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Outline of trikonic: Diagrammatic Trichotomic
City University of New Yorkgaryrichmond@rcn.com
New York, USA
Copyright © bySpringer-Verlag
Abstract.C. S. Peirce’s research into a possible applied science of whatever might be trichotomically analyzed has here been developed as trikonic – that is, diagrammatic trichotomic. Such a science (and its proposed adjunct tool) may have significant implications for the work of testbed collaboratories and other virtual communities concerned with pragmatically directed inquiry and knowledge representation.
In a fragment entitled "Trichotomic" Charles S. Peirce, the originator of the science, introduces trichotomic as "the art of making three-fold divisions" [EP I: 280]. This new applied science rests upon Peirce’s triadic categorial distinctions, which he outlines in many places, for example, in a letter to Victoria Lady Welby:
"Firstness is the mode of being of that which is such as it is, positively and without reference to anything else.
Secondness is the mode of being of that which is such as it is, with respect to a second but regardless of any third.
Thirdness is the mode of being of that which is such as it is, in bringing a second and third into relation to each other." [CP 8.328]
Working from suggestions in the "Trichotomic" manuscript and in such related writings of this period as "A Guess at the Riddle" and "One, Two, Three," the author has developed and here introduces trikonic, |>*k, method and proposed tool for trichotomic analysis-synthesis in an internet age, directed especially towards catalyzing the evolution of virtual communities.
Triadic divisions figure in nearly every discipline within Peirce’s purview of theoretical science, prominently in his scientific and coenoscopic philosophy, and most especially within his phenomenology and logic interpreted as semeiotic (a tripartite science – theoretical grammar, critical logic, and theoretical rhetoric – all directed to the analysis of semiosis, or "sign action" in one form or another). Additionally, for a very few years (1885-1888), Peirce made an intensive study of a practical science allied to, but significantly different from, the theoretical ones which tended to dominate his interests. Although he returned to theoretical researches in the years following, the development of an applied science of trichotomic – facilitating inquiry into anything which might be so analyzed – was seen by him as essential to the growth of a scientific humanity, a change which he imagined it might help usher in. In one of the drafts of "A Guess at the Riddle," he comments that his fully developed trichotomic analysis would constitute "one of the births of time." [CP 1.354 Fn 1 p 181]
Yet, even in those communities which value Peirce’s contribution to the development of a scientific philosophy, research into an applied trichotomic science remains largely untouched by contemporary knowledge representation workers, whereas from one perspective it would seem to constitute a quasi-necessary element in the realization of Peirce’s pragmatism in collaboratory testbeds and other virtual projects which incorporate this pragmatic perspective. It will be argued here that there are many places where |>*k should help in facilitating certain researches, conversations, and decision-making processes of vital interest and value to a virtual community.
2. |>*k: diagrammatic facilitation of trichotomic analysis-synthesis
Peirce held that all thinking was essentially diagrammatic and that perhaps the most important diagrams were visual ones [CP 2.778]. The principal purpose of |>*k is to facilitate diagrammatic thinking concerning trichotomic relations of importance to a given community’s research interests, making such diagrams available for (virtual) group manipulation, commentary, etc. It is certain that not all things can be trikonically analyzed – yet, in consideration of the Peircean reduction thesis– which holds that all relations of more than three elements are reducible to triadic relations, but triadic relations are not reducible to dyadic and monadic relations [CP 3.483, 1] – there is probably not much of potential importance to a professional community which could not be incorporated into such an analysis, especially since provisions are also made within |>*k for important dyadic (and other –adic) relations where these need be considered.
Regarding such pragmatic contexts as testbed collaboratories one might argue that, given the self- and hetero-critical nature of effective collaborative work and the decision-making processes needed in preparing for and accomplishing the system- and tool-centered inquiries needed for the achievement of virtual community goals, |>*k could prove to be of some considerable value in catalyzing movement towards what has been called a Pragmatic Web [4, 5], in which definitions are provided of the context and purpose of information that might be handled by the continuously developing Semantic Web. It is maintained here that a tool for electronic trichotomic could expand pragmatic analysis even beyond those domains which Peirce considered in his trichotomic, and exactly in the direction of contemporary collaboratory needs in the projected suite of electronic tools that will support the growth of such virtual communities (and in consideration of, for example, proposals of methods and tools, discovery of the extent of participant agreement in principle and practice regarding them, identification of specific areas of consensus and disagreement, establishment of criteria for making decisions on issues relating to personnel, etc.).
