C. S. Peirce and the Quest for Gamma Graphs

Peter Øhrstrøm
Department of Communication
Aalborg University
Langagervej 8
9220 Aalborg Øst
e-mail: poe@hum.auc.dk


Summary: This paper deals with some aspects of the history of C. S. Peirce's Existential Graphs. In his construction of this graphical method during 1896-1897 Peirce was motivated by some interesting considerations regarding diagrammatical reasoning. In the present paper this motivation will be briefly discussed. Whereas Peirce managed to bring the graphical systems of Alpha and Beta Graphs to a high degree of perfection, his treatment of the Gamma Graphs remained tentative and unfinished. Some of his suggestions can also be shown to be mistaken. It is, however, clear that Peirce with his Gamma Graphs was aiming at a complicated system in which one can deal with a number of interesting problems regarding various kinds of modality. Peirce, himself, was well aware of the shortcomings of his treatment of the Gamma Graphs, and he mainly concentrated on the formulation of a Gamma agenda for his followers.

As Mary Keeler and Christian Kloesel [1996] have pointed out Charles Sanders Peirce struggled for over twenty years with his system of graphical logic called Existential Graphs. The question of dating the invention is nevertheless interesting. At least, Peirce himself paid considerable attention to it. He stated that he had invented his Existential Graphs in January 1897, although he did not publish these new ideas until October 1906 [CP 4.618] in The Monist. It should be mentioned that this exact dating of the invention of Existential Graphs is a bit questionable. In a lecture on "The Logic of Relatives" given on February 17, 1898, he stated: "Finally about two years ago, I developed two intimately connected graphical methods which I call Entitative and Existential Graphs." [Ketner, p.151] Given that the former statement is from a paper published in 1908, it seems reasonable to see the latter statement as the more reliable in the sense that he probably was working with the problems already in 1896 (and perhaps even earlier). One very likely scenario appears to be as follows: During 1896 Peirce was very interested in diagrammatical reasoning, which led him to the formulation of the basic ideas involved in Existential Graphs. A first version of the new theory may have been ready by January 1897. Actually, he wanted the system to be called "the Existential System of 1897", since he in this year wrote an account of the system and offered it for publication to the editor of The Monist, who nevertheless did not want to publish it in the form in which it appeared at that time [CP 4.422].

It is obvious that the invention of the Existential Graphs can be seen as a natural continuation of Peirce's work with Venn Diagrams and Euler Circles. His interesting improvements of these classical methods have been carefully studied by Eric Hammer, who has convincingly emphasised the importance of the fact that Peirce provided "syntactic diagram-to-digram rules of transformation for reasoning with diagrams" [Hammer, 1995]. It is likely that it was these efforts which made him aware of the great power of diagrammatical reasoning.

Peirce had earlier worked intensively with the establishment of an algebraic approach to logic. But during 1897 he came to prefer the diagrammatical approach as clearly superior when compared with the algebraic approach to logic [CP 3.456]. He later came to consider diagrammatical reasoning as "the only really fertile reasoning", from which not only logic but every science could benefit [CP 4.571]. According to Peirce the use of diagrams in logic can be compared with the use of experiments in chemistry. Just as experimentation in chemistry can be described as "the putting of questions to Nature", the experiments upon diagrams may be understood as "questions put to the Nature of the relations concerned". [CP 4.530] In this way diagrammatical reasoning may be seen as some sort of game. In fact, as Robert W. Burch [1994] has argued, Peirce regarded the system of Existential Graphs as inseparable from a rather game-like activity which is carried out by two fictitious persons, the Graphist and the Grapheus. The two persons are very different. The Graphist is an ordinary logician, and Grapheus is the creator of the universe of discourse, who also makes continuous additions to it from time to time [CP 4.431].

