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Definition of formal language in wikipedia : formal language is a set of strings of symbols that may be constrained by rules that are specific to it.

For example, let's take (meaningful) English words, symbols. Then, give the English grammar rule to form strings. So, English is a formal language.

What's wrong here? Or English a really formal language?

Rubertos
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  • From a mathematical standpoint, the problem is the same as if you speak of "the set $A$ of all tasty apples." Membership in the set is not defined in a mathematically precise way. Likewise, while there may be some good approximations of a formal definition of what English is, these are not exactly accurate. – user204305 Jan 03 '15 at 09:49
  • @user204305 Why should the membership be precise? Isn't any subset of the set of all finite strings of symbols a formal language? – Rubertos Jan 03 '15 at 09:54
  • For one thing, it can be logically problematic to consider a subset of a given set that is defined by a vague property. For example, let $n$ be the minimum of the set $A$ of all natural numbers that "cannot be defined in fewer than thirty English words." If $A$ were in fact a set, the existence of $n$ would clearly lead to a contradiction. – user204305 Jan 03 '15 at 10:01
  • @user204305 Hmm... I'm asking whether a natural language is a formal language, not that it is a concsistent logical system. Since this is completely my guess, please don't get offended..:) I'm a beginner. Here is how I think. First, we define what formal language is, then put truth values and other stuffs to make it a formal logical system. Am I wrong? – Rubertos Jan 03 '15 at 10:22
  • Look, I may have misunderstood your question. I thought you were saying "How can something as imprecisely defined as English be a formal language?" But actually, it seems you're saying something like "Suppose we can write down a complete, finite description of what constitutes an English sentence. Then is English a formal language?" In this case, yes. But if you are saying that we should take English as it really is, then I would have to say that this isn't a mathematical question. – user204305 Jan 03 '15 at 10:28
  • @user204305 yes this is not a mathematical question.. Indeed philosophical. I just realized that there is a philosophy.stackexchange.com.. Sorry for this and thank you:) – Rubertos Jan 03 '15 at 10:31
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    In your quotation you have omitted the start : "In mathematics, computer science, and linguistics ...". Chomsky's theory, called Generative grammar was aimed at identifying a finite set of rules sufficient to produce all meaningful sentence of the human language; in this sense, it is linked to the mathematical study of formal languages. – Mauro ALLEGRANZA Jan 03 '15 at 17:54

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The definition on Wikipedia is now improved: "a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. "

No one has managed to identify a set of rules that draw a clear line between grammatical and ungrammatical English, even for a specific dialect. So English is not know to be a formal language, and almost certainly is not.

Note also that this definition is about syntax, not meaning.

chrishmorris
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    I'm not sure whether the proper conclusion is "English is not a formal language" or just "English is a formal language, but nobody has managed to identify which formal language it is". – Misha Lavrov Mar 15 '19 at 14:50
  • At the end, was it now or as you pointed; know? I think it is known – Mikasa Jun 13 '23 at 14:16
  • The problem is not mainly that no one has supplied the grammar. The problem is more that no one has a robust definition of "English". I doubt whether my idiolect is a formal language, but even if it is, the set of sentences you make is different than the set of sentences I make. – chrishmorris Jun 14 '23 at 14:44