In short, it will be argued that there are significant places within a pragmatically focused community where a trichotomic analysis-synthesis would prove valuable. Such analysis could be facilitated by |>*k as applied science and electronic tool for analysis-synthesis concerning matters involving such three-fold divisions as those to be considered below. While the tool itself is still in the planning stage, the terminology, symbol system, diagrammatic approach, etc. have been fairly fully fleshed out. However, only an outline and overview of |>*k will be presented here.
Despite the necessary omission for now of a further explication of the purpose, design, structure and content of the proposed trikonic tool, a primary goal of |>*k is the application of the theory and diagrammatic system to virtual inquiry and collaboratory practice, so that >*k is proposed as a small, but possibly integral component of a quasi-necessary suite of principles/meta-principles, tools/meta-tools necessary for realizing the goals of collaboratories modeled on Peircean principles.
3. |>*k analysis and Peircean category theory
As its theory and practice inextricably interpenetrate, it is perhaps not possible to introduce |>*k as diagrammatic system without at the same time introducing fundamental categorial distinctions and relations pertaining to it. This shall, therefore, quasi-necessarily be the approach followed below: basic trichotomic relations and terminology will be introduced as the system itself is explicated. The paper introduces a few convenient notational conventions associated with |>*k beginning with the trikon symbol itself, diagramming trikonic relations in relation to the three Peircean categories.
|>, the trikon symbol, represents a trichotomic relationship showing the three categorial elements of any object under consideration and, in its vectorial part (to be discussed later), the six possible paths through the three categorial elements.
Three Peircean categories are at the core of trikonic analysis-synthesis:
1ns = firstness
|>3ns = thirdness
2ns = secondness
When there is no vectorial movement, one should read the three categorial divisions around-the-trikon as if occurring all-at-once, each necessitating each other, no one more fundamental than the others. They stand in a genuinetrichotomic relationship, where genuine is used as a technical term opposed to degenerate (in the mathematical sense of these expressions).
The trikon symbol, |>, which rather resembles the outline of a "forward" button on an electronic device (suggesting, perhaps, process or evolving structure or time’s arrow, etc.), is devised to hint at the characteristic of each categorial element within Peirce’s three Universes of Experience
In such mature work on theory of inquiry as his "The Neglected Argument," Peirce characterizes the Universes of Experience in this way:
Of the three Universes of Experience familiar to us all, the first comprises all mere Ideas, those airy nothings . . . their Being consist[ing] in mere capability of getting thought, not in anybody's Actually thinking them. . .The second Universe is that of the Brute Actuality of things and facts. I am confident that their Being consists in reactions against Brute forces. . . The third Universe comprises everything whose being consists in active power to establish connections between different objects, especially between objects in different Universes.
Such is everything which is essentially a Sign – not the mere body of the Sign . . . but, so to speak, the Sign’s Soul, which has its Being in its power of serving as intermediary between its Object and a Mind. Such, too, is a living consciousness, and such the life, the power of growth, of a plant. Such is a living constitution – a daily newspaper, a great fortune, a social "movement." [CP 6.455]
Upon such conceptions, the categories are placed around the trikon as follows:
and 2ns into relationship in futuro, so "to the right")
1ns, ideas ("airy nothings," mere possibility of being actualized, so, "floating" at the top)
|>3ns, habits (tending to bring 1ns
2ns, events (brute actions and reactions, existential and earthbound, so, "sinking" to the bottom of the diagram)
These, in turn, yield Peirce’s three Universal Categories:
From an even more abstract perspective, one can also see the categories as representing first, something, second, other, third, that which brings something and another into relationship:
what is in itself
|>bringing into relation (mediate)
correlate to another
There are naturally lively associations for each of the categories, of which this selection, found in the "Trichotomic" manuscript (EP I: 280), is representative: "First is the beginning, that which is fresh, original, spontaneous, free. Second is that which is determined, terminated, ended, correlative, object, necessitated, reacting. Third is the medium, becoming, developing, bringing about."
In addition to the universal categories, there are equivalent existential categories:
The colon after "thought" above signals that that element will itself now be trikonically analyzed.
So, within thought, one can identify three logical modalities. These are more or less equivalent to the logical quantifiers:
may be == vague,
|> will necessarily be if. . . == general, "
actually is == specific, $!
There are indeed many other important authentic trichotomic relations operative in virtually all the theoretical sciences and, no doubt, even beyond these in the vast trichotomic semeiotic universe Peirce conceives.
