CONCEPTUAL GRAPHS AND PEIRCE LOGIC
CHAPTER 2 – SYSTEM ALPHA
Basic Ideas
This short chapter discusses the combination of Conceptual Graphs with Peirce logic. Since Peirce's System Alpha deals with whole propositions there is very little to say about the adaptation of Conceptual Graphs to System Alpha.
System Alpha And Conceptual Graphs
There is one aspect of a proposition written as a conceptual graphs that differs from a proposition written as a character string or other unit representation. That is, whilst the contents of the nodes of the conceptual graph are not available for performing operations with the joined nodes of a conceptual graph can be separated and the proposition broken down to a degree. Thus the proposition “There is a cat sitting on the mat”, which as an English sentence, is just a character string, could be written as a conceptual graphs in one of the following forms:
Fig. 2.1
If a proposition were represented as either of the first two graphs in this diagram then there would be nothing that could be done with them but if represented as the third graph, a simple graph, then the example above could be altered in various ways. If the graph is evenly enclosed parts of it can be removed, as long as the remainder is well formed, and if oddly enclosed then additional parts can be added.
The Rules
We summarise the rules, including minor modifications to allow the erasure and insertion of partial graphs.
Double Negation – a double cut can be drawn around or removed from any empty part of the sheet of assertion or from any graph or graphs.
Erasure – any graph or any part of a graph may be removed from any evenly enclosed context.
Insertion – any graph or any part of a graph may be drawn in any oddly enclosed context.
Iteration – any graph or any part of a graph may be copied into the same context or any cut that is nested at any depth in the same context as the graph.
Deiteration – any graph or part of a graph, g, which is a copy of a graph in the same context or any context in which g is nested may be removed.
A “part of a graph” is either a relation node plus all attached arcs or a concept node plus all attached arcs. Where the removal of either of these types of part leaves a relation node with insufficient arcs then that node plus all attached arcs must be removed. Where a concept is added it may be left as a single, unattached concept. Where a relation node is added it must include all necessary arcs attached to all necessary concepts, which may need to be added as single concepts beforehand.
With the ability to add or remove parts of graphs the following is a valid proof in System Alpha with conceptual graphs:
Fig.
2.2
Since the referent of the oddly enclosed [ CAT ] concept is generic the referent of the evenly enclosed iteration of it is also generic and therefore the two concepts are not coreferent. In System Alpha there is no means of making the link. Had the oddly enclosed [ CAT ] concept been instead an individual concept then that individual referent would have been iterated with the iterated concept and the link preserved but in this case the resulting graph would only apply to one particular cat. System Beta solves such problems.
Updated
30th December, 2006