Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
And if you are looking at this in color, we have used the colors green and blue to indicate the 6 quantifiers
Q that are decreasing in the sense that if p ∈ Q and r ⊆ q then r ∈ Q. That is, they are closed under the
subset relation:
{} {{}} {{j}, {}} {{m}, {}} {{j}, {m}, {}} {{j,m}, {j}, {m}, {}}.
(If you don’t have color, you should mark these yourself, to see where they are.) Notice that nothing denotes a
decreasing quantifier.
The color blue indicates the 2 quantifiers that are both increasing and decreasing, namely the top and
bottom:
{} {{j,m}, {j}, {m}, {}}.
The first of these could be denoted by the expression something and nothing, and the second by the expression
something or nothing.
12.4 Binary relations among things
The denotations of binary predicates will be relations among things, which we will identify “extensionally,”
as the sets of pairs of things. When E is the set above, there are 16 different binary relations, namely,
{,,,}
{,,} {,,} {,,} {,,}
{,} {,} {,} {,} {,} {,}
{} {} {} {}
So if only John loves Mary and Mary loves John, and no other love is happening, we will interpret loves as the
property {〈j,m〉, 〈m, j〉},
[[loves]] ={〈j,m〉, 〈m, j〉}.
And the property of “being the very same thing as”
{}
[[is]] ={〈j,j〉, 〈m, m〉}.
It is a good exercise to think about how each of these properties could be named, e.g.
[[loves and doesn ′ tlove]] ={},
[[loves or doesn ′ tlove]] ={〈j,j〉, 〈j,m〉, 〈m, j〉, 〈m, m〉}.
Notice that some of the binary predicates are increasing, some are decreasing, and some are neither.
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