4. Triadic semeiotic
Yet, as important as these singular trikonic relations are, an even more significant set of relations, at least from the standpoint of what might be called an evolutionary pragmatism, is made up of the various groups and complexes of trikons – "trikons of trikons," so to speak, such as those which figure prominently in the analysis of semeiotic grammar. Here trichotomic analysis begins to take on a richness and suggestiveness which hint at the evolutionary movement possible within a semeiotic reality which Peirce explicates in his metaphysical writings (see, for example, RLT). Radically different from such dyadic semiotics as, for example, Saussure’s semiology and much of the twentieth century semiotics deriving from it, Peirce’s semeiotic posits three essential elements within an integral whole, the semeiotic triad abstracting sign, object, and interpretant, showing relationships holding between the sign and the object as it is represented for an interpretant sign (not necessarily an actually existing interpreting person: an interpretant could, for example, be a computer, or a future interpreter, etc.). This trichotomy exemplifies the "life of the sign," its vitality and semiotic movement. The |> always represents a genuine semeiotic relationship:
From the standpoint of Peirce’s categorial perspective, this suggests that the sign itself is a mere possibility (e.g., "cat" could be "chat" or "gato" or "gatto"), representing a brute actuality "(feline-being-in-the-world"), to some possible or actual interpretant, (e.g., a computer program generating an ontology of concepts related to "feline being"). In section 5 below, we will see how the semeiotic triadic "always-already" implies semiosis, or "sign action" as soon as vectorial movement is considered. But, for now, analyzing the sign/object/interpretant more abstractly and, as it were, "statically," the three sign elements above yield a nonadic trikonic group, the 9-adic diagram of possible relations of these three in consideration of certain categorial constraints – three tripartite "parameters" for every sign.
Combining these to form embodied signs types, where each sign has an actual (triadic) relationship to the object, the sign in itself, and the interpretant, and in consideration of what Liszka has termed the "qualification rule,"Peirce constructs the well known 10-adic Classification of Signs. In other words, the "parameters," as they are referred to here, are not completely independent of one another, and the combinatorial upshot is not 27 kinds of sign but ten. Here Peirce analyzes ten classes of embodied signs in what he characterizes as the order of involution. It is a matter of some debate, but this researcher would suggest that a complete |>*k analysis strongly supports the notion that the 9-adic diagram presents only the types of relationships possible for yet to be embodied sign classes. In a word, the nine sign "parametric" choices do not themselves represent embodied signs, whereas the ten classes do; i.e., not the nine, but Peirce’s ten classes of sign, are the exhaustive set of sign classes fully determinate in terms of this level of classification.
These ten classes themselves naturally form three "trikons of trikons" with an additional single central |> placed within, further illustrating the deep trichotomic nature of semeiotic as Peirce conceived it.
5. |>*k process: vector analysis-synthesis
At the heart of |>*k or, better, as its methodological advance guard, are six vectors.
In a passage that points both toward trichotomic vector analysis AND the "reduction thesis," Peirce writes.
. . . .Now the triad . . . has not for its principal element merely a certain unanalyzable quality sui generis. It makes [to be sure] a certain feeling in us. [But] the formal rule governing the triad is that it remains equally true for all six permutations of A, B, C; and further, if D is in the same relation at once to A and B and to A and C, it is in the same relation to B and C; etc. [CP 1.471]
In |>*k these six vectors (or, directions of movement through the trikon) factor significantly, all of them being authentic permutations of logical – and in some cases, temporal – paths of the relations holding between the 1ns, 2ns, 3nsof the object under consideration. For instance, Parmentier  has noted that a vector of determination (whereas the object determines a sign for an interpretant) and a vector of representation (whereas the interpretant creates a sign to represent an object) move in opposite directions.
2/1/3|> …for an interpretant.
The object determines…
…creates a sign (for example, a diagram)…
3/1/2 |> An interpreter (say, a scientist)…
…of a complex object.
