The British journal Science Progress has published a paper by William Abler that tries to derive language origins (and much else) from first principles, the way Euclid deduced his geometry and Newton set forth his Principia. Einstein's first relativity paper also argued from a couple of axioms. I mention Euclid, Newton, and Einstein to show that science can be based on assertions as well as experiments, and also as a caution. It takes a mighty big brain to begin with a priori laws and still get the science right. Abler is betting the farm that he belongs in that class.
Just to be persnickety, I'll point out that there is another way to approach this confusion: ask questions about empirical facts. Several are possible. How is it that human children quickly develop the ability to speak their parents' language? Why don't apes use language in the wild since experiments have proven they are smart enough to be trained to use sign language, at least in its simplest form? Why don't any other animals cooperate on the basis of a speech triangle?
Having said that there is no agreement on whether there is or is not a universal grammar, Abler performs a thought experiment and asks, "What if there were ten supposed language universals and 10 languages were found, each lacking a different one of the 10 supposed universals in language. Would that show that there are no universals of language?" 
He answers, "Probably not. It would show, instead, that language is not a machine, like a watch that stops functioning when one of its parts is removed." I like that and suspect that I will probably steal it on future occasions. He goes on, "Language is a system that continues to function because a single missing part can be inferred from the other nine." Apparently, by language, Abler is referring not just to French or Sanskrit, but instances of usage as well. Thus, "John told Tom to cut the grass, and Jim the weeds," is a case of inference even though the second clause lacks a subject [Tom] a verb [told] and a secondary verb [to cut].
When we understand language that broadly, it seems as though no universal universals are going to be found. Language, especially in its oral form, is subject to all kinds of ad hoc shortcuts. But then Adler does propose a universal universal, "the property of discreteness, which characterizes language at every level of its organization."  I don't think he will get much argument on the point. The fact that every language comprises combinations of a small set of sounds well known. I have previously reported that any learned system of vocalizations is combinatorial, even if the learner starts with holistic sounds. (See: Apes Don't Imitate; People Do) Abler asserts, "On the basis of discreteness, then, we may reasonably expect to place the scientific study of language onto a theoretical basis."  Give the man credit for building on an unquestioned fact: phonemes give us morphemes which give us words which give us sentences which give us paragraphs which give us stories which give us literature.
Having identified the scaffolding of literate civilization, Abler then abandons it with very little cause that I can see. He tells us that "language and mind must have their source in some more ancient natural system…" wait a minute! Where did mind enter the story?
Well, bler is really interested in mind, not language. He titles his paper, "The Human Mind: Origin in Geometry," and asserts that, "any theory of mind and brain must begin with a theory of language."  Is that true? Certainly many theories of brain don't start with language, and until rather recently in the history of philosophy theories of mind didn't start there either. However, I'm going to move on because it would be perfectly acceptable for Abler to have said, "I want to understand the human mind, and my gateway will be through understanding language first."
Yet I'm still balking: "must have their source…" Why not sources? Must language and mind have a common source? I proceed with trepidation. But I'm encouraged by the fact that he is looking toward a natural history explanation. Language (I'm skipping mind in this discussion) "must be derived, through a process of descent with modification, from some other system that already existed …"  Good. I'm with him.
What system might that be, he wonders. And quotes from Pinker and Bloom's famous paper, pointing to simpler languages like pidgin, baby talk, etc. Abler protests that these cannot really be the source of language, and mostly I agree. The Pinker and Bloom paper legitimated the search for language origins and triggered much research that has moved beyond their starting point. A standard reaction has been, "Pinker and Bloom say X, but that cannot be right. So what is right?" And more research follows.
Abler's reaction is more unusual. He rejects the problem of looking for sources of descent through modification! "Since language is not a mechanical device like a watch, it is probably not an organ, like an eye, that is built up gradually as its component parts take shape. Instead, it is probably a system, like geometry, whose structure shows evidence of design, and which emerges only a little at a time (Euclid, ca 300 BC), but did not evolve by natural selection. Internal processes in the formation of sentences remain completely hidden." 
What!? Earlier when he contemplated universal he said that language isn't like a watch because users can infer missing elements. Now he talks about the gradual build up of component parts. What happened to phonemes, morphemes, words, sentences, paragraphs, stories, literature? Those are built up from parts. I thought he was going to build his theory from them?
Then we get, "Instead, it is probably a system, like geometry…" Probably! At best I could tolerate, "Instead, it might be a system like geometry…" But that is hardly the only choice. What happened to, instead it might be a learned behavior, like birdsong; or, instead it might be a suite of competencies, like playing baseball; or, instead it might be a human invention, like baby slings? Abler may well have his reasons for saying it is probably a system, but he has to explain it a bit better. I kept on reading, but mainly in the hope of learning why he thinks language did not evolve by natural selection.
No such luck. Abler has irretrievably jumped to a new track and instead of deriving a theory of language and mind from the build up of discrete elements, he talks (very nicely) about the relation between algebra and geometry, then the relationship between algebra and language (formalism galore), and then says the antecedents of language "do not lie in behaviour or psychology… [but] in the pre-biological properties of geometry and arithmetic." 
It often happens that basic biological structure reflects some sort of geometric simplicity. Why are daisy petals arranged as they are? Look not to adaptation but geometry; yet even there the petals have some history concerning function, genes, and evolution. The biological history of the geometry of language—assuming there is such a thing—still needs to be accounted for.
Too bad, because I liked seeing see his original proposition laid out so explicitly: any idea about the evolution of language has to account for the appearance of phonemes, then words, then sentences, then more.