# Peirce’s Graphs

Jay Zeman

Department of Philosophy
University of Florida

Over a decade ago, John Sowa (1984) did the AI community the great service of introducing it to the Existential Graphs of Charles Sanders Peirce. EG is a formalism which lends itself well to the kinds of thing that Conceptual Graphs are aimed at. But it is far more; it is a central element in the mathematical, logical, and philosophical thought of Peirce; this thought is fruitful in ways that are seldom evident when we first encounter it. In one of his major works on Existential Graphs, Peirce remarks that

one can make exact experiments upon uniform diagrams; and when one does so, one must keep a bright lookout for unintended and unexpected changes thereby brought about in the relations of different significant parts of the diagram to one another. Such operations upon diagrams, whether external or imaginary, take the place of the experiments upon real things that one performs in chemical and physical research. Chemists have ere now, I need not say, described experimentation as the putting of questions to Nature. Just so, experiments upon diagrams are questions put to the Nature of the relations concerned (4.530).[1]

That Peirce’s Graphs are likely to be a fruitful formalism in the context of Conceptual Graphs is a given; I would like to discuss some of the aspects of EG and of the Mathematical, Semiotical, and Logical thought in which Peirce embedded them. This may enhance the fruitful application of Peirce’s logic in CG, and indeed, may provide a perspective which is helpful in work with Conceptual Graphs in general.

Arguably by 1880 and certainly by 1885 Peirce had developed the algebra of  logic to something we can recognize as equivalent to the Classical algebraic logic of today; he had independently discovered and given his own twist to much of what Frege (1879) had developed across the Atlantic (Zeman 1986). Yet he grew dissatisfied with the algebra of logic, and in the last decade of the last century, he worked out a mathematical formalism which differed greatly from the algebraic, but which was mathematically equivalent to it. Why’d he do that? Wasn’t it enough to invent the standard logic, without going into this graphical stuff? Seeking the answer to this will take us in the direction of understanding the fruitfulness of his approach.

Peirce, who considered himself primarily a logician, had a very clear idea of what he meant by logic:

The different aspects which the algebra of logic will assume for the [logician in contrast to the mathematician] is instructive …. The mathematician asks what value this algebra has as a calculus. Can it be applied to unravelling a complicated question? Will it, at one stroke, produce a remote consequence? The logician does not wish the algebra to have that character. On the contrary, the greater number of distinct logical steps, into which the algebra breaks up an inference, will for him constitute a superiority of it over another which moves more swiftly to its conclusions. He demands that the algebra shall analyze a reasoning into its last elementary steps. Thus, that which is a merit in a logical algebra for one of these students is a demerit in the eyes of the other. The one studies the science of drawing conclusions, the other the science which draws necessary conclusions (4.239).

For Peirce, deductive logic is the study of a process, the process of reasoning necessarily. It is an empirical science, but it is not to be confused with psychology; as he comments,

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Logic is not the science of how we do think; but, in such sense as it can be said to deal with thinking at all, it only determines how we ought to think; nor how we ought to think in conformity with usage, but how we ought to think in order to think what is true. That a premiss should be pertinent to such a conclusion, it is requisite that it should relate, not to how we think, but to the necessary connections of different sorts of fact (2.52).

Logic, then, is not just a science, but a normative science. Note that Peirce speaks of deductive logic as exploring the “necessary connections of different sorts of fact”; his approach here is not to be confused with that of many thinkers on the topic, who imagine that logical truth emerges, somehow, full-panoplied from the head of the thinker, bearing an automatic and unchallengeable normative relationship to fact; this kind of apriorism is strongly at odds with the scientific thought of Peirce. Note him as he comments further upon the relationship between logic and mathematics:

It might, indeed, very easily be supposed that even pure mathematics itself would have need of one department of philosophy; that is to say, of logic. Yet a little reflection would show, what the history of science confirms, that that is not true. Logic will, indeed, like every other science, have its mathematical parts. There will be a mathematical logic just as there is a mathematical physics and a mathematical economics. If there is any part of logic of which mathematics stands in need—logic being a science of fact and mathematics only a science of the consequences of hypotheses—it can only be that very part of logic which consists merely in an application of mathematics, so that the appeal will be, not of mathematics to a prior science of logic, but of mathematics to mathematics (2.247).