The former, the vector of determination, figures prominently in many of Peirce’s semeiotic analyses. Indeed, one finds descriptions of all six vectors ("orders") scattered through Peirce’s voluminous writings, including analyses of pairs of vectors. For example, in a discussion of the categories in "The Logic of Mathematics" paper [CP1.417-520] Peirce analyzes Hegelian dialectic as following a vector of order, whereas thesis (1ns) is followed by antithesis (2ns) thus leading to synthesis (3ns). He opposes to this (Hegelian) "order of evolution" an inverse "order of involution or analysis" (vector of analysis) by which, indeed, the three universal categories are in a sense themselves derived, Peirce first positing 3ns which involves 2ns which in turn involves 1ns. Peirce held that Hegel’s order, as logically significant a it undoubtedly is, yet gains its generative power and effective meaning only when seen in the light of the categories. So, despite the heuristic value of Hegel’s dialectical insight, Peirce’s own analysis of evolution as such follows a different order (vector of process) whereas chance sporting (1ns) leads to patterns of habit-formation (3ns) culminating in, say, some actual structural change in an organism (2ns). The last of the six paths (vector of aspiration) represents in a sense the unique character of specifically human evolution, that is, the evolution of consciousness which, from a Peircean perspective, is dependent upon critical self-control.
While it is expected that the six expressions used here to name the vectors will for the most part hold, they are nevertheless offered as mere temporary expedients until the development of a trikonic tool will allow for such semeiotic inquiry as might bring about agreement concerning optimal terminology in vectorial analysis. There would indeed seem to be considerable creative potential for a semeiotic which could integrate the generative power of interpenetrating and transactional vectorial movement.
The thematization and exploration of the six vectors (especially as they interpenetrate in hierarchies of constraint, dependence/independence, etc.) aims at bringing new light, and ultimately a more systematic treatment, to some of the difficult issues which arise especially in semeiotic analysis. In short, the vector issue invites treatments involving graphically logical aspects of dependence and constraint, correlation, as well as the "living" reflection of the categories by the semeiotic triad. The proposed trikonic tool is meant to facilitate such treatment in forms suitable and editable for purposes of ongoing dialogue and comparison among many participants in any given collaborative inquiry (or activity).
While it is certainly possible that an individual analyst with nothing more than pen and paper might benefit from using Peirce’s trichotomic theory and the diagrammatic system here outlined, yet the thrust of |>*k is towards empowering virtual communities employing it. It is anticipated that the approach will eventually contribute to an increase in shared understanding by the members of a given virtual community in any number of matters of interest and importance to it. Agreement in any matter so considered would in some sense represent the synthesis of the trikonic analysis in which not only various trikonic elements and relations (and complexes of trikons embedded in yet other trikonic relations, and strings of such relations, etc.) are identified, but especially as the results of these analyses are directed to the concerns of virtual communities in their attempts to reach their goals collaboratively. Of course, not all six vectors need be considered in any given analysis (or synthesis), although it is likely that several or all are operative in some ways and to some extent in most complex semioses.
Consider, for example, the forms of inference as given in the syllogism. Associating the syllogism’s rule, case, result with the three categories results in the following: rule is naturally connected to 3nsas expressing lawfulness, case (as existential case) with 2ns, and result with the idea, character, feeling, image, possibility, etc. which is being considered, so 1ns :
result (these beans are white)
|>rule (all the beans from this bag are white)
case (these beans are from this bag)
The three types of inference:
abduction (representation vector):
this handful of beans that I find on the table are white;
3/1/2 |>All the beans from this particular bag are white,
this handful of beans are possibly from this bag.
deduction (analytical vector):
these beans will necessarily be white.
3/2/1|> All of the beans from this bag are white,
these beans are drawn from this bag;
induction (determination vector):
all these beans are white;
2/1/3 |> all the beans in the bag are probably white.
These beans are drawn from this bag,
In a further step these three inference patterns taken together can be seen to have their own vectorial relationship when considered as elements of an inquiry (for example, an experiment conducted on scientific principles) where, in Peircean inquiry at least, the following three steps occur: (1) an abduction – in experimental science, retroduction to an hypothesis – gives rise to (2) the deduction of what would necessarily follow if the hypothesis were correct, and upon which an experiment may be constructed, followed by (3) the actual inductive testing of the hypothesis. (Such inquiry expresses, nearly to perfection, the process vector).
abduction (the case is possible)
1/3/2|> deduction (the result is necessary)
induction (the rule is probable)
The point here (and by way of example) is that trikonic vector analysis in consideration of such complex semeiotical processes as inquiry—with its characteristic sequence of three stages following the vector of process (with each of these three stages being a unique form of inference having its own characteristic vectorial path)—might contribute to the advance of both theoretical and applied logic of inquiry.
