THEORY OF CONCEPTUAL GRAPHS
CHAPTER 7 – CONTEXTS, MODALITIES AND THEORIES
Basic Ideas
So far we have dealt with isolated small graphs, which is satisfactory when huge numbers of graphs of the same pattern make up a traditional relational database for example. In reality graphs exist as groups or sets and different sets of graphs are related to each other in various ways. This chapter describes the various ways sets of graphs can be related, or put into context.
Sheet Of Assertion
There is one set of graphs in any knowledge base that is made up of the whole of the graphs that exist in that knowledge base. Since we are using the logic of Peirce we must use his terminology and therefore we can say that, for a model of some universe of discourse, every graph is written, or scribed, on the “sheet of assertion”.
The sheet of assertion is finite but potentially huge. The reason that it is finite is that it represents the total contents of a finite mind and the reason that it's huge is that the human mind can store vast quantities of information.
The content of the sheet of assertion must be taken to be self consistent. Everything on the sheet of assertion is true, at least in the particular universe of discourse. If ever a false graph appeared on the sheet of assertion then the whole knowledge base would be inconsistent. New graphs may be written on the sheet of assertion as long as they are true. Such graphs are called “proper axioms”.
The boundaries of the sheet of assertion are shown by braces { }. All graphs are placed between the braces. It is also possible to “fence off” an area of a sheet of assertion with braces and to use this area as a workspace. In other words { { } } <=> { }.
Contexts
We can say that any set of graphs which describe a particular situation is a “context”. The set will be bounded by braces. Contexts may be nested so we might have something like:
{ context1 { context2 } { context3 { context4 } } ... }
However, since the braces only represent an arbitrary area of the sheet of assertion they have no logical purpose and serve to delimit groups of graphs to make them easier for humans to see. The nest of contexts shown above can just as easily be written as:
context1 context2 context3 context4 ...
It is not possible to refer to a context of this kind. For that we need to delve a bit deeper.
Modalities
Modalities, or modal propositions or contexts, are those propositions that are taken as entities in their own right. An example of a nonmodal proposition might be the sentence “John is travelling to London by train.” whereas a modal proposition might be the sentence “Mary believes that John is travelling to London by train.”. In the second sentence the object of Mary's belief is the entire sentence describing John's journey and yet whilst it is true that Mary hold this belief the believed sentence may not actually be true. The second sentence in conjunction with the sentence “John is travelling to London by bus.” is not a contradiction. We now look at two ways of representing modal propositions, starting with Sowa's notation and then looking at our own.
Sowa's Notation
Sowa places the modal proposition into the referent of a “context” or “modal concept” which is a concept with the type label PROPOSITION. The sentence expressing Mary's belief about John can be represented as:
Fig. 7.1
We believe that this notation suffers from semantic difficulties. In particular it is not possible to refer to this concept except by referring to the whole of the referent in exactly the form shown above. Any query system that contained the above graph could not answer general queries about who was travelling or where they were going without the query containing the whole nested context because any other version of the nested context would be a different referent. In a realistic system a nested context is likely to be a huge graph and it is impractical to use such a graph as a referent. Also, the type label does not add any information because all graphs are propositions anyway. More will be said about this in the chapter on the notation of negation.
An Alternative Notation
In the context
of Mary's beliefs it is true that John is travelling to London by
train. Mary has a concept of John's journey. This notion suggests the
means to represent this, and other, modal propositions. The modal
context goes into the type field of a concept to give a “modal
concept”. However, unlike type definitions, there is no
coreference link between the referent of the modal concept and a
concept in the type field. In other words the modal concept's
referent refers to the whole modal context. For example:
Fig. 7.2
This graph shows Mary's belief is represented and the * in the modal concept's referent refers to the whole context nested inside the modal concept's type field. The linear form of this is exactly the same as that of a type definition except that there is no requirement for, indeed a requirement for not, a coreference link from the modal concept's referent to the nested context:
[ WOMAN : mary]->( BELIEF )->[ { [ TRAVEL ]-
( AGENT )->[ PERSON : john ],
( DESTINATION )->[ CITY : london ],
( INSTRUMENT)->[ TRAIN : * ] } : * ].
Of course any concept within a nested context could itself be a modal concept. Thus we could have a graph representing the sentence “Peter thinks that Mary believes that John is travelling to London by train”. The drawing of a display form graph and writing of its linear form are left as exercises!
The Generic Context
The generic context is a generic marker, which represents all the graphs within a context, enclosed in braces: { * }. As with ordinary generic markers it can have a coreference link to another generic marker. The braces indicate that the generic marker stands for a set of graphs and not the name of a set of graphs and that upon instantiation it is the set of graphs that becomes the value of the marker.
Theories
Situations described by modal concepts are theories or hypotheses. Since modal concepts do not contain a coreferent link into the nested context nothing of the nested context is asserted to be true, except that the theory described by it exists. The distinction between the sentences “These statements are true” and “These statements exist” is very important. Contrast this with the type definitions we saw in an earlier section in which there is a coreference link into the definition. Although we wanted the differentiae of the definition to have some hypothetical status we are nevertheless stating the existence of the genus.
It is possible to compute theoretical results from such theories and to compare the results of one with those of another. It is also possible to perform generalisations over several such theories. These computational aspects are discussed elsewhere.
Updated 29th December, 2006