Working with his "Application to the Carnegie Institution" for support for his research in logic (dated July 15, 1902) Peirce established the following interesting defintion of diagrammatical reasoning:

In the same draft Peirce maintained that "all necessary reasoning is diagrammatic". However, Peirce was not quite clear on the question of generality of his diagrammatical approach. In a letter to Lady Welby, dated March 9, 1906, Peirce made clear that there is some limitation to the system of Existential Graphs. He found it hard to see how, for instance, a piece of music or a command from a military officer could be represented in terms of graphs. On the other hand, in the introducing statement in his Monist paper (1906), he was very firm on the question of generality: One possible way of explaining this tension is that although Peirce felt sure that all kinds of human reasoning can be represented in terms of Existential Graphs, he understood that no representation can be perfect or complete. It cannot "directly exhibit all the dimensions of its object, be this physical or psychic." [MS 291; 1905] Every representation will show its object only in a certain light, i.e. from a certain perspective. For this reason, Peirce maintained that no sign can be "perfectly determinate" [CP 4.583]. But on the other hand he stressed that the system of Existential Graphs is "a rough and generalized diagram of the Mind" [CP 4.582].

For the understanding of the Peircean position the notion of a diagram obviously becomes fundamental. According to Peirce a diagram should mainly be understood as "an Icon of intelligible relations" [CP 4.531]. It is, however, very interesting that Peirce appears to have related diagrams of Existential Graphs to the passage of time. He pointed out that Existential Graphs can represent "propositions, on a single sheet, and arguments on a succession of sheets, presented in temporal succession" [Roberts 1992, p.662], and that the system of Existential Graphs may "be characterized with great truth as presenting before our eyes a moving picture of thought." [MS 291; 1905]

Roberta Kevelson has argued that Peirce's reference to time plays an important rôle as "intensification in his explanation of modality in the Existential Graphs" [Kevelson, 1987, p.102]. This is true for all Existential Graphs. As Mary Keeler [1994] has argued, Peirce's Existential Graphs can naturally be viewed as an instrument for investigating semiotic continuity. Like Robert W. Burch's analysis [1994] of Existential Graphs in terms of game-theoretical semantics, Mary Keeler's historical investigations emphasise that the Existential Graphs should be understood in relation to time, human experience and communication. These temporal aspects are particularly relevant for the Gamma Graphs, which can in many cases be interpreted in terms of temporal logic. After all, in the Peircean context time should be viewed as one of the most important sorts of modality.

The Rules of the Gamma Graphs

Peirce's so-called Alpha graphs correspond to the ordinary propositional calculus, whereas his Beta graphs correspond to first-order predicate calculus. In what he called 'The Gamma Part of Existential Graphs' [CP 4.510 ff.], he put forth some interesting suggestions regarding modal logic. He wanted to apply his logical graphs to modality in general - that is, to use them for representing any kind of modality. However, he was aware of the great complexity in which a full-fledged logic involving modal and temporal modifications would result. This is probably one of the reasons why Peirce's presentations of the Gamma graphs remained tentative and unfinished. In the following I intend to explain some of the problems he was facing and suggest some ideas regarding the possible continua-tion of his project.

According to Don. D. Roberts, Peirce began working with the Gamma Graphs already in 1898, and both in 1903 and 1906 he dealt intensively with them [Roberts 1992]. As Peirce himself pointed out, the system of Gamma Graphs is "characterized by a great wealth of new signs; but it has no sign of an essentially different kind from those of the alpha and beta part." [CP 4.512] The most important new graphical elements in the Gamma Graphs are the `broken cuts' and `tinctures' corresponding to various kinds of modality. However, as long as only one kind of modality is involved we can do with just one new graphical element, the broken cut. This is in fact what Peirce himself did in the main parts of his "Apology for Pragmaticism" in the Monist (1906). Here he presented a logic based on four rules (or "Permissions") for Gamma Graphs, that involve exactly one kind of modality (corresponding to one tincture). In the following I shall discuss these Gamma rules.

Peirce considered a "Phemic Sheet", which refers to a universe of discourse. The two sides are recto and verso, respectively. Graphs on the recto are posited affirmatively and graphs that are "negatived" are scribed on the verso [CP 4.555]. It is important to note that writing something on the verso is not only a matter of negating the graph in question, it may also involve some kind of modality corresponding to one of the tinctures. Peirce did not specify what kind of modality he had in mind in his formulation of the rules.