I call your attention to two of the features of logic as Peirce discusses it in the above; first of all, that logic is “a science of fact,” and secondly that “Logic will, indeed, like every other science, have its mathematical parts. There will be a mathematical logic just as there is a mathematical physics and a mathematical economics.” This view of logic and of the relationship of logic to mathematics is at odds with what is the opinion of many, probably most, philosophers. The (philosophically) prevalent view is roughly that of Russell and Whitehead (1910), as derived from Frege and Peano; on that view (as Frege put it)

arithmetic would be only a further developed logic, every arithmetic theorem a logical law, albeit a more developed one. (Frege 1964, 107)

(Now, Philosophers may believe that, but very few mathematicians do!) Peirce sees the relationship between logic and math as analogous to that between physics and math—an adequate study of logic demands mathematics as a tool, no less than does an adequate study of physics.

And just as physics is a science of fact, so too is logic. Say What? How can that be? Just as an example, consider the relationship between logic and probability theory (e.g., Zeman 1978a, Foulis & Randall 1972, 1973, Jammer 1974, 340-416). A sample-space is, essentially, a set of evidence, evidence which may be taken as confirming or refuting propositions (in standard logical terminology, we would speak of a confirmed proposition as true, and a refuted one as false). a entails b, then, would mean that the evidence for (confirming) a is included in the evidence for b. Empirical aspects of logic so examined emerge when we ask about the conditions of confirmation and refutation. Classically speaking, no matter how complicated an empirical situation may appear, we may always think of ourselves as employing one and only one physical operation to gather evidence; the Classical Probability Theory of Kolmogorov (1950), in fact, effectively identifies this physical operation with the sample space. And it is precisely this postulation which forces the logic of Classical sample-spaces to be Boolean. But empirical (not a priori) developments around the turn of the century (19th to 20th, that is) forced us to recognize that the Classical view is not general enough—the empirical development was, of course, the advent of Quantum Mechanics, and its upshot for Probability Theory was that there are situations for we cannot make the classical assumption of the refinability of all physical operations for a sample space into one.[2] The result is that not only is the physics of the subatomic different—more general than!—the physics of large objects, but the logic as well is more general (being orthomodular rather than just Boolean).

The empirical aspects of logic might manifest in a number of ways; Peirce saw the empirical in logic emerging in the “experiments on diagrams [which] are questions put to the Nature of the relations concerned” (4.530). There is here an intricate and intimate interplay with mathematics (which, as we have noted, plays a vital role in Peircean deductive logic). Mathematics, the “Science which reasons necessarily,” does its reasoning by diagrams. Creative mathematical work deals with these diagrams, and does so by a process of inquiry involving Abduction, Deduction, and Induction within the domain of these diagrams; indeed, the necessity of inquiring thus within this domain is the origin of  creativity in the area of necessary reasoning. But by changing the slant of our inquiry in the domain of diagrams, we may make it as well the locus of creative work in Logic; the difference is as Peirce stated earlier between what he would call a “Calculus” and what he would call a “Logic”; the Calculus is aimed at getting to a conclusion as rapidly (and of course accurately) as possible: “Can it be applied to unravelling a complicated question?” The “Logic,” while just as interested in accuracy as the Calculus, is not so concerned with the conclusion as it is with how we got there: the Logic “shall analyze a reasoning into its last elementary steps.” But both mathematics and logic operate within the field of diagrams (note, by the way, that “diagram” here is very general—it doesn’t necessarily imply “graphics” or Cartesian coordinates; the formulas used in algebra are diagrams, for example, as are the very numerals which name numbers).

