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Towards the end of this essay on formal systems and human language the author states:

Chomsky and Halle and many other linguists assume that language IS such a [formal] system... These are powerful assumptions that not only are not fully defended, they appear to have no evidence at all except the intuitions of linguists.

I'm inclined to agree with the author. Formal systems seem far too rigid and deterministic to encompass natural languages. However I'm fairly ignorant about the literature on this topic.

What are some arguments for and against language as a formal system? Is there a middle ground or better options? Is there any consensus or schools of thought within the Linguistics community?

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The author has mistaken language and grammar, and that criticism isn't valid for any period of generative phonology. Grammar is a cognitive ability which can be modeled as a particular kind of formal system. Language is a kind of behavior that is the product of a number of cognitive and physical systems, one of which is grammar. The viewpoint expressed by Chomsky & Halle (1968) recognizes a broad division between "competence" (grammar) and "performance" (the other stuff), but also imputes to grammar a lot of stuff that is not really grammar (such as the ability to quantitatively judge the extent to which some string of sounds is "close to English"). Since 1968, there has been great vacillation in how much is to be attributed to grammar as opposed to other cognitive faculties: formal theorists of phonology are inclined to call much less of language behavior "grammar". See Hale & Reiss (2008) The phonological enterprise for a general overview of the formalist approach.

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  • Why do you say that "ability to quantitatively judge the extent to which some string of sounds is "close to English"" is not really grammar? – Greg Lee Dec 30 '19 at 19:16
  • It's behavioral meta-knowledge about language, not part of the system that generates the language. It's like my knowledge that "father" is Germanic and "paternal" is Latinate. Classic competence vs. performance – except that Chomsky had a, well, erroneous view of how much is the computational system. Thus decades of errors conflating "grammar" and "language". – user6726 Dec 30 '19 at 19:31
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    By that reasoning, it would seem that assimilating loan words (further) into a language could not be undertaken by naive native speakers who have not studied etymology. Isn't this a little silly? – Greg Lee Dec 30 '19 at 21:34
  • Why do you think that follows? It's a fundamental principle of formal grammar that grammar is only a contributing factor to behavior: not all behavior is governed by grammar. You're attributing to me premises that I reject. Language behavior includes much more than grammatical knowledge. – user6726 Dec 31 '19 at 20:26
  • Adapting foreign words into their own language is something naive speakers know how to do, obviously, because they do it. For instance, loan words into Turkish are ofter harmonized -- changed so they come closer to having all front vowels in a word, or having all back vowels. The changes happen just as you would expect, starting with harmonizing the second syllable to the first and working their way to the end of the word, since Backness Harmony is progressive in Turkish. In your view, are these loan word changes independent of the phonology of Turkish? I hardly think so. ... – Greg Lee Dec 31 '19 at 21:34
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No, natural language is not a formal system. Some rather interesting theories about natural langages are formal systems. But to confuse a theory with the phenomena that it is a theory of is incoherent, in my opinion. Did Newton propose that our solar system was a differential calculus? Of course not. He invented the latter to describe the former.

Is it true that Chomsky and Halle claimed natural language is a formal system? Not so far as I'm aware. Could be, though, I suppose. Chomsky says some things I find difficult to understand. I would have to see a direct quote on this, however. (Richard Montague apparently regarded English as a formal system.)

There are a number of technical problems with the handout you refer us to. Formal systems needn't be deterministic, and they may have infinities of elements, for instance. Countable infinities, at least.

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  • Yeah, to Montague it was all spelled words, no language, all logic. That is surely a formal system; but calling it "English" is not really useful. – jlawler Dec 31 '19 at 19:02

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