Folks, (01)
I've been traveling for the past two weeks, and I haven't
had a chance to participate in the recent discussions.
But I'd like to make a few comments on some of the points. (02)
Re sets and types: If we agree on a suitable version of
logic L (which Common Logic aficionados may assume is
something like CL), we can adopt the following definitions: (03)
1. A type T may be specified in L by a monadic predicate T(x),
which is defined by one or more axioms written in L that
make some assertions about T. These axioms may be said
to define the _intension_ of T. (04)
2. For any given model M of L, the _extension_ of T in M is
the set of all x in M for which T is true: {x | T(x)}.
Every x such that T(x) is called an _instance_ of T in M. (05)
This definition can be used to clarify some of the issues
that were raised in the email notes. For example, Chris Menzel
observed that there may be a problem with the extension of SET.
However, if the model M is fixed in advance, then the extension
of SET in M could use the same math used to specify M itself. (06)
There was also a question about timeless sets and whether one
could say that a set that includes people alive today could
be said to exist before there were any people on earth (or
even before the earth itself was formed). This question could
also be answered in terms of the above definition: (07)
1. First construct (or specify) the model M. (08)
2. If you want to talk about entities that come into existence
and pass out of existence at various times, you might prefer
to use a 4D ontology. But you could also use a 3D ontology
in which you partition all the entities into "snapshots"
or "time-slices" or "situations" at each time t. (09)
3. With the snapshot approach, there would be a large domain D
of everything in M that could be mentioned in the language L,
and there would also be a dyadic predicate occurs(x,t), which
would mean that entity x occurs in the snapshot at time t.
(I'll use the word "occur" to avoid confusion with the backward
E for "exists".) (010)
4. However, you could also make statements that compare entities
in D at different time slices. For example, you might want to
say that most adult men alive today are taller than Napoleon
was at the beginning of the 19th century. In general, you
could make statements about individuals that might occur at
different times and create sets that include entities that
occur in widely separated snapshots or time slices. (011)
There was also a discussion between Barry and Matthew about a
type that happened to be specified in terms of an arbitrary
collection of individuals. The definition above does not rule
out that possibility. For example, you could pick any set
{a,b,c,d} and define the type T by the axiom (012)
For all x, T(x) iff x=a or x=b or x=c or x=d. (013)
You might, for example, let a be the integer 7; b, President Bush;
c, the moon; and d, a tetrahedron. Since it seems odd to call
this a type, many people have argued for various principles of
goodness, coherence, or generality for distinguishing which
axioms are permissible in the specification of a type. (014)
One possibility is to prohibit constants that identify particular
individuals, such as President Bush, in the axioms that define
a type. However, the earth itself is an individual, and you
might want to define the type whose extension is the set of all
living organisms that have ever existed on planet earth. For
that matter, 4 is also an individual, and you might want to
define the type Quadrilateral as a four-sided polygon. So the
issues cannot be resolved by such a simple restriction. (015)
In any case, a definition such as the one at the beginning of this
note can clarify many of the issues, and it provides a starting point
for talking about principles of coherence or goodness for types. (016)
John Sowa (017)
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