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

Stabler - Lx 185/209 2003
{j}
{j,m}
{}
{m}
So if only John sings, we will interpret sings as the property {j},
[[sings]] ={j}.
12.3 Unary quantifiers, properties of properties of things
Now we turn to unary quantifiers like something, everything, nothing, John, Mary,…. Thesearewillproperties
of properties, which we will identify with sets of sets of properties, ℘(℘(E)). When E is the set above, there
are 16 different unary quantifiers, namely,
{{j,m},{j},{m},{}}
something:{{j,m},{j},{m}} {{j,m},{j},{}} {{j,m},{m},{}} {{j},{m},{}}
john:{{j,m},{j}} mary:{{j,m},{m}} {{j,m},{}} {{j},{m}} {{j},{}} {{m},{}}
everything:{{j,m}} {{j}} {{m}} nothing:{{}}
Notice that the English words that denote some of the unary quantifiers are shown here. Notice in particular
that we are treating names like John as the set of properties that John has.
If you are looking at this in color, we have used the colors red and blue to indicate the 6 quantifiers Q that
are increasing in the sense that if p ∈ Q and r ⊇ q then r ∈ Q. That is, they are closed under the superset
relation:
{} {{j,m}} {{j,m}, {j}} {{j,m}, {m}} {{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, on this view,
names like John denote increasing quantifiers, as do something and everything.
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{}