Pierre and Chris P, (01)
I base those definitions on semantics of the CL core,
which is "typeless" -- i.e., there is only one type
of entity and no formal distinction between individuals
or relations. (02)
PG> Although I gather that if the original motivation is
> ontological the idea is that instance_of is a primitive
> (that would not ask for a definition). (03)
True, and I believe that is the motivation Barry Smith
would prefer. However, my point is that we need to adopt
a methodology that will enable us to deliver a solution
to the ONTAC WG without resolving the debates between
nominalists and realists. (04)
Therefore, my proposal is agnostic with respect to that
debate: adopt a version of logic, such as CL, and define
instance_of in terms of the syntactic features of CL.
Nominalists can accept that solution as stated, and
realists can think of it as a "quick-and-dirty" method
of avoiding a commitment. (05)
PG> But the main point might just be that the types are
> individuals. That point only should make the definition
> you're suggesting fail, because (y x) would either
> yield non-sense or in the best case falsehood. (06)
CL is truly agnostic. (y x) is *never* nonsense (i.e.,
something that would cause a syntax error). If y is
not a monadic relation, the result is simply false.
And there is no reason why you can't write (07)
(and (y x) (x y)) (08)
You can even write (x x). CL provides ways of interpreting
all such statements. If they're contradictory, that's
your problem, not CL's. (09)
CP> The issue is whether the identity of intentional
> types/instances is determined by their definition (rather
> than their extension). So if two types have different
> definitions, they have different intensions and so they
> are then different types. (010)
This point came up a couple of days ago, and my recommendation
was to follow the solution proposed by Alonzo Church: (011)
http://www.jfsowa.com/logic/alonzo.htm (012)
In short, two types are equal by intension iff the rules for
converting definitions allow the definition of one to be
converted to the definition of the other. See the excerpt. (013)
John (014)
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