**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,

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

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

wouldhappen 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 “ifAis true,Bis true,” such that should we hereafter learn something of which we are now ignorant, namely thatAis true, then, by virtue of this rule, we shall find that we know something else, namely, thatBis 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 whichAis true,Bis true too. The hypothetical proposition may therefore be falsified by a single state of things, but only by one in whichAis true whileBis 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 19^{th} Century seeking an adequate treatment of
the *Hypothetica*l 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 tomydoctrine, 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, orpossible case, orpossible state of things. In its primitive state, that which ispossibleis 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

(1) |
Possibly , p |
i.e. M: p |
M holds at world px iff
y(Rxy & holds at py) |

(2) |
Necessarily ,p |
i.e., L p |
L holds at world px iff y(Rxy
holds at py) |

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

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

, 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:g

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 propositionsas 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

to be true and yet would not know another graphgto be true. We shall naturally express this by Figure 5.h

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

Bfollows after the state of informationA, 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.