This opens up, by the way,  another area of Peirce’s thought which is integral to the matters we are discussing: the Semiotic.[3] Although the theory of signs has connections throughout Peirce’s logic, we must here advert to some of his classifications of signs, in particular, to how the sign represents its object. The sign as so representing may be

Icon, Index, [or] Symbol. The Icon has no dynamical connection with the object it represents; it simply happens that its qualities resemble those of that object, and excite analogous sensations in the mind for which it is a likeness. But it really stands unconnected with them. The index is physically connected with its object; they make an organic pair, but the interpreting mind has nothing to do with this connection, except remarking it, after it is established. The symbol is connected with its object by virtue of the idea of the symbol‑using mind, without which no such connection would exist (2.299).

Most to the point here are signs considered as icons. The resemblance which constitutes an icon is very general. In fact, the best mathematical characterization of iconicity is in the notion of a mapping. And Peirce has something considerable in mind; note in what follows that he has broadened his terminology regarding the Graphs. A Pheme is a sentence, though conceived as including interrogative and imperative as well as indicative signs (4.538). Thus the “Phemic Sheet” is the Sheet of Assertion, but in a broader sense than in the simple Alpha-Beta Existential Graphs. The Leaf is a sign which might be thought of as a container for Phemic Sheets. Peirce here takes us beyond the simple two-valued logic in the same way that possible-worlds semantics does in the study of modal logic; Peirce’s vision, however is even broader than that of the possible-worlds approach. He comments:

The entire Phemic Sheet and indeed the whole Leaf  is an image of the universal field of interconnected Thought (for, of course, all thoughts are interconnected). The field of Thought, in its turn, is in every thought, confessed to be a sign of that great external power, that Universe, the Truth (4.553 n2).

So Peirce is aiming here at a mapping, an icon, of some of the important features of “mind”; I believe that he was able to carry this project considerably further than has generally been recognized (the text in question is that from which we have been quoting, his Prolegomena to an Apology for Pragmaticism, 1906 (4.530-572)).

Peirce in his work on EG is trying to set up a mathematical logic which will enable the appropriate description and analysis of deductive reasoning; as I have indicated, he was endeavoring here to be quite comprehensive in this effort. Although he is not explicitly aiming his logical endeavors at a technology of reasoning, I must remind you that theoretical physics is not aimed explicitly at a technology to control and manipulate the physical world, either. Physics is an effort to  understand that world. But that understanding has been most fruitful in the generation of such a technology. I would suggest that the theoretical study of EG may well have the same result in the development of a technology of mind. So it seems to me that we could do worse than to follow his efforts and attempt to understand logic as he did, and to take him as seriously in his efforts to construct Gamma graphs as you have in his work on Alpha and Beta. And by the way, I believe that a major part of this must involve the study of his broader thought; and Existential Graphs is only a part of that thought. This study would include, ideally, an examination of his Semiotic, his Phenomenology, and his Pragmatism as well as his mathematical logic.

Before I go into any technical matters concerning the graphs, I would like to address myself briefly to a question I raised earlier: why did Peirce, who had developed a successful algebraic logic as early as 1880 go through the additional effort of developing EG at all? He answers this question himself. In his discussions of EG, he introduces an alternative notation to his Lines of Identity; objects in the universe of discourse may be represented by what he calls selectives as well as by the distinctive Line of Identity. The Selective is more like a conventional variable in algebraic logic, though like the Line of Identity, its quantification is implicit (Zeman 1964, 1967); thus

X is red
X is round

says the same thing as does

 Figure 1

But Peirce preferred the latter Line-of-Identity notation. His reasons for doing so will also be the essential reasons why he preferred the Graphs to the Algebras as a notation for Logic, “the science of [that investigates] necessary reasoning.”

[The] purpose of the System of Existential Graphs … [is] to afford a method (1) as simple as possible (that is to say, with as small a number of arbitrary conventions as possible), for representing propositions (2) as iconically, or diagrammatically and (3) as analytically as possible. … These three essential aims of the system are, every one of them, missed by Selectives (4.563 n1).