I shall use readings like "it is possible that not p" and "it must be that p" corresponding to the modal expressions M~p and Np and corresponding to the following graphs:


Peirce formulated the first two permissions (rules) for Gamma Graphs in this way:

Peirce argued that the rule of Deletion and Insertion is evident, and that the rule of Iteration and Deiteration "will be seen instantly by students of any form of Logical Algebra" [CP 4.566]. As an illustration Peirce showed how one can deduce the proposition "Every catholic must adore some woman" from the proposition "There is a woman, whom every catholic must adore". His deduction is carried out in the following way. The premise corresponds to this diagram:


Using iteration one may deduce:


According to Peirce, this may by deletion be transformed into the diagram, which is to be proved:


One may wonder how the last step is carried out. It appears that it does in fact involve two steps. First, a deletion of a part of a line of identity according to the rule. This leads to the diagram:


In order to arrive at the above conclusion, it might seem that the following extra principle is needed:

This principle is well known from the logic of Beta Graphs, i.e. for unbroken cuts (See [Øhrstrøm et al. 1994]). However, one can establish a generalisation of this rule to broken cuts. In the above example the retraction of the ligature may be proved from the above diagram by deleting parts of the ligature from the recto area and by adding the unattached line of identity is to the diagram. It should be noted that this addition of the unattached line of identity is just one of the two axioms in the system of Beta Graphs.

The result of these operations is the diagram:


From the above diagram one can then get the concluding diagram by deiteration and deletion of the unattached line of identity.

It is very likely that Peirce had this combination of the rules in mind, but the series of operations is not as simple and straightforward as one may imagine from the reading of Peirce's text.

Expressed in modern modal logic Peirce's deduction takes us from

to the conclusion The crucial step in this deduction appears to be based on the truth of the following theorem: or equivalently where N is short for ~M~, and 'a' is an arbitrary predicate. As explained in [Øhrstrøm et al. 1994], this theorem is provable in any standard modal logic with standard quantification theory. This may also be expected from the fact that the rule of retraction of ligatures can be proved from the two graphical rules mentioned above, taken together with the rules of the Beta Graphs.

The third rule mentioned by Peirce in his 1906 paper is the following one:

Peirce demonstrated how this rule together with the two other rules mentioned above can be used in order to establish a proof of the modal syllogism: The premises correspond to the following two diagrams:


By iteration we easily find:


By the rule of insertion:


By the rule of deiteration we find:


By collapse of the two cuts:


By the rule of deletion, we find:


The rule of the double cut is, however, very problematic. If the double cuts in question may be any combination of two broken or unbroken cuts, then a number of rather unattractive propositions can be proved. One may, for instance, prove the theorem:

The reason is that given the unrestricted rule of the double cut one can from the diagram:


deduce the diagram:


It is likely, judging from his formulations in [CP 4.567], that Peirce was considering the possibility of excluding the rule of the double cut from his system, or at least giving it some secondary status. It is understandable that he gave the rule a place in his system, since it may appear difficult to imagine any usable alternative. But he is not likely to have realised that the use of an unrestricted rule of the double cut would in fact undermine the modal distinctions themselves in the Gamma Graph System. For this reason it is obvious that in order to save the Peircean project we have to restrict the rule of the double cut to unbroken cuts only, and construct some additional rules which can account for reasonable deductions such as the one in the above syllogism. Harmen van den Berg [1993] has argued that it would be natural within the Peircean context to introduce the following rule:

Using this rule one can transform the diagram


into the diagram


Using the rule of the double cut (on unbroken cuts) this leads to the diagram:


This means that the rule of modal conversion together with the restricted rule of the double cut can in fact do the job of establishing a proof of Peirce's modal syllogism.

The fourth rule which Peirce introduced in his 1906 paper is the following one:

This rule seems to be based on the following observation, which may be called `the existential disjunction theorem': Peirce illustrated the fourth rule by means of the implication from the proposition ÒThere is a man x and a man y, such that if x is bankrupt, then y must commit suicideÓ to ÒThere is a man x, such that if x is bankrupt, then x must commit suicideÓ. Stated graphically, it follows from the rule applied to this example that the proposition corresponding to the diagram:


implies the proposition corresponding to the diagram:


According to Charles Hartshorne and Paul Weiss, Peirce in a letter to F.A. Woods in 1913 expressed scepticism as to the universal validity of the fourth permission [CP 4.569, note]. In the National Academy of Science meeting in Washington, April 1906, he had even called this permission "quite out of place and unacceptable" [CP 4.580] On the other hand, in the same context he also stated that he found himself unable to refute the rule, but he suggested that we may have to reject the idea that "every conditional proposition whose antecedent does not happen to be realized is true." [CP 4.580].

