# Understanding why Antisymmetry gives us SVO and not SOV

I'm trying to understand the theory of antisymmetry as described here. I know the slogan that it treats every language as underlyingly SVO and derives other surface orders through movement.

I don't understand why the conditions imposed on nodes that asymmetrically c-command each other leads to SVO linear order.

I'm not even sure why the Linear Correspondence Axiom produces a unique order of terminal symbols. More specifically, I'm wondering how we would prove that the order SOV is incompatible with the LCA or what assumptions we need to make in order to rule out SOV.

First, I think we have at-most-binary syntax trees. A syntax tree is a rooted directed acyclic graph. The vertices have labels but the edges do not. I'm not treating a syntax tree as inherently equipped with a linear order. I use vertex and node interchangeably. The out-degree of any vertex can be 0 (a terminal node), 1 (a non-branching node) or 2 (a branching node).

Given two nodes A and B, A dominates B if and only if there's a directed path form A to B. A also dominates A, for all nodes A.

I nonstandardly define the branching parent of A to be minimum of all the branching nodes that strictly dominate A. The branching parent of A is undefined if no such node exists.

For the definition of c-command, I use the definition from this Wikipedia article. A c-commands B if and only if A does not dominate B, B does not dominate A, and the branching parent of A dominates B.

A asymmetrically c-commands B if and only if A c-commands B but B does not c-command A.

According to the LCA, if A asymmetrically c-commands B, then all the non-terminals under A precede all of the non-terminals under B.

Next, I'll define a ternary relation R. R(A, B, C) is true if and only if the nearest common ancestor of A and B is dominated by and not equal to the nearest common ancestor of A, B, and C.

If we have a transitive sentence, I think it is fairly noncontroversial to assume R(verb, object, subject), i.e. the verb and object form a constituent that is strictly dominated by the clause as a whole.

The S node itself, being a single node, would have the clause as a whole as its branching parent, and therefore would be forced by LCA to appear to the left of the common ancestor of the O and V nodes, since S asymmetrically c-commands O and S asymmetrically c-commands V.

Okay, so far, so good.

However, I don't understand how we rule out SOV. It seems like, in order to do this, we would somehow need to make object "branchier" than the verb in order to force the verb to be placed first ... and that if the verb were somehow "branchier" then it would appear last.

It feels like I'm missing something fundamental.