CONCEPTUAL GRAPHS AND PEIRCE LOGIC

CHAPTER 3 – SYSTEM BETA

Basic Ideas

This chapter deals with the adaptation of Peirce's beta rules to the Conceptual Graphs version of Peirce's line of identity, the main feature of System Beta. It goes further than this and also deals with the extension of Peirce's beta rules with respect to type and relation labels. This is new to the Conceptual Graphs adaptation of System Beta and is necessitated by the fact that Conceptual Graphs is a typed logic with type and relation hierarchies.

Line Of Identity

Lines of identity are shown in either of two ways: generic markers and individual markers. Where the identity of an individual is known then its individual marker is placed in all referents of concepts which refer to it. In this way the line of identity can be visualised as a line connecting all these individual markers. Where the identity of an individual is not known then the generic marker is placed in all referents of concepts that refer to it. In this case, if there are more than one such concept, each copy of the associated generic marker must be given a coreference marker so that the line of identity can be identified amongst all others.

The generic marker is a free variable and can acquire values, which may be individual markers or other generic markers. When this happens all copies of the generic marker on the line of identity that are contained within the context of the generic marker will also acquire the same marker.

All this is based on the notion that information may be added to oddly enclosed contexts and removed from evenly enclosed contexts.

Type And Relation Labels

In an oddly enclosed context any type label or relation label may be replaced by any of its subtypes or subrelations. This is a consequence of the fact that in an oddly enclosed context information may be added at will and the replacement of labels with “sublabels” does just that. It is not necessary that the referent of a concept that has acquired a more specialised type label conforms to the new type label since the whole context is false anyway.

In an evenly enclosed context any type label or relation label may be replaced by any of its supertypes or superrelations. This is a consequence of the fact that in an evenly enclosed context information may be removed at will and the replacement of labels with “superlabels” does just that. It is not necessary that the referent of a concept that has acquired a more generalised type label is tested for conformity to the new type label since any referent automatically conforms to all supertypes of its type.

The Rules

• 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 – Graphs: Any graph may be removed from any evenly enclosed context.
  Lines of Identity: Any line of identity fully contained in an evenly enclosed context may be removed, any evenly enclosed individual referent may be replaced by a generic marker, any evenly enclosed coreference marker may be removed.
  Labels: Any evenly enclosed type or relation label may be replaced with any of its supertypes or superrelations.

• Insertion – Graphs: Any graph may be drawn in any oddly enclosed context.
  Lines of Identity: A line of identity may be drawn between any appropriate graphs contained within the same oddly enclosed context, any oddly enclosed generic marker may be replaced by any individual marker, any oddly enclosed generic marker may be given a coreference marker.
  Labels: Any oddly enclosed type or relation label may be replaced by any of its subtypes or subrelations.

• Iteration – Graphs: Any graph may be copied into the same context or any cut that is nested at any depth in the same context as the graph.
  Lines of Identity: Any line of identity may be extended to any graph which is, or could be, an iteration of the graph from which the line of identity originates, any generic marker that could be the result of iteration of another generic marker may be given any coreference marker associated with the original generic marker, any individual marker that is given to a generic marker can be copied to all coreferent copies of the generic marker that are contained in the same context or any context nested within that of the original generic marker.

• Deiteration – Graphs: Any graph g which is a copy of a graph in the same context or any context in which g is nested may be removed.
  Lines of Identity: Any line of identity may be removed from any graph which is, or could be, an iteration of the graph from which the line of identity originates, a coreference marker may be removed from any generic marker that could be the result of iteration of some other generic marker, any individual marker that could be the result of iteration of another individual marker may be replaced with a generic marker.

With these rules we have a system capable of the whole of first order logic with equality. Given that we have already seen that we can have graphs of higher orders we also have a complete system of logic for any order whatever. It is also possible, by the use of modal concepts, to axiomatise any modal logic. Nevertheless, Peirce started to develop his System Gamma but it was not complete before his death. This is the subject of the next chapter.

Examples Of Proofs

Since we have arrived at our complete system of logic we could and should give a number of examples of proof that show, in particular, the way that System Beta deals with lines of identity, type labels and relation labels. This we now do:

Fig. 3.1

This diagram shows a selection of ways in which the evenly enclosed graph on the left may be modified by the rule of erasure. By this rule any information may be removed. This means that any type label may be replaced by that of a supertype and any relation label may be replaced by that of a superrelation. Additionally, any referent may be replaced by a generic marker. There are many other possible erasures that can be carried out. Although not shown, one of these is to remove relations from the graph.

Fig. 3.2

This diagram shows a selection of ways in which the oddly enclosed graph of the left can be modified by the rule of insertion. By this rule any information may be added. This means that any type label may be replaced by that of a subtype and any relation label may be replaced by that of a subrelation. Additionally, any generic referent may be replaced by an individual marker. There are many other possible insertions that can be carried out. Although not shown, one of these is to add extra relations to the graph.

Fig. 3.3

This series of graphs starts with the double cut, as all derivations of tautologies must, and then performs insertions, erasures, iterations and deiterations to follow a possible line of reasoning that ends with the self evidently true statement that if the boy John has a sister Mary and a father Peter then there exists a man Peter.

Updated 31^st December, 2006