In the present context, we can readily see Peirce’s call for simplicity as fulfilled in EG as opposed to ordinary logical notation; this is, I believe one of the things that has made Existential Graphs so attractive in CG work. Iconicity and Analyticity of representation might be considered together; he notes that the analytic purpose of a logic “is infringed by selectives” (and so also by the variables of ordinary algebraic logic);

Selectives are not as analytical as they might be, and therefore ought to be … in representing identity. The identity of the two [X's in the red-round diagram] above is only symbolically expressed. . . . Iconically, they appear to be merely coexistent; but by the special convention they are interpreted as identical, though identity is not a matter of interpretation … but is an assertion of unity of Object …. The two [X's] are instances of one symbol, and that of so peculiar a kind that they are interpreted as signifying, and not merely denoting, one individual. There is here no analysis of identity. The suggestion, at least, is, quite decidedly, that identity is a simple relation. But the line of identity which [is in the lower diagram] substituted for the selectives very explicitly represents Identity to belong to the genus Continuity and to the species Linear Continuity (ibid.).

Peirce goes on to comment on other aspects of iconicity and analyticalness exhibited by important signs of EG. It is very clear that he sees EG as serving the purpose of a Logic far better than does the standard algebraic logical calculus; it aids us to follow the process of reasoning with greater ease and acuity. I note that the aim of CG seems, in large part, to be movement toward the ideal of Artificial Intelligence; the employment of a deductive system fitting Peirce’s norms for a Logic (as does EG) would seem to be far more appropriate for this than the narrower aim of a specific-purpose “Calculus.” So again, it would seem that attention to Peirce’s broader thought is likely to prove most fruitful in the development of Conceptual Graphs.

I now wish to look at a theme in Peirce’s thought which—as do so many of his ideas—anticipates contemporary developments in mathematical logic, and which receives, in his presentations of Existential Graphs, a treatment and a twist which goes beyond what most contemporary logicians have done. Peirce had been concerned with what he called “hypothetical propositions” and their relationship to the de inesse (truth-functional) conditional (see Zeman 1997b) since at least 1880;  in 1885 we find him commenting that

The question is what is the sense which is most usefully attached to the hypothetical proposition in logic?  Now the peculiarity of the hypothetical proposition is that it goes out beyond the actual state of things and declares what would happen were things other than they are or may be. The utility of this is that it puts us in possession of a rule, say that “if A is true, B is true,” such that should we hereafter learn something of which we are now ignorant, namely that A is true, then, by virtue of this rule, we shall find that we know something else, namely, that B is true.  There can be no doubt that the Possible, in its primary meaning, is that which may be true for aught we know, that whose falsity we do not know. The purpose is subserved, then, if throughout the whole range of possibility, in every state of things in which A is true, B is true too.  The hypothetical proposition may therefore be falsified by a single state of things, but only by one in which A is true while B is false (3.374).

This is a theme which we see in Peirce’s logical work for the rest of his life. Here it takes the form of a contrast between “If-then” as what he calls a “Hypothetical,” and “If-then” as truth-functional. The truth-functional If-then is an instantiation, just one concrete instance of the If-then as Hypothetical; the Hypothetical is a general, a universal. While Peirce’s Non-Relative algebraic logic, and later his Alpha Existential Graphs, gives an adequate treatment of the truth-functional de inesse conditional, Peirce the logician spent a great deal of his energy in the last part of the 19th Century seeking an adequate treatment of the Hypothetical If-then and related matters. A central theme here is that of a range of possible situations. Peirce finds at least a partial answer to the logic of the Hypothetical in such a framework in quantification as he develops it in his Logic of Relatives. In 1902 he comments that

In a paper which I published in 1880, I gave an imperfect account of the algebra of the copula.  I there expressly mentioned the necessity of quantifying the possible case to which a conditional or independential proposition refers. But having at that time no familiarity with the signs of quantification which I developed later, the bulk of the chapter treated of simple consequences de inesse. Professor Schröder accepts this first essay as a satisfactory treatment of hypotheticals; and assumes, quite contrary to my doctrine, that the possible cases considered in hypotheticals have no multitudinous universe. This takes away from  hypotheticals their most characteristic feature (2.349),

which is that they are generals, that they represent a quantifiable range of situations of which each instantiation would be a de inesse conditional! Note that the domain with which we deal here isn’t one of people, books, machines, and other such ordinary individuals, but of situations, of what Peirce elsewhere called “States of Information” (see, for example, 4.517). And this will suggest immediately the contemporary semantics of modal logic, which involves domains whose members are usually called “Possible Worlds.”

