CONCEPTUAL GRAPHS AND PEIRCE LOGIC

CHAPTER 1 – EXTENSIONS TO NOTATION

Basic Ideas

Sowa adopted Peirce logic as the logical basis of Conceptual Graphs and therefore, by implication, Peirce's notation. Conceptual graphs are not the same as Peirce's graphs because conceptual graphs make different kinds of node, relation and concept, distinct. They also include type labels and relation labels in a more versatile form. However, a Peircean graph can be replaced by an equivalent conceptual graph without compromising the semantics or validity of the proposition it represents.

Negation in Peirce logic is a primitive of the system and is therefore given its own symbol, the cut. For Conceptual Graphs Sowa suggested the use of a monadic NEG relation, with the context to be negated placed in the referent of the concept to which the NEG is attached. We will discuss this form of expressing negation and we will argue that it is not the best way to do it. We will then introduce a more Peircean cut into the Conceptual Graphs notation.

Sowa's Negation

Sowa places contexts in the referent field of a modal concept of type PROPOSITION. The semantics of such concepts has already been discussed and will not be repeated here. To negate the proposition Sowa adds the monadic NEG relation to the modal concept. Therefore the negation of the proposition P would look like:

Fig. 1.1

This suffers from the difficulties of reference discussed previously for modal contexts. We could instead use our own notation for modal graphs, in which case we would have:

Fig. 1.2

This fixes the referential problem and allows lots of other similar propositions to be negated and referred to. There is, however, a more fundamental problem with either of these attempts at a notation for negation. Since in each of these examples the complete graphs are evenly enclose any part of them may be removed by Peirce's rule of erasure. In the second example this could give us:

Fig. 1.3

In other words, legitimate use of one of Peirce's rules has changed the truth value of the modal proposition. This cannot be correct. Sowa gets round this objection by stating that “special” relations, such as NEG, must have their own rules and special cases but this seems to complicate the matter.

Sowa also uses a shorthand notation for negation. He regards the type label PROPOSITION as the default type label for modal concepts and so it may be omitted. He contracts the (NEG)-> relation into either ¬ or ~. With this shorthand we get:

Fig. 1.4

In linear notation this is shown as ¬[ P ] or ~[ P ]. This looks more like Peirce's cut, which it is intended to model, but since it is merely a shorthand for the fuller form its semantics must be assumed to be those of the fuller form. Note that we have not included the braces in the linear form. They are optional as since the boundaries of the negated context are fixed by the concept brackets the braces are redundant.

Peircean Negation

Since we feel that Sowa's notation for negation is not quite what we would like we must provide an alternative. To do this we refer to Peirce's own cut and observe that it is nothing more than an oval drawn around the graphs to be negated. We do exactly the same. This means that we have added a new symbol to Sowa's original set but we feel that this is justified because negation is a primitive of the system and as such requires its own symbol. With this we would draw the following:

Fig. 1.5

In the linear form we might write ( P ) but this would be ambiguous because this would also represent a 0-adic relation. To resolve this we write either of: ¬( P ) or ~( P ). This notation also allows us to define further types of context in the future. In particular we will see this with System Gamma.

Negations may be arbitrarily nested, as with Peirce's notation and since they are not special cases of concepts there is no need to provide extra rules for their treatment.

Updated 30^th December, 2006