Date: Mon, 31 Aug 1992 13:18:35 -0700
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Subject: Contexts
> I agree that the Principia logic, like many non-N.D. logics, is quite awful
> in use, and one of the main reasons is that its rules apply only at the
> 'top' level, so that to apply obvious inferences below that level, as in
> your example, requires longwinded, artificial and unnatural processes of
> unpacking and repacking. (As an aside, I also note that our university
> system spends a fair fraction of its effort teaching undergraduates to
> manipulate these artificial notations, cf. any logic textbook. I tried to
> get Rochester to do something different and was suppressed: on asking why,
> they told me it was good mental discipline to put the kids through. Like
> Latin in England.)

There are many disciplines that can strengthen the mental faculties,
but at least Latin has practical applications.  While you were at
Rochester, it's too bad that you hadn't crossed the lake to visit
Don Roberts at the U. of Waterloo.  He has been teaching introductory
logic using Peirce's graphs and then teaching Principia notation at
the end of the semester as an alternate version.  Interesting point:
Roberts and another instructor each taught a section of introductory
logic with Roberts using P's graphs and the other instructor using
a standard textbook.  At the end, all students took the same final
exam, which was completely in Principia notation.  And Roberts's
students had a higher average, with their best scores on proofs.
See Roberts (1973) op. cit. for a brief discussion.  (Note: If you
study Latin, you can say things like op. cit.  But I don't know
anyone who ever dropped a Principia-style proof into a conversation.)

> However, that was not my main point, but to look more pragmatically at what
> could be done on machines right now. And to return to 'contexts', I think
> you are contributing to the confusion surrounding this term. You say that:

>> I use the word "context" in a very narrow sense -- I mean it as nothing
>> more than a notation for packaging a collection of graphs. ....  A [context]
>> could contain all the world's knowledge or it could contain just one simple
>> atom.

> but a few lines later we get a shadow of the more exotic idea again:

>>  For such systems [CYC-ish], it is important
>> to analyze the permissible operations for moving information in and
>> out of various contexts (i.e. packages), reasoning within one of those
>> packages, and then exporting an answer to another package.

> But this notation doesn't preserve any structure whatever, as you have just
> said: it gives complete freedom to move anything in and out of these
> scope-contexts.

Yes, but as we have agreed, the current literature is replete with a
mixture of ideas that haven't been sorted out.  What I was trying to do
is to distinguish the following three notions:

 1. Notation:  I use the term "context" to mean a box for holding a
    collection of propositions (in my system, the propositions are
    stated by graphs, but you could adapt the ideas to any notation --
    even to sentences in ordinary English).

 2. Classification of uses:  Once you have the notion of context as a
    box or package, then you can start to talk about the application of
    the propositions inside the box:  Are they being used to describe
    something, or are they merely being mentioned (cf. Quine's use-
    mention distinction)?  If they are being used for some purpose,
    then what is the purpose?  Can you give a complete list of all
    possible uses?  Or at least a set of guidelines for recognizing
    a use if you see one?  Or other guidelines for recognizing a mere
    mention instead of a use?

 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."

> There is a claim here: that there is a significant idea of a 'context',
> which is something which plays a nontrivial role in complex tasks of
> large-scale knowledge representation. This idea, or rather collection of
> ideas, is new and now being gradually got clear by McCarthy, Guha and
> others. But its not a notational idea. It could probably be realised in
> just about any logical notation you like. To identify these semantically
> important CYC-contexts with Pierce's bracket-scopes (or anyone else's way
> of indicating the scope of quantifiers or connectives) misses their point,
> and if taken seriously is likely to be far too restrictive on our semantic
> imagination.

Yes, the notation by itself does not solve the problem, but at least, it
lets you begin to formulate it in a perspicuous way.  In one of my talks,
I showed an example of a birthday party described in conceptual graphs
in which there are 5 boxes that represent contexts:  one for the party
itself, one for a process that occurs at the party, and three for the
components of the process:  a state (candles burning while guests sing
"Happy Birthday"), an event (the birthday person blows out the candles),
and a state (the candles generating smoke).  When you translate that
example into predicate calculus by the operator phi, you get a formula
with 89 pairs of parentheses; 5 of those pairs correspond to the 5 boxes
in the CG form.  The 5 boxes in CG notation are obvious, but trying to
find the corresponding pairs of parentheses is a nontrivial exercise.

> But I expect we agree, really. Its hard to disagree with:

>> a proof procedure that preserves the package structure can
>> be very helpful.

I think we agree on most of the technical issues.  I would just like
to stress the importance of notation:  Without a good notation
for contexts, it is very hard to see what is going on, very hard to
classify possible uses for contexts, and very hard to formulate the
rules for the permissible operations on contexts.

We may agree that the boxes do not solve the problems, but they at least
allow you to formulate them clearly.  And I do think that Peirce should
be credited with making a good start on classifying contexts and even
formulating rules for operating with different kinds of contexts.
Peirce himself credited his graphs with helping him in that regard:
"My Existential Graphs have a remarkable likeness to my thoughts about
any topic....  I do not think I ever *reflect* in words; I employ
visual diagrams, firstly because this way of thinking is my natural
language of self-communion, and secondly, because I am convinced that
it is the best system for the purpose"  (quoted by Roberts, p. 126).

In his notation of 1896, which I described in my earlier note, he used
the ovals only for negation.  But he later developed a system of colors
or tinctures for distinguishing contexts and classifying them.  On p. 94
of Roberts's book, there is a table of 12 tinctures classified in three
groups:  metal for actual states (argent, or, fer, plomb); color for
modal states (azure, gules, vert, purpure); and fur for various
intensional states (sable, ermine, vair, potent).  He also stated
rules, such as

   For the interpretation of a line of identity which extends from metal
   to color or from metal to fur, metal takes precedence:  that is, the
   line does not denote the abstraction (represented by the color or fur)
   but denotes an existing individual to whom the abstraction pertains.

Note:  the term "line of identity" was P's term for what I call a
collection of concepts connected by coreference links (dotted lines or
variables).  This rule means that the end of a line in an actual (metal)
context fixes the referent of an existential quantifier, which may be
referred to in a modal (color) or intensional (fur) context.

Besides using his graphs for talking about other subjects, Peirce
also developed them as a metalanguage for talking about graphs.  He
even stated all the rules of inference for existential graphs in
existential graphs.  Note:  This is quite different from Russell's
use of higher order logic in his theory of types.  Instead, it is
a version of first-order logic where the domain of discourse consists
of the elements of the notation.  In a way, it is comparable to
Goedel numbering; but instead of assigning integers to each element
of the notation, Peirce assigned special squiggles.

For more detail, see Roberts's book.  Even better, see the thousands of
pages of manuscripts in the Harvard archives, many in glorious colors.
You can imagine why publishers in 1906 were loathe to print such stuff.