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Provably (or by sufficiently rigorous arguments), what are the limits of language (natural & formal)? What can a language not speak about? If there is something non-senseless which language cannot speak of, is that limit imposed by the structure of the language OR by our rules of interpretation of the language?

Also, can we think i.e. mentally reason about the thing which cannot be expressed by a language?

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    You do realize that the question is nonsensical? You are asking us to speak about what language can not speak about. If this is intended please explain what sort of answer you are looking for. For Wittgenstein's take see If “the limits of my language are the limits of my world”, then how can it be that “what can be shown, cannot be said”?
    – Conifold
    Jun 8 '19 at 23:24
  • Wittgenstein summarized the problem. However it's detailed exegesis Is better done by Whorf (and Sapir Jacobson and other linguists) Jun 9 '19 at 1:24
  • May add: If you insist on "prove rigorously" instead of say "indicate" "explain" "elaborate" the question becomes untenable as @Conifold suggests. Jun 9 '19 at 2:05
  • @Conifold Yes, in that way it is quite nonsensical. While we may not be able to write a sensical statement which language cannot speak of, but, by using natural language, I think we can certainly talk about it or describe it. That is the sort of answer I am looking for. What is the nature of such statements, if they indeed exist.
    – Ajax
    Jun 9 '19 at 5:48
  • I think you're looking for concepts that we know about but that our language cannot describe. But how could we communicate such concepts here then? Jun 9 '19 at 7:30
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Comprehension abilities determine the limits of language.

Imagining that a hypothetical language has representations for neuronal firings, any thought emerging from the firings can be communicated through the language.

Communicating yellow to a color blind person is an example of limit of human language. See qualia for more examples.

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Language is limited to the things and events we can put names to, or at least apply referent signifiers to. But as a rule, names and referent signifiers are abstract tokens for complex entities or events, which cannot themselves be fully expressed in language. Making any utterance is a bit like throwing a ball, where we expect the listener to actively catch it within the rules and contexts of the particularly language game we're playing. For instance, if we make an utterance like:

Amy walked to the barn

We assume that the listener knows, in context:

  • whether Amy is a person, horse, or cat
  • what range of behaviors might be considered 'walking' for a thing of Amy's type
  • if the destination is a hay barn, a Pottery Barn, or a nightclub called 'The Barn'

If the listener knows those things, they can effectively 'catch' the meaning we've tossed at them. If they do not, we find ourselves having to explain or describe the ambiguous terms until the listener can catch the meaning. But explanation and description suffer from diminishing returns; the less common experience we share with the listener, the more time, effort, and detail we need to convey meaning.

Much of our understanding of the world is carried in non-linguistic forms. For instance, most of us have a sense of what it means to 'walk', 'sashay', or 'bumble' — we have an image of it in our heads — but few of us could explain or describe those acts in a way that would make sense to someone who has no idea what they mean. We'd have a far easier time demonstrating than explaining, just as we'd have a far easier time introducing someone to Amy than trying to give a complete, unique, and accurate description of her. Language is limited by this onerous and pragmatic necessity of building common understandings (a 'soft' limit), not by any 'hard' limit precluding something from entering into language in the first place.

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Well, one limitation is that a language (formally described) is a countable set of strings. This means that there is some way to write down all the sentences in the language, one by one, so that every sentence in the language will appear at some definite position in your list. Your list will have a beginning, but not an end; you will keep writing it forever, much like the digits of pi.

Since language is a countable set of strings, it can at most name a countable set of phenomena.

But some phenomena are not countable. The set of real numbers between 0 and 1, which we can name by the interval (0, 1), is not a countable set; there are "too many" to write them in a list like that. See Cantor's diagonal argument.

As a consequence, there are real numbers between 0 and 1, that have no sentence or formula naming that specific real number. In fact, another theorem says the set of specific real numbers that we can name has measure 0. This means that greater than 99.999999999% (as many 9s as you like) of the interval (0, 1) is not specifically nameable by any language. We can talk about sets of numbers that happen to include these un-nameable numbers, but we can't name each un-nameable number individually.

Many natural phenomena may be continuous. This means that language can never precisely say where an electron is, or even what the electron's probability distribution of locations is; language can be as precise as desired, but never perfectly precise.

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