Constituent Structure - Part 24 | 30 PRELIMINARIES b Terminal node A node that dominates nothing except itself. c Non-terminal node A node that dominates something other than itself. Axiomization of dominance In an early article on the mathematics of constituent trees Zwicky and Isard 1963 sketch a series of definitions and axioms that specify the properties of structural relations. These axioms were updated in Wall 1972 and Partee ter Meulen and Wall 1990 and discussed at length in Huck 1985 Higginbotham 1982 1985 and McCawley 1982 more recent axiomizations can be found in Blevins 1990 5 Blackburn Gardent and Meyer-Viol 1993 Rogers 1994 1998 Backofen Rogers and Vijay-Shanker 1995 6 Kolb 1999 and Palm 1999 .7 8 The axioms while not universally adopted provide a precise characterization of the essential properties of the dominance relation. Trees are taken to be mathematical objects with at least the following parts based on Huck 1985 6 a a set N of nodes b a set L of labels c the binary dominance relation D 8 on N x y represents the pair hx y where x dominates y d the labeling function Q from N into L. 5 Blevin s axioms actually exclude some of the principles discussed below especially those that disallow multidomination and tangling line crossing . We will discuss Blevin s proposals in Chapter 10. 6 Backofen Rogers and Vijay-Shanker 1995 actually argue that first-order axiomization is impossible for finite but unbounded trees they propose a second-order account that captures the relevant properties more accurately. The proposal there is too complex to repeat here and since for the most part first-order description will suffice to express the intuitive and basic properties of trees we leave it at this level. 7 In this book I have kept the logical notation to familiar first-order logic. These latter citations make use of a more expressive logic namely weak monadic second-order logic MSO which allows quantification not only over variables that range over individuals but also over variables that
