it forms an ordered pair with the head directly, , or S i-precedes H. Invoking i-precedence prevents more than two elements from merging with H; only one element can i-precede H (the specifier), and H can i-precede only one element (the complement). Kayne notes that is not mappable to a tree structure, since H would have two mothers, and that it has the consequence that and would seem to be constituents. He suggests that is replaced by , "with an ordered triple replacing the two ordered pairs and then being mappable to a ternary-branching tree" (pp. 17). Kayne goes on to say, "This would lead to seeing my [(1981)] arguments for binary branching to have two subcomponents, the first being the claim that syntax is n-ary branching with n having a single value, the second being that that value is 2. Mapping [ to ] would retain the first subcomponent and replace 2 by 3 in the second, arguably with no loss in restrictiveness." Theoretical arguments Antisymmetry theory rejects the head-directionality parameter as such: it claims that at an underlying level, all languages are head-initial. In fact, it argued that all languages have the underlying order Specifier-Head-Complement. Deviations from this order are accounted for by different syntactic movements applied by languages. Kayne argues that a theory that allows both directionalities would that languages are symmetric, whereas in fact languages are found to be asymmetrical in many respects. Examples of linguistic asymmetries which may be cited in support of the theory (although they do not concern head direction) are: Hanging topics appear at the start of sentences, as in "Henry – I've known that guy for a long time". They are not attested at the end of sentences. Number agreement is stronger when the noun phrase precedes the verb (Greenberg's Universal 33). Examples of this are found in English sentences such as There's books on the table, where the verb frequently fails to agree with the following plural noun, and in French and Italian compound tenses, where the past participle may agree with a preceding direct object but not with the following one. Relative clauses that precede the noun (as in Chinese and Japanese) tend to differ from those that follow the noun: they more often lack complementizers (akin to English that) or relative pronouns and are more likely to be non-finite (this can be found, for example, in Quechuan languages.)