THEORY OF PEIRCE LOGIC

CHAPTER 2 – SYSTEM BETA

Basic Ideas

Peirce's system beta takes system alpha one step further by decomposing propositions so that each entity and each action done by or state experienced by them can be dealt with separately. A system which separates all the entities of a proposition and can reason about them is a first order system. It also allows the statement that two apparently different individuals are actually the same one, and so we have equality. System beta is therefore a version of First Order Logic With Equality and is a very powerful formalism.

System Beta

There are two additions to system alpha that we must consider. The first is relations and the second is identity. A third addition is also worth considering – contexts- although this will not add anything to the rules of inference.

Relations

A relation consists of two types of node: the relation node and the associated entity nodes. In system beta these are very simply shown as words. Each one represents precisely one instance of its type and nothing else. Therefore a propositions such as “There is a cat sitting on the mat.” refers to three things: a cat, the mat and an act of sitting. These might be given the labels CAT, MAT and SIT. In this way the proposition has been split into its components unlike when we met it in the previous chapter where it was taken as a unit.

Splitting the proposition into its parts is not enough because we have lost the connection between the parts. With just the labels CAT, MAT and SIT we do not know whether the cat was sitting on the mat or the mat was catting on the sit. For the humans who chose the meaningful labels it is obvious but the computer will not know. Hence we need another symbol.

Identity

To show the connections of an individual to its relation nodes Peirce used the ligature, also known as the line of identity, which is a line, possibly branched, that joins the individual to each relation that it is playing a part in. This line is a self contained, self identifying line that does not join any other such line (although in complex diagrams it may need to cross another line). Thus the proposition “There is a cat sitting on the mat.” might give us the following beta graph:

Fig. 2.1

In this diagram the left hand line represents the cat and the right hand line the mat. A certain knowledge of the meanings of the labels is necessary but system beta does not provide any syntax to distinguish which of the nodes attached to the line represents the entity identified by the line and which the relation.

Such a graph can be negated by the use of a cut. It is also possible to construct graphs with nested cuts where an individual is referred to in both the oddly enclosed and evenly enclosed areas. An example might be:

Fig. 2.2

In the graph a branch of the animal's line of identity crosses the inner cut. There is no restriction on where, in a true graph, a line of identity can go, except that it cannot join onto a cut.

Quantifiers

In the discussion on system alpha we introduced the existential quantifier but we can now be a bit more precise and say that a line of identity is an existential quantifier or existentially quantified variable. Similarly, since a line of identity uniquely identifies a single individual, we can say that the line is existentially quantified.

The graph in Fig. 2.2 can be read as shown above but if this reading is true then so is the following: “It is false that there exists an animal sitting on the mat which is not a cat” or “All animals sitting on the mat are cats.”. The use of the words “false that there exists” and “all” here indicates the presence of the universal quantifier. The universal quantifier allows us to say things about whole classes of objects, events, situations, etc.. In standard logic the universal quantifier is given the symbol ".

Universal quantification is not a primitive of system beta because it is constructed from the existential quantifier and negation. However it is a very important concept. It allows us to formulate rules about the world and is the basis of all rule based systems. Peirce's original Entiative Graphs used universal quantification as one of its primitives but natural languages refer more to examples and instances and so existential quantification is more natural.

Contexts

Peirce provided a notation for contexts which are boundaries around entire propositions. The context can be joined to a line of identity and will thus say something about the individual represented by that line. However, Peirce also said that a context does not assert the enclosed graphs but merely asserts some statement, made by relations attached to the context, about those graphs. Since contexts of various sorts are the subject of system gamma we will not say any more about them here except that Peirce's notation for them was a “lightly drawn oval”.

Primitives

For system alpha we defined the following primitives: the blank sheet of assertion, existential quantification, conjunction and negation. We must also add:

• The line of identity – identifies and shows the relationships between an individual and its properties. The line of identity is the existential quantifier.

This primitive converts system alpha into system beta, which is equivalent to first order logic with equality. Although not previously mentioned Peirce introduced a further primitive, the point of teridentity which is a point to which three ligatures may be attached. Generally two of the ligatures already exist, as a single ligature. The point of teridentity is a syntactic device that allows a branch to be made in a ligature. Any line of identity may contain arbitrarily many points of teridentity arbitrarily close together. All this simply means that a line of identity may be arbitrarily branched.

The Rules

The addition of the line of identity to system alpha in order to produce system beta requires the augmentation of the rules of inference to accommodate them. We will do this largely without justification since this can be found elsewhere and the justifications for the alpha rules in the previous chapter can be extended to include the line of identity. The only thing we will say is that the addition of a line of identity adds information and the removal of a line of identity removes information.

Since propositions in system beta are split into their component parts each part may be treated as a separate proposition in its own right, represented by its own graph. There is therefore no need to augment the rules to deal with this because the alpha rules apply to the parts of a proposition in the same way as to the whole original proposition. Therefore in the following summary of the rules the term “graph” refers to any partial or “complete” graph.

• 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.

• 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.

• 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.

• 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.

Strictly speaking the rules that add or remove part of a line of identity should be formulated in such way that a segment of the line of identity should be added or removed by adding or removing a point of teridentity before or after the addition or removal of the ligature. Since a point of teridentity cannot exist without the third ligature attached to it its existence depends entirely on the existence of the ligature.

We will conclude this chapter with a selection of proofs using system beta:

Fig. 2.3

This small selection of proofs gives the idea. Note that iteration of a node such as CAT does not imply iteration of the line of identity. The two operations are entirely separate and without the iteration of the line of identity the graph produced by the iteration on CAT is completely meaningful. Note also that the line of identity could have been iterated before the CAT node as long as when the CAT node was iterated it was attached to the “dangling” iteration of the line of identity.

Similarly, had the example shown deiteration, the deiterations of the line of identity and the CAT node would have been completely separate and could have occurred in either order.

Updated 28^th December, 2006