# All non-head material must be phrasal. — Why's this principle nice idea from a mathematical point of view?

Andrew Carnie. Syntax A Generative Introduction (3 ed, 2012). p 208.

Pls see red underline. I never took math after high school! I don't know calculus. What author mean by "nice idea from a mathematical point of view"? This is linguistics not math book!

He's using "mathematical" loosely in this case. What he probably means is that it's more parsimonious/simple to only allow one head per phrase, which is in line with principles of Economy for linguistic theory. It all basically comes down to Occam's Razor: all things being equal, choose the simplest solution (or however you want to instantiate it). Since 1 head per phrase is simpler than 2 heads, it's "mathematically" nice.

• I suggest explaining your analysis of simplicity: what do you think "simpler" means, and how does this make something (what thing?) simpler. I argue that it makes things more complicated. – user6726 Jan 22 '20 at 17:42
• λxy . ∃d[simple(d)(x) ∧ d > max(λd1. simple(d1)(y))] I probably messed up somewhere, so someone smarter please correct me. – Rinzuu Jan 22 '20 at 20:18

This seems to be an appeal to generative power, in a loose sense. Throughout the history of generative grammar, there has been a concern with / interest in the class of "languages" that can be generated using a given theory. The underlying premise is that the best theory is one that allows the description of all attested languages, and nothing else (often justified in terms of Popperian falsifiability and the quasi-Popperian claim that a theory should not predict the existence of a thing not known to exist).

What is not expressly stated is the alternative theory: that non-head material can be anything. The theory in question says that all non-head material must be phrasal, therefore non-head material cannot be non-phrasal (obviously). The alternative theory allows both outcomes. The implication is that therefore the class of languages that can be generated by the theory in question is a proper subset of the class of languages that can be generated by the (unnamed, implicit) alternative theory. Given the aforementioned premise about "exactly the attested languages", the proposed theory is "mathematically nicer".

There are a few problems with this line of reasoning, investigation of which would take us too far afield. Briefly, though, (1) the "smallest set of languages" premise is quite questionable as a logical principle, (2) it remains to be proven that there is any actual difference in the set of languages that the two theories allow (the proposed theory probably can describe any imaginable language, it just does it a different way). A competing meta-principle is that the best theory is the one that posits the fewest theoretical entities (classical Occam's Razor), in which case adding a stipulation "must be phrasal" complicates the theory, with no concomittant benefit.

• I believe another way of thinking about the mathematical virtues of a grammar in which nonhead material is necessarily phrasal is in terms of Emonds' Structure Preservation Principle. If syntactic trees by definition are such that (i) they contain both phrasal and nonphrasal nodes, and (ii) their phrasal and nonphrasal nodes are in complementary distribution, then you can have a rather constrained transformational component. It's important to keep in mind people working in Transformational Grammar (TG) all through the 1960s and 1970s obsessed over how to constrain syntactic movement. – Deep_Television Feb 22 '20 at 8:22
• Peters & Ritchie had shown TG was equivalent to a Turing Machine, which meant language did not belong to the 'nice', 'learnable' complexity classes and that the problem of acquisition was unsolvable in terms of familiar computational lines of fracture (Gold's Theorem). Since acquisition is now widely believed to be driven mostly by innate principles, the computational approach is not in vogue anymore. Incidentally, in current models of syntax such as Bare Phrase Structure, Emonds' Structure Preservation Principle is also meaningless: categories can be both heads and phrases (e.g. clitics). – Deep_Television Feb 22 '20 at 8:23