Tree of numbers

Definition
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|.

Examples
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.

Links

 * Utrecht Lexicon of Linguistics