From a modern point of view, Peirce's reaction regarding the fourth rule appears to be quite understandable. The motivating theorem mentioned above (from [CP 4.569]) can be formulated in terms of symbolic (and non-modal) logic in the following way:

It is interesting that a similar disjunction theorem holds in modal logic with any modal operator N: This means that the fourth rule holds provided that the cuts mentioned in the above formulation are all unbroken as indeed they are in Peirce's own example. A generalisation to broken cuts would correspond to claims like the following equivalence: It is easy to see that this equivalence would presuppose something like Barcan's formula, i.e. It is, however, well known that this equivalence can not in general can be assumed as valid, although it is valid in some systems, for instance the system S5 (with standard quantification theory).

Peirce pointed out that an implication from the proposition "there is a man every dollar of whose indebtedness will be paid by one man", corresponding to the diagram


to the proposition "there is a man every dollar of whose indebtedness will be paid by the same man", corresponding to the diagram


is invalid. With respect to such an implication the fourth rule cannot be applied, since the smallest cut which wholly contains a ligature connecting the two graphs `owes' and `will pay' is the same as the smallest cut that contains those two graphs. In terms of modern symbolic logic this difference between the two propositions comes out as

It is obvious that because of the occurrence of the universal quantification the existential disjunction theorem cannot be used here. (Incidentally, the preoccupation with pecuniary examples may well stem from Peirce's financial plight at this time!)

Towards a Logic for Gamma Graphs

The logic of the Gamma Graphs is the logic of modality. Peirce suggested a multi-modal approach to the Gramma Graphs. In fact, he proposed a system involving 12 different tinctures in the diagrams. It is not clear why he suggested exactly that number, but it is important to stress the fact that he wanted the tinctures to stand for various kinds of possibility, intention, and actuality. The specific understanding of a diagram involving tinctures will depend on the relevant interpretation.

Although Peirce suggested a multi-modal system, there is - as far as can be seen from the printed sources - no indication in his works of a study of the ways in which various kinds of modality may interact with each other. It seems that this is just one of the many themes which he left for his followers to pursue. I have shown elsewhere [Øhrstrøm 1996] how the Peircean ideas of tinctures and graphs can give rise to a tense logic with two basic tense operators.

Peirce himself realised that there is a lot to do in order to bring the logical system of the Gamma Graphs into a satisfactory form:

Although Peirce's account of the logic of Gamma Graphs was incomplete, and although it contained some inaccuracies and mistakes, it did in fact define a paradigm for further research. Towards the end of his life he was very much aware of this aspect. Regarding his Existential Graphs he wrote: "I am now working desperately to get written before I die a book on Logic that shall attract some good minds through whom I may do some real good." [Semiotic and Significs; quoted from Keeler 1994].

Based on the Peircean ideas it is rather obvious what kind of rules one should study in order to extend a modern system of Existential Graphs, such as the one developed at Loughborough University [Heaton 1994, Kocura 1996], to cover not only Alpha and Beta Graphs, but also Gamma Graphs.

According to Sun-Joo Shin [1994], one of the main reasons why the Existential Graphs never gained great popularity among logicians is the fact that the method is rather complicated. Taking for instance the above mentioned 4th rule for Gamma Graphs into consideration, Sun-Joo Shin's view seems appears reasonable. On the other hand, it is likely that the valid version of this rule as well as other Peircean rules can be reformulated in simpler ways that can make them easier to use. There may even be hints of how to do this in the still unpublished papers by C. S. Peirce. This kind of work seems to be very relevant in the modern context of computer science. As John Sowa has argued for many years, and as clearly demonstrated in a new book edited by Gerard Allwein and Jon Barwise [1996], the increasing use of visual displays suggests the introduction of tools for logical reasoning with diagrams.


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