[CL] CG: Update to Annex B

[John F. Sowa]

Chris,

In my note to Pat, I mentioned the basic idea
briefly, which I'll state in more detail:

  1. CGIF is a serialized form of a graph, which has
     no intrinsic order.  Therefore, it consists of
     a set of nodes and a set of arcs, where the only
     relationships among the nodes are determined by
     the explicit connections of the arcs.

  2. Like the AI semantic net gang of the 1960s,
     Peirce dabbled with graph logic in the 1880s,
     but didn't have a good way of showing scope.

  3. But in 1896, he hit on the bright idea of using
     an oval enclosure to limit the scope of quantifiers
     and operators such as negation.

  4. Since he only used one quantifier type (universal
     in 1896, but he switched to the dual, existential,
     in 1897), the scope of the any quantifier anywhere
     inside an oval covered the whole oval.

  5. But if you had ovals within ovals, quantifiers in
     outer ovals governed quantifiers in the inner ovals,
     but not vice-versa.

  6. When I adapted Peirce's system to CGs, I squared off
     the ovals, adopted existentials for the core, and
     defined universals in terms of existentials.

  7. Since I allow both universals and existentials within
     any context, I distinguish the scope by giving higher
     precedence to the universals; e.g., if you have the
     set {(Ex),(Ay), (Az), (Ew)} placed anywhere inside
     a context box, the logical effect is the same as the
     linear order:  (Ay)(Az)(Ex)(Ew).

  8. But if you want more levels of nested quantifiers,
     you can nest context boxes within context boxes as
     deeply as you like.

 > I don't understand this at all.  How do you propose to
 > "move all universals to the front" in a CGIF sentence
 > equivalent to "(exists (x) (forall (y) (Foo x y)))"?
 > Do you employ some sort of skolemization...

No.  I just meant that for any context, you sort the
unordered set of nodes:  put all the universal nodes
up front, then the existential nodes, then all the
ordinary concepts and relations, and finally all the
nested contexts.

The idea of having unordered sets of quantifiers governing
an entire area, by the way, is very useful for handling
natural language discourse.  When new sentences come up,
anaphoric references often link up with quantifiers in
preceding sentences, so you want to allow new chunks of
logic to be inserted into one box or another during the
process of discourse analysis.

This does not, of course, magically solve the problem of
resolving any ambiguities, but it gives you a convenient
set of pigeonholes for parceling out the chunks of logic
as you translate them from the input source.

John