And here we have, I think, yet another reason that Peirce preferred the Graphs as a notation for logic. His algebras of logic did not offer a medium for the simple, iconic, and analytic presentation of modality, of the realms of possibility and necessity. The Graphs, on the other hand, had some features which made them most suitable to this purpose. Although he experiments with graphical analogs of modal operators (note his “broken cut” (4.515 ff.) which means “possibly not”), his emphasis is on possible worlds; before introducing the broken cut, he writes of what amounts to a “universe of universes” of possibilities and of fact, and

in order to represent to our minds the relation between the universe of possibilities and the universe of actual existent facts, if we are going to think of the latter as a surface, we must think of the former as three‑dimensional space in which any surface would represent all the facts that might exist in one existential universe (4.514).

Clearly, a representation of such a universe might be found in, say, a book of Sheets of Assertion. Peirce did indeed explore such representations. And he does this explicitly, stating that for the Gamma Graphs,

in place of a sheet of assertion, we have a book of separate sheets, tacked together at points, if not otherwise connected. For our alpha sheet, as a whole, represents simply a universe of existent individuals, and the different parts of the sheet represent facts or true assertions made concerning that universe. At the cuts we pass into other areas, areas of conceived propositions which are not realized. In these areas there may be cuts where we pass into worlds which, in the imaginary worlds of the outer cuts, are themselves represented to be imaginary and false, but which may, for all that, be true (4.512).

Again, we have material suggestive of present-day possible-world semantics for modal logic (a source is Zeman 1973) which then must be considered a rediscovery of something that Peirce did a century ago. Peirce worked with the “Book of Sheets” model and variations thereof in a number of locations. As was his wont, however, he also tried out other ways of representing this material. One of the most interesting, and I think, one with a great deal of applicability to Conceptual Graphs, is in the work I quoted at the start of this paper, his 1906 “Prolegomena to an Apology for Pragmaticism.” In this paper, he develops what he calls “Tinctured Existential Graphs.” “Tincture” in this sense is a technical term of heraldry. The designers of coats-of-arms needed a way of representing the appearance of their product in a day when the exact picturing of colors, metals, and furs on paper was impossible, or at least difficult and expensive. Thus the “Tinctures” used in the graphical description of coats-of-arms. Peirce employs the tinctures for similar reasons (lest we think of how benighted those times were, let us reflect on the fact that the use of color is restricted in many publications even today!); the Tinctures were a way of indicating in black and white what could far better be done in color. In fact, we may for present think of the Tinctures as Colors; “Color” was one of the “Modes of Tincture” which Peirce wished to employ; this mode was the mode in which Possibility and Necessity would be dealt with, and this is our prime interest today.

As we have noted, from very early on, Peirce saw “states of information” as values of quantified variables; note that the “quantified subject of a hypothetical proposition” he refers to below might just as well be a “possible world” in the sense of contemporary logic:

the quantified subject of a hypothetical proposition is a possibility, or possible case, or possible state of things.  In its primitive state, that which is possible is a hypothesis which in a given state of information is not known, and cannot certainly be inferred, to be false. The assumed state of information may be the actual state of the speaker, or it may be a state of greater or less information.  Thus arise various kinds of possibility (2.347).

