THEORY OF PEIRCE LOGIC

CHAPTER 3 - GAMMA

Basic Ideas

System Gamma is all about the representation of the properties of whole propositions and provides a form of modal logic. Peirce produced a complex, and yet incomplete, system whereby he enclosed propositions in oval of various forms (dotted, solid) and colours, or tinctures. Each combination or variant of forms represented some modality or other.

The main difficulty with modal logic is that there are probably infinitely many modes and even if they are finite in number it is not realistic for each to have its own symbol. Nevertheless he provided a form for deontic logic, the logic of possibility and necessity, that we will discuss.

General Modal Logic

Peirce's system gamma is too large to be dealt with here and the reader is referred to the bibliography for more information. However, Peirce himself provided an alternative representation for modal propositions that we can adopt in place of his more elaborate system gamma. These are his contexts, as mentioned in the previous chapter.

For Peirce a context is represented by the enclosure of the modal proposition within a lightly drawn oval, to distinguish it from the, presumably, more heavily drawn cut. He allows a line of identity to be joined to it so that the entire proposition can be referred to as a single unit. In this way the proposition stands for itself and the writing of a proposition within a context does not assert the content of the proposition but just the existence of the proposition. Examples of modal propositions are:

Fig. 3.1

In these examples none of the propositions nested within the “lightly coloured” ovals asserts that their content is true. What is true in each case is the property that is indicated by the label at the other end of the attached lines of identity. In the last case the modal graph is itself nested inside a modal context which means that the graph expressing his belief about Mary's belief is true whether or not Mary really holds the stated belief or whether John likes or hates doughnuts.

This formalism is very powerful and this summary should be adequate to give the idea of how it is used.

Deontic Logic

Deontic logic introduces two new truth values and as such represents something of a refinement or augmentation of one of the primitives of Existential Graphs. Peirce introduced the mode of possibly false. This he represented by the broken cut, which is an oval constructed from dashes. From this primitive he then constructed the two modalities of deontic logic: possibility and necessity. These are constructed from possibly false and the ordinary cut as follows:

Fig. 3.2

The modes of possibility and necessity are shown in standard logic with the symbols  for possibility and  for necessity. If p is possible it is written p and if p is necessary it is written p. The following diagram gives a list of the possible variations of possibility, necessity and falseness:

Fig. 3.3

Rules Of Inference

There is one rule of inference that operates on the broken cut and that is the rule of cut conversion. Cut conversion has two forms depending on the depth of nesting of the cut to be converted.

Evenly Enclosed Cuts

For an evenly enclosed cut that asserts some proposition p to be false then we can assume that if p is false it is possible for p to be false. Therefore we can say that false implies possibly false and that the evenly enclosed ordinary cut can be converted to the broken cut.

Oddly Enclosed Cuts

For a proposition p which is enclosed in a broken cut within an evenly enclosed normal cut say that p is not possibly false. If p is not possibly false it must be true, which is equivalent to its being enclosed in a double normal cut. Therefore the oddly enclosed broken cut can be converted to a normal cut. This is shown below:

Fig. 3.4

The above diagram does not show all possible conversions of even the starting points shown. This demonstrates the complexity of the whole system gamma. Such complexity makes it very difficult to produce computerised systems without the use of very good heuristics.

Updated 28^th December, 2006