I recently began looking at languages from a mathematical perspective. From a mathematical perspective a formal language is the widest definition of a language I have found. However, there is a caveat in the wikipedia article:

In computer science and mathematics, which do not usually deal with natural languages, the adjective "formal" is often omitted as redundant.

The key constraint of a formal language is that there is a set (possibly infinite) of words which are "well formed." The above quote indicates that this is not true of natural languages. Is there an example of a language which does not meet this rule?

So far, I assume there are two directions to approach this probme:

  • Languages where "well formed-ness" is not a simple true or false binary value
  • Languages where the words are a class, not a set.

Unfortunately , I am looking at this from a computer science perspective. In CS, formal languages are king, so I'm finding it difficult to remind myself of characteristics of natural languages which cannot be found in formal languages. I'd appreciate some help!

closed as off-topic by Keelan, Swami Vishwananda, jeroenk, James Kingsbery, virmaior Aug 13 '15 at 0:06

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    I'm voting to close this question as off-topic because it belongs on the Computer Science SE. – Keelan Aug 8 '15 at 7:06
  • @Keelan I avoided putting it there. As quoted in the article, in CS the term "formal" is often considered redundant, making it a poor place to seek a good answer. I'm looking more at the "what else is possibly out there" side of things, which is more of a philosophical question, not a "what fits within these bounds" which is more of a CS question. – Cort Ammon Aug 8 '15 at 16:03
  • I disagree. A definition defines both what is inside a scope and what is outside it (namely everything but what's inside). And this is, as you said, a definition from CS. How they call that definition is unimportant. But there is a good answer here, so it's okay. – Keelan Aug 8 '15 at 16:48
  • Almost every natural language is capable of expressing "there is a tree outside of my window", and no computer language that I know of is. "Computer languages" are restricted to expressing instructions, while natural languages can also make declarative statements and more. "Set of strings of symbols that may be constrained by rules" is wider only syntactically, the main difference between formal and natural is in semantics, natural languages express meanings. Formation of natural words follows formation of meanings, and those can not be formed through syntax. – Conifold Aug 9 '15 at 1:19
  • @Conifold I think you may underestimate computer languages. While the argument of semantics vs syntax is good, the example of "there is a tree outside of my window" is trivial to express in nearly every computer language I know of, other than perhaps the pronoun "my" which may require a little finagling. – Cort Ammon Aug 9 '15 at 3:02

Any natural language allows introducing new words defined entirely by context clues.

One is allowed language moves like "We call one such as him a 'neebledy gitner'." Where no one has seen this sequence of letters used this way before its introduction, and what specific aspects of 'one such as him' are germane to the newly named status of 'gitnerhood' or 'neeblediness' remains to be determined entirely by inference.

I can form a nonce-word at will without marking it out in any way as a newly minted word, and it will be understood. Speakers will just incorporate it into their lexicon provisionally for the duration of the context, and for that period, it is a valid word. These words do not follow any formation rules or usage constraints beyond those of normal words, and they do not remain a permanent addition to the language.

So even if you allow for adding words by parsing a formal definition, or by allowing a generic form in a specific context, there is no single grammar that produces all and only English words. The list changes by the minute.

If there were some true definition of all English words, it could not possibly accommodate this level of flexibility, where words are temporarily or locally valid simply because we use them, but the exact same lexeme elsewhere would not be valid, for lack of a proper introduction.

  • Thank you, that is exactly the sort of thing I knew just had to be there, but could not put my finger on! And that example is perfect for a confused mind such as mine! – Cort Ammon Aug 8 '15 at 1:42
  • A shorter answer I almost gave would just be that languages evolve, not being able to predict the future, you cannot enumerate all English words. But this kind of thing points up the deeper perversity of natural language habits, we make important bits up as we go along and get understood anyway, somehow. – jobermark Aug 9 '15 at 0:11
  • Can interesting and useful philosophical analysis be done on an 'informal basis' or must every term used be defensible as being of proper syntax and semantics. Can an informal language for analysis be used in philisophy or is there such a fear of error every idea expressed has to be , with a mutually agreed upon style of communication , beyond reproach? – 201044 Aug 16 '15 at 14:06
  • @201044 I don't see how this is in context here. Obviously, we discuss philosophy in natural language almost all the time, and very, very few people even try to give a real definition of the terms they are using, they devolve on understanding by layering meaning together informally. So what is your complaint? Who is it that displays this fear? Argument is the soul of philosophy. So, any statement beyond reproach is probably not productive. If we can all agree, we should probably be looking elsewhere for more interesting material. – jobermark Aug 16 '15 at 14:58
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    @201044 If you want to ask Can interesting and useful philosophical analysis be done on an 'informal basis' or must every term used be defensible as being of proper syntax and semantics. Can an informal language for analysis be used in philisophy or is there such a fear of error every idea expressed has to be , with a mutually agreed upon style of communication , beyond reproach?, then start a new question. (I've deleted all comments related to that. / I agree with jobermark that it's a completely separate question. – virmaior Aug 18 '15 at 10:45

This is an interesting question. Sometimes natural languages are taken to be "non-formal" but many linguistics, starting with Montague, don't formally distinguish between human and "formal" languages. The main difference is that artificial (computer) languages don't have ambiguous grammars. Indeed any natural language can be described by a (context-sensitive) grammar. From this perspective, a nonformal language would be one whose words don't form a recursive set (over an alphabet).

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