Let us examine very quickly the mechanisms of Possible-Worlds semantics; the familiar treatments of this go back to the well-known work of Kripke (Beginning with Kripke 1959), as well as to that of Prior (Beginning with Prior 1957). The earliest and best-known such systems are what Segerberg (1971) later called “relational” modal logics. Relational modal semantics works with a pair <W,R>; W is a set of what may be called “possible worlds”; intuitively, if x   W, then it makes sense to speak of a proposition, say p, being “true at x” or “true relative to x” or some such; if (as is one common interpretation) x, y   W are “instants of time,” then we can see how p might be true at x, say, but false at y; what “possible world” means intuitively will vary tremendously, depending on the features of the semantics in question, and the correlative concept of modality (possibility and necessity) that goes with it. Now note that in the “instants of time” example, features of the semantics are dependent on how these instants are related. The relation in question is the second member of the pair <W,R>. R is such that Rxy makes sense for x, y   W; in the temporal example, Rxy is most often the relation “y is in the future of x (or is the same as x).” We note that R so construed will have definite properties; it will be reflexive and transitive, anyway. R is commonly called “The accessibility relation”; note that temporal access as construed above will differ from, say, spatial access (which we would probably think of as being symmetrical as well as reflexive and transitive). We then can associate modality with possible worlds by the expressions:

 (1) Possibly p, i.e. Mp: Mp holds at world x iff y(Rxy & p holds at y) (2) Necessarily p, i.e., Lp Lp holds at world x iff  y(Rxy   p holds at y)

A central feature of Modal Logic as interpreted in Relational Semantics is that the meaning of modality—of possibility and necessity—is intimately and precisely linked to the properties of the accessibility relation. Thus, reflexive and transitive access give us a semantics appropriate to the well-known modal logic S4, while adding symmetry gives us the modal logic S5. The modal logic S5 is a limiting case of this type of modality; it is the system in which Necessarily p can be taken to mean simply that p holds in all possible worlds.

Peirce had had the notion of “possible world” well before he got into EG; his development of Gamma Graphs, however, supplied him with the basis of a mechanism for handling the relations between these possible worlds, and so of a treatment of modality which could be integrated with his logic as a whole. In the Beta Graphs, quantification is handled, remarkably, without explicit quantifiers (this in spite of the fact that Peirce was co-inventor of the quantifier). Peirce’s preferred method of handling quantified variables is by the Line of Identity (more generally, by the Ligature). This is not, as we have already noted, because of mathematical deficiencies in the alternative representation—selectives—but because of reasons relating to the representation vis-a-vis Logic: the Line of Identity does a better job of showing us what’s going on with quantified variables. The Line of Identity will have an analog in the realm of possible worlds within the Graphs, but it may be best to start off with a concept of Selective here.

Peirce had experimented with representations of possible worlds as definite individuals—Peircean seconds. We see this in his notion that

in the gamma part of [Existential Graphs] all the old kinds of signs take new forms. … Thus in place of a sheet of assertion, we have a book of separate sheets, tacked together at points, if not otherwise connected. For our alpha sheet, as a whole, represents simply a universe of existent individuals, and the different parts of the sheet represent facts or true assertions made concerning that universe. At the cuts we pass into other areas, areas of conceived propositions which are not realized. In these areas there may be cuts where we pass into worlds which, in the imaginary worlds of the outer cuts, are themselves represented to be imaginary and false, but which may, for all that, be true (5.512)

But as far back as 1880, as we have noted, he was aware that an adequate treatment of the topic required not just examination of definite individual worlds, but of a quantifiable range of possible states of affairs (see 2.349); and he had experimented with representations for such; in discussing one of his graphical modal operators (the “broken cut”) he remarks that

You thus perceive that we should fall into inextricable confusion in dealing with the broken cut if we did not attach to it a sign to distinguish the particular state of information to which it refers.  And a similar sign has then to be attached to the simple g, which refers to the state of information at the time of learning that graph to be true.  I use for this purpose cross marks below, thus:

 Figure 2 Figure 3 Figure 4

These selectives are very peculiar in that they refer to states of information as if they were individual objects.  They have, besides, the additional peculiarity of having a definite order of succession, and we have the rule that from Figure 3 we can infer Figure 4.

