceive the world. They are questions on which linguists, philosophers, anthropologists, psycholgists, neuroscientists and whoever else has thought about them have yet to agree on an answer.

Concentrating on whether language can completely describe the world, language is an act of classification. To be useful and manageable it has to restrict its terms. The world is an infinite number of things, events and possibilities and no language can possibly have words for them all. To that extent therefore there is a limit to language, but I am not sure you can say what the limit is.


----------



## berndf

Hulalessar said:


> The second, not completely unrelated to the first, is whether language limits thought and/or the way we perceive the world. They are questions on which linguists, philosophers, anthropologists, psycholgists, neuroscientists and whoever else has thought about them have yet to agree on an answer.


The obvious answer at any point in time is: yes, _language is limited_ (as to what we can communicate) and _language limits_ (as to which concepts and ideas we can formulate; sufficiently complex ideas and theories cannot be created _purely_ non-verbally; of course, we have to include mathematical and logical formulae and other formal language into a broadened concept of language and verbal communication).
 
Today, most epistemological schools would agree: _Language can evolve as our knowledge and theories about the world evolve and, hence, jointly overcome the limitations of language and of thought_. Example: Kepler's laws of planetary movement and Newton's laws of mechanics reached the limits of what you could express with the language of 17th century science. Maxwell's electrodynamics, e.g., could not have been formulated (and therefore not have been conceived) with the language of that time. It requires the fully developed differential calculus _and the language associated with it_. But this would not have been developed, hadn't the new 16th and 17th century theories of physics created the need for such a calculus and a mathematical language which goes beyond traditional language of philosophy.


----------



## jonquiliser

berndf said:


> The obvious answer at any point in time is: yes, _language is limited_ ...
> 
> Today, most epistemological schools would agree: _Language can evolve as our knowledge and theories about the world evolve and, hence, jointly overcome the limitations of language and of thought_. Example: Kepler's laws of planetary movement and Newton's laws of mechanics reached the limits of what you could express with the language of 17th century science. Maxwell's electrodynamics, e.g., could not have been formulated (and therefore not have been conceived) with the language of that time. It requires the fully developed differential calculus _and the language associated with it_. But this would not have been developed, hadn't the new 16th and 17th century theories of physics created the need for such a calculus and a mathematical language which goes beyond traditional language of philosophy.



I beg to differ. I would say that language can express pretty much anything we can think or know - that's a prerequisite. Language isn't formed only by concepts or terms. When empirical phenomenon are discovered, recorded or explained, they may lack names - but certainly they can be described, talked about, discussed, understood, thought.


----------



## Teach & Learn

Voice sounds and written symbols are covered so far, but I don't see "body language" mentioned here. It's better to define "language" before talking about "limits" of it.


----------



## berndf

jonquiliser said:


> I beg to differ. I would say that language can express pretty much anything we can think or know - that's a prerequisite. Language isn't formed only by concepts or terms. When empirical phenomenon are discovered, recorded or explained, they may lack names - but certainly they can be described, talked about, discussed, understood, thought.


I think the history of modern science proves you wrong. I am not agreeing with logical positivists like Carnap who claimed that further progress of science necessitated a completely new language. But if you read the giants of early modern science, like Descartes, Galileo, Kepler, Brahe or Newton, you will notice two things: how far there use of language as such (not just individual words) differs from that of medieval philosophers but also how much is missing to be able to talk about the concepts of contemporary science. New word views require new ways of thought and new ways of thought require different ways to use language.

Regarding your statement "but certainly they can be described, talked about, discussed, understood, thought", I can only repeat my earlier example: Electrodynamics cannot be "discussed, understood and thought" without the differential calculus and modern algebra which in turn cannot be "discussed, understood and thought" without modern mathematical formula-language. See e.g. how bulky and incomprehensible Kepler's own formulation of his laws of planetary motion was as he had only the Latin language at his disposal and not modern formula-language though these laws are by modern standards of rather low complexity.


