Tree of numbers
SEMANTICS: in a sentence of the form [S [NP D CN] VP] the set A of entities denoted by the common noun CN can be divided into a subset with elements that belong to the set B of entities represented by VP, and a subset with elements that don't belong to that set, i.e. A intersect B and A - B, respectively. In a domain with n dogs, the dogs can be divided over these two subsets in n+1 ways, each of which is represented by an ordered pair x,y where x = |A intersect B| and y = |A - B|. The tree of numbers is a complete representation of all these pairs of numbers for each possible size of A:
(i) |A|=0 0,0 |A|=1 1,0 0,1 |A|=2 2,0 1,1 0,2 |A|=3 3,0 2,1 1,2 0,3 |A|=4 4,0 3,1 2,2 1,3 0,4 |A|=5 5,0 4,1 3,2 2,3 1,4 0,5 ... ...The meaning of a determiner D can be represented as a subset of a tree of numbers. The determiner every, for example corresponds to the x,0 pairs on each row:
(ii) |A|=0 + |A|=1 + - |A|=2 + - - |A|=3 + - - - ... ...Many properties of determiners (like upward monotonicity and downward monotonicity) and relations between determiners (like negation) can be clarified in the tree of numbers.
|LIT.||Gamut, L.T.F. (1991)|