These signs are of great use in cleaning up the confused doctrine of modal propositions as well as the subject of logical breadth and depth (4.518).

My suggestion (first made in Zeman 1997a) is that the Tinctures of 4.530 ff. may be regarded as selectives rather than as representations of definite individual possible worlds. A line-of-identity representation is only a short hop away; in fact, we find Peirce making this hop in his continuation of the above:

Now suppose we wish to assert that there is a conceivable state of information in which the knower would know g to be true and yet would not know another graph h to be true.  We shall naturally express this by Figure 5.

 Figure 5

We have a new kind of ligature, which will follow all the rules of ligatures.  We have here a most important addition to the system of graphs.  There will be some peculiar and interesting little rules, owing to the fact that what one knows, one has the means of knowing what one knows—which is sometimes incorrectly stated in the form that whatever one knows, one knows that one knows, which is manifestly false (CP 4.521).

And he develops this even further:

The truth is that it is necessary to have a graph to signify that one state of information follows after another. If we scribe

 Figure 6

to express that the state of information B follows after the state of information A, we shall have

 Figure 7

(CP 4.522).

This last is a version—employing lines of identity (“ligatures”) for states of information—of the rule of necessitation which is a feature of the most commonly studied modal logics.

So we see that Peirce did do explicit graphical work with concepts that are familiar to the contemporary modal logician. The Tinctured Existential Graphs present a medium for the extension of this work. For simplicity, let us think of Tinctures only in terms of colors for now; two areas of the “Phemic sheet” which are the same color are thought of as continuous with each other. We can even picture them as “cross-sections”of a special Line of Identity (just as the “tic-mark” selectives Peirce uses above are like cross-sections of a Ligature for States of Information); the sameness of color of a given tincture ties in with the continuity of that LI—which has to be embedded in dimensions beyond our usual three. This line of identity will then be a quantified variable for possible worlds. The rules for such Tinctures can be worked out if we understand the Beta rules; I do this explicitly in Zeman 1997a. A Tincture as bearer of modality is structured: it actually involves two colors; although they are closely related, as we shall note. From this perspective, a Sheet of Assertion (which, of course, we associate with a “world,” with a locus where propositions can be true or false) has two “sides”: a recto, or true side and a verso, or false side; the verso is “seen” through the Cuts (which Peirce often describes as actually-cut-through-the-paper-and-turned-over). If the recto is a color in the R-G-B model, the verso will be its color complement—a Tincture whose recto is Red <255,0,0> will have a verso of Cyan <0,255,255>. But the pictured Sheet of Assertion will not represent a specific member of a domain of possible worlds; rather, it is a “cross-section” of a continuum (which, of course, would require an extra dimension or more for its representation). The continuum is a Line of Identity (more generally, a Ligature) and the pictured SA is a Selective associated with that line—thus a variable for possible worlds rather than a constant name for a possible world.

And the Tinctures as Ligatures/Selectives operate by the same rules for implicit quantification laid out by Peirce in the Beta Graphs (see Zeman 1997a for the specifics of this). There is more; as we have noted, we must be able to deal with an “accessibility relation” between possible worlds to get the well-known contemporary treatments of possible-worlds semantics. The ability to do this is provided by the colors involved in the Tinctures. In the notation we introduced earlier, we interpret Rxy as meaning that x has access to y; with the tinctures, this would hold iff the recto color for x is <a,b,c> (in the R-G-B model), the recto color for y is <d,e,f>, and all of the following hold: a   d, b   e, and c   f. This gives an accessibility relation which is essentially a partial order (and so would have the Lewis-Modal S4 as its basic system), but which is open to many different variations for specific purposes (of course, White would be the Supremum in this p.o., but recall that each Tincture involves the complement of its recto color, making the basic p.o. of Tinctures a tree (a semilattice).

It seems to me that the Gamma graphs of Peirce as we have been examining them present us with great opportunities for the enrichment of the study of CG. The precise directions that this enrichment will take is dependent on the ingenuity of researchers in the area.