----------



## Hulalessar

Thought and language are inextricably linked. Is thought* possible without language? Is it is possible to have a new thought unless the language is there to express it? Does the language to express a new thought exist before the new thought does? Is it the case that new concepts and the means of expressing them arise mutually?

*I exclude here non-verbal modes of thought such as music.


----------



## darnil

Meyer Wolfsheim said:


> I have a feeling this question might be out of the scope of these forums, so if any of you have any works you could recommend me to I would greatly appreciate it.


 
We're dealing here with the issue of language complexity. And I remember having read a very interesting discussion on this topic in _Linguistic Typology_, 5 2/3, even though it was limited to the creole languages. There are many ongoing discussions in these days on the issue, but I'm no expert. Maybe some other foreros can help you much better.


----------



## ccesarjj

There are several entangled issues here that need to be focused one at a time so that they can be discussed fruitfully. First of all, as it has already been remarked the scope and usage of highly polisemic terms such as "limits" and "language" must be specified. 

1) if the term "language" is to be understood in a morphosyntactically restricted fashion and limits are thought to be just a matter of  numbering possible grammatical utterances, the question becomes, as its has been pointed out, a trivial mapping of some mathematical discussions about cardinals.

2) if we are rather concerned with how we fathom a formally defined semantic realm by using systematically several syntactical devices structured following a clearly defined set of morphological rules, we are dealing then with the problem of comparing the expresiveness of various formal languages: lots of logicians, mathematicians, philosophers, and linguists have addressed this issue and we just have to take a look at some of the classical and current discussions in the fields of model theory and universal logic (Regardless of their 'infinite' scope, there certainly are languages which are more expressive than others).

3) It could be argued that the limits been asked for are not a matter of *cardinality* or *expressiveness*, but are located beyond any formal description or beyond the hierarchy of all possible *metalanguages*, they  are intended to be some sort of 'transcendental' limits of humans' symbolic activities. Well then, I'd say we might be mapping linguistically some classical philosophical discussions addressed by philosophers such as Kant, Hegel and Wittgenstein regarding the *scope of human thought *and that cannot obvioulsy be settled in a few lines. Moreover, I'd say those philosophical issues lie beyond pure syntax or semantics and require a broader pragmatical and historical approach. Thus, addressing them properly requires dropping aside the main assumptions made at the beginning of this thread.


----------



## ccesarjj

You can find some information about some of these issues on http://plato.stanford.edu 
If you're rather concerned with some of the "transcendental aspects of language", here's an interesting paper to take a look at, written by Martin Stokhof of the Universiteit van Amsterdam http://staff.science.uva.nl/~stokhof/papers.html
(a bit technical and requires a little background on contemporary semantics and philosophy)


----------



## ccesarjj

I would also like to add that we must recall our verbal expressions are not only used to "talk about" something (i.e. to denote objects), but "to do" many other things (We "do things with words", as J.L. Austin said): greeting, promising, praying, chanting, etc. Thus, an appropriate account of the "limits" of a "natural" language might also need to take into account how to set "boundaries" to our possible "actions". Anybody interesting in accomplishing the goal?


----------



## ccesarjj

By the way, if a thought is to be identified as such on linguistic grounds, then there might be several systematic ways of tackling this seemingly puzzling questions (i.e.dissolving some of them). 



Hulalessar said:


> Thought and language are inextricably linked. Is thought* possible without language? Is it is possible to have a new thought unless the language is there to express it? Does the language to express a new thought exist before the new thought does? Is it the case that new concepts and the means of expressing them arise mutually?
> 
> *I exclude here non-verbal modes of thought such as music.



However, good arguments have to be provided in order to restrict ourselves to such a conception of 'thought'. Anyway that would certainly be a matter of another thread.

Regards,
CC


----------



## boadicea7

To follow a  bit of a structuralist line, Language (like culture) provides a field of action that makes an enourmous range of expression posible but within that  range that to a certain degree determines one's thought, or at least expression


----------

