CONCEPTUAL GRAPHS AND PEIRCE LOGIC
CHAPTER 4 - SYSTEM GAMMA
Basic Ideas
Peirce's System Gamma was to be a huge system of logic but was unfinished at his death. He had ovals of all sorts of different colours, or tinctures, to represent a wide range of modalities. The range of possible modalities is huge and may be infinite and so maybe Peirce's system was doomed. However, we will consider adding one modality to our logic since it may link into logics that deal with degrees of certainty and truth. This modality is the modality of possibility or necessity, the modality of deontic logic.
Deontic Logic
Deontic logic deals with two notions: that of possibility and that of necessity. Possibility is the quality of some proposition that may be true but is not necessarily so. For instance it is possible that some ball is red. Necessity is the quality of some proposition that must be true and cannot possibly be false. For instance it cannot be false that one's children are younger than oneself. Peirce formulated deontic logic in terms of the operator possibly false which he represented as an oval made of dashes and which he called the broken cut.
With the possibly false operator it is possible to define p is POSSIBLE as possibly false that p is false and p is necessarily true as p is not possibly false.
Notation And Rules Of Inference
Peirce's notation is the "broken cut" - an oval made of dashes. Examples of Peirce's notation for can be found in Chapter 3 of the Peirce Logic section. We suggest that the linear form notation based on this idea, and linked to our notation for negation (~( )), is :( ). Therefore in our notation the proposition that p is possibly false is shown as :( p ). For p is necessary, i.e. p is not possibly false, we have ~( :( p ) ) and for p is possible we have :( ~( p ) ).
The rules of inference on the broken cut were described in Chapter 3 so this will not be repeated. In the linear form the rule of cut conversion allows an evenly enclosed ~( ) to be converted into :( ) and it allows an oddly enclosed :( ) to be converted into ~( ).
Peirce did not allow iteration or deiteration across a broken cut. However, insertion and erasure appear sound. The following, in graphical form, must also be sound:
Fig. 4.1
Although each proof can be completed in two standard steps it makes sense in an efficient system to perform the above conversions directly in one step.
Updated
8th January, 2007