Keenan's account of the DEFINITENESS EFFECT associated with the English there-construction based on the notion of INTERSECTIVE DETERMINER is well-known. In Part 1 of this paper, I will consider a similar kind of effect in Japanese constructions. In particular, I shall show that a very general form of Quantifier Float and the Head Internal Relative Clause, two phenomena particularly prominent in Japanese syntax, allow us to extend the idea of the Definiteness Effect to predicates with more than one argument. I will then show in Part 2 that this idea provides an empirical motivation for extending Keenan's idea of intersective determiners to 2-dimensional (transitive) spaces. Part 2 thus concerns a mathematical extension of the classical mathematical theory of determiners to 2-dimensional spaces with its empirical grounding in Japanese syntax. Part 3 generalizes the mathematical theory introduced in Part 2 to n-dimensional spaces in general, without any empirical concern.
Part 1, Part 2 and Part 3 can in principle be read independently. Part 1 primarily concerns Japanese and is empirical and descriptive. The reader who is not particularly interested in mathematics may wish to read only Part 1. Those who are interested in the "mathematics of language" but not particularly in details of Japanese syntax may wish to skim through Part 1 and start careful reading from Part 2. On the other hand, those who are only interested in mathematics and do not care about, or wish not to be bothered by, empirical facts may wish to read only Part 3. Due to the intended relative independence of the three Parts, the reader who wishes to read through the paper from Part 1 through Part 3 may encounter some redundancy through the paper.
Part 1, sections 1-5, is a slightly revised version of the first five sections of Kuroda (2007). I wish to express my gratitude to the CSLI Publications for granting me a permission to reproduce this portion in this paper.