The c-command matrix CM is
(25)
CM =
|
• |
A |
B |
C |
D |
|
A |
1 |
1 |
1 |
0 |
|
B |
0 |
1 |
1 |
0 |
|
C |
0 |
1 |
1 |
0 |
|
D |
1 |
1 |
1 |
1 |
where 1 indicates A c-commands A, and 0 indicates that it does not, similarly
for B, C, D.
5.3 Definitions of C-Domain & Governs
Definition [13] p.134
The total of all the nodes c-commanded by an element is the c-domain of
that element.
Definition [13] p.135
A governs B iff:
i) A is a governor,
ii)A c-commands B and B c-commands A.
Remarks:
The first requirement is a restriction on the set A (in linguistic terminology the
category A). A governor is a part of speech which generalizes the notion of
a verb governing an object; unfortunately there does not seem to be a formal
definition of it. The second requirement is that A and B should be sufficiently
’close’.
5.4 Definitions of CU-Domain & CU-Command
Now let D(A) be the set of all the ultrametric distances to other nodes at the
same height and let M(A) be the set of these which have the smallest value.
Call M(A) the cu-domain of A and say A cu-commands all BεM(A) (in
words B is a member of M(A). This is illustrated by Figure 14.
5.5 Theorem showing the identity between C-Domain &
CU-Domain
Theorem:
The sets A c-commands B and A cu-commands B are identical, likewise the
c-domain and the cu-domain.
Proof:
From the i) part of the definition of c-command h(A) = h(B), so that we are
only concerned with nodes at the same height h(A) = i. Let the first branching
node above A be F, with h(F) = i + k. Let H be any node dominating F, with
h(H) = i+l. Let E be the subsidiary node dominating B and C and dominated
by F, with h(E) = i + j. The closest nodes to A are B and C both with an
ultrametric distance k. The sets D(A) and M(A) are D(A)={A,B,C,D},
16