Contexts, lifting, and Peirce

sowa@watson.ibm.com
Message-id: <199209302159.AA00825@quark.isi.edu>
Date: Wed, 30 Sep 92 17:42:43 EDT
From: sowa@watson.ibm.com
To: jmc@cs.standord.edu
Cc: srkb@ISI.EDU, interlingua@ISI.EDU, cg@cs.umn.edu, buvac@cs.stanford.edu,
costello@cs.stanford.edu, guha@mcc.com
Subject: Contexts, lifting, and Peirce

John,

The version of contexts that I have been using has been influenced by
various sources, including your work, Guha's work, Kamp's DRS's, etc.
But the basic approach is derived from Peirce's existential graphs.
His rules of inference were all stated in terms of the conditions for
moving graphs (or formulas) in and out of contexts (which he called
"enclosures").

I'll say more about Peirce below, but I wanted to comment on some of

> The contexts that Guha and Sasa Buvac and I have studied are a lot
> more ambitious than any of the three possibilities you list, and
> I think this greater ambition is appropriate to Interlingua/KIF.

I fully agree that a rich context mechanism is necessary for KIF.
But by giving a few simple examples in my note, I didn't mean to
exclude richer systems.

>  1. A packaging mechanism for enclosing a collection of formulas and
>     allowing them to be named and referenced as a single unit.
>
>  2. The contents of that package, which could be called anything from
>     "quoted formula" to "microtheory".
>
>  3. The permissible operations on the formulas in the package.  These
>     operations could be defined by a set of axioms in a larger package
>     that encloses the one under discussion.

> You choose the packaging mechanism alternative and give the
> following examples....

I didn't consider those three points to be mutually exclusive.  I just
wanted to say that I use the term "context" in the same way as Peirce
used the term "enclosure".  That makes the notion of "context" very
simple, and the richness comes from the kinds of axioms that one might
assume under Point #3.

> None of the six examples relate formulas in the box to formulas
> outside the box.  Our work is entirely based on {\it lifting formulas}
> that relate a formula in an inner context to formulas outside
> the box.

Although my last note didn't give any examples of "lifting rules"
for moving statements in and out of contexts, all of Peirce's rules
of inference for existential graphs are based on such rules.  In my
note to Pat Hayes on August 31st (also copied to these email lists),
I gave an example of such rules in an extended version of Point #3:

>  3. Operations:  Once you have classified all (or at least a few)
>     possible uses, you can state rules for determining the permissible
>     operations with the boxes -- i.e. under what conditions can you move
>     graphs (or sentences) in and out of the boxes, combine them with
>     other graphs by rules of inference, etc.  For example, suppose you
>     have four boxes with the following labels:
>
>     B -- Mary's beliefs.
>
>     K -- Mary's knowledge.
>
>     C -- Commonsense knowledge about the world.
>
>     E -- Esoteric knowledge found in encyclopedias and reference books.
>
>     Then you could formulate rules that allow a reasoning system to
>     move things from K or C to B, but not vice-versa.  You could also
>     have rules that say "If p is in E and Mary reads p, then you can
>     put p in K and B."

I completely agree with the need for lifting rules, but as I said in
my last note, I would prefer to formulate such rules as axioms in the
metalanguage (or meta-meta-meta...language).  In the framework that I
summarized in my note, I said that the universe of discourse for each
metalanguae was the union of the UoD's of all the nested contexts plus
the languages of those nested contexts plus the contexts themselves.
That formulation gives you all the license you need to state a very
wide range of lifting rules.

Of course, such enormous power gives you the ability to do very
undisciplined things.  But it also give you the ability to make up
creative new disciplines for restating older theories in the new
framework.

As an example, I mentioned how you can formulate a theory of modality
by introducing two contexts:  L for the laws and F for the facts.  Then
necessity is defined as provability from the laws, and possibility is
defined as consistency with the laws.  But that only provides a way of
defining simple modes where you have only one modal operator per formula.
It doesn't define the iterated modalities such as "possibly possible"
or "necessarily possibly necessary".  But by going to the metalevel,
you can define axioms that state the lifting rules for moving formulas
in and out of the context L.  The strong axiom system S5, for example,
results from a metarule that prohibits additions or deletions from
the context L of laws.  The weaker axiom systems result from different
constraints on moving formulas between the laws L and the facts F.

Similar techniques can be used to formulate theories about the knowledge
and beliefs of interacting agents who can "tell" things to one another
or generate their own theories about other agents' knowledge and beliefs.

Note about Peirce:  I strongly recommend Peirce's work on existential
graphs as an important source of ideas about contexts.  My notion of
context as a very lightweight "enclosure" came directly from Peirce.
All his rules of inference are variations of different kinds of
lifting rules for moving graphs in and out of different enclosures
under different conditions.  Peirce used colors to distinguish contexts
for different modalities, intensionalities, etc., and he even translated
his rules of inference for existential graphs into existential graphs.

Unfortunately, a lot of Peirce's most interesting work on contexts
is buried in thousands of pages of unpublished manuscripts in the
Harvard library.  The most accessible introduction is the book by
Don Roberts, _The Existential Graphs of Charles S. Peirce_, Mouton,
The Hague, 1973.  As an example, Roberts (p. 84) gives one of Peirce's
graphs that states the rule of excluded middle:

Take any graph g; either g may be placed on the Sheet of Assertion
or g may be placed on a [negative] enclosure which is on the
Sheet of Assertion.

(See Roberts' book, since I can't send Peirce's graph by email.
Conceptual graphs do have a linear form that can be represented
in the ASCII character set, but Peirce's original graphs do not.)

To distinguish different kinds of contexts, Peirce colored his
enclosures when he generalized them for the "tinctured existential
graphs".  Following is a statement by Peirce quoted by Roberts (p. 102):

The nature of the universe or universes of discourse (for several
may be referred to in a single assertion) in the rather unusual
cases in which such precision is required, is denoted either by
using modifications of the tinctures, marked in something like the
usual manner in pale ink upon the surface, or by inscribing the
graphs in colored ink.

As this quote suggests, Peirce's work is not just a historical oddity,
but a gold mine of insights into the nature of contexts and the kinds
of operations that may be performed on them.  Back in 1911, he was
not only thinking about multiple "universes of discourse", but he
was developing a graphical formalism that was capable of relating
multiple universes of discourse in a single assertion.

John Sowa