Dear Matthew and Pat, (01)
I strongly disagree with that point: (02)
MW> Types are sets, OK. Why not just call them 'sets', then?
> I suspect the reason is that while all types are sets, not
> all sets are types. (03)
For every type t and every domain d (or possible world, if
you want to use that metaphor) there exists a set of all
entities of type t in the domain d. (04)
In the terminology of worlds, a common term for that set
is the *denotation* of the type t in the world w. (05)
PH> Then I REALLY don't understand why you don't just use the
> word "set", which is about as common and as thoroughly described
> as any concept has ever been in the history of mankind. (06)
I agree with Pat. If you have a purely extensional approach,
you don't need any notion other than 'set'. To avoid confusion,
you should just use the word 'set'. A set is uniquely defined
by its members, and two sets with the same members are identical. (07)
On the other hand, there are good reasons for using the word 'type'
for intensional distinctions: (08)
- Since the empty set is a subset of every set (including itself),
the set of all unicorns is a subset of the set of all cows. (09)
- But the type Unicorn, according to traditional descriptions,
is definitely not a subtype of Cow. It is possible that somebody
could genetically engineer a kind of mammal with a single horn,
which would conform to the usual descriptions of unicorns. In
that world or domain, the set of unicorns would not be empty,
and it would not be a subset of the cows. (010)
There are many other reasons for types, but they all treat types
intensionally: two types can have different definitions (Unicorn
and Griffin, for example) but have the same extensions in the
real world. (011)
John (012)
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