6. Applications to testbeds and other pragmatically informed virtual communities
The principal purpose of |>*k vector analysis is to explicate this sort of vectorial movement both theoretically and as it might appear in any actual semiosis, most notably those activities directed towards the pragmatic realization of complex virtual communicative projects. Trikonic analysis – especially in conjunction with such a consensus-building instrument as Aldo de Moor’s GRASS tool  – could lead to a kind of trikonic synthesis catalyzing the further evolution of virtual community development. How all this is to be accomplished can hardly be considered here. However, one can say with de Moor that it ought to be a "user-driven" model, participants together legitimatizing the specifications concerning the meaning of the information they consider together, the collaborative processes to be employed within their own communities, and so forth. 
Collaboratories built on pragmatic principles are virtual communities where users ought to play an active role, especially as many significant decisions need be made concerning, for example: the role of each member, the nature of and timing of each member’s work in relation to the other’s and to the project(s), the selection of tools to be employed for use individually and together, as well as individual and group reflection upon the process as a whole (including the structure and goals of the collaboratory itself). It is maintained here that all of these aspects of collaboratoriality would be facilitated and catalyzed by employing a trikonic approach to testbed inquiry, practice, decision making, etc. Wherever Peircean pragmatic logic, principles and methods factor emphatically within a virtual project, |>*k promises to be of significant intellectual and social benefit. In such collaboratories, pragmatic guidance would seem to be invaluable in support of virtual community development, for it is at the pragmatic (and not the semantic) level that such conversations, inquiries, and decisions are made. |>*k, facilitating such pragmatic guidance, could have a significant role to play in whole-system development.
Possible applications of |>*k to such virtual community development may be suggested by considering two examples of vectorial analysis as they relate to collaboratory activity. For example, the process vector might be applied strategically to testbed operations employing pragmatic approaches to inquiry and experimentation (see section 5). In this case (presented here somewhat abstractly), the retroduction of hypotheses of interest and concern to a collaboratory, such as a proposed tool being considered for use, 1ns, leads to the explication of the implications for testing as this relates to the suite of tools already being used, as well as to personnel to be involved, financial considerations, etc. 3ns, followed, perhaps, by the decision to test, and the actual testing (under agreed upon conditions), 2ns.
At a later stage, and in further consideration of virtual community development, a given virtual collaborative community, 2ns, having selected and tested a certain complete suite of tools, etc., 3ns, finds that it begins to more and more fully achieve its collaborative goals, 1ns.
These unit, abstract analyses can, however, only hint at the rich possibilities for employing |>*k in actual pragmatic inquiries within virtual communities.
7. Summary, conclusion and further prospects:
Recapitulating some of the basic themes and goals of |>*k:
Theoretically, it is meant to explicate:
- basic trichotomies.
- trikons composed of trikons
- trikons in series
- trikonic vector analysis
Pragmatically, and following upon the theoretical analysis, especially vector analysis, it is meant to support:
- collaborative inquiry
- testbeds and other collaborative projects
- consensual evolution of consciousness
As previously noted, although not everything under the sun can be trikonically analyzed, yet significant dyadic (and other –adic) relationships are also dealt with in a chapter of |>*k not considered in this brief introduction. |>*k may be capable of providing rather complex and complete analyses in areas of vital concern to virtual communities.
Peirce is considered by some to be the most original and creative genius that America has produced. His contributions – to phenomenology, to logic as semeiotic, and to a rather large number of other sciences – have been seminal in conceptual structures research. Peirce’s contributions can be seen to include especially his pragmatism, semeiotic, critical and graphical logic, most notably existential graphs as transformed into conceptual graphs by John Sowa.
In recent work with Arum Majumdar on analogical reasoning , Sowa has again demonstrated how Peirce’s logical ideas are a veritable prerequisite for progress towards NLP and in establishing the conditions necessary for the further progress towards a Semantic Web.
Similarly, Mary Keeler has insisted that normative science and, especially, pragmatism as method be tested in collaboratories, paradigmatically in a project appropriately involving Peirce’s own work, the projected digitizing of his manuscripts as the PORT project.
Both Keeler and Sowa have stressed how quintessential the theory and practice of Peirce’s pragmatism is to the possible success of virtual community projects as are represented by such communities as ours. Both these researchers have allied virtual community development directly to the evolutionary, and ultimately, co-evolutionary pragmatism of Peirce. It is anticipated that Peirce’s category theory – upon which the |>*k project is based – will begin to demonstrate its value to research, and to the other creative work of virtual communities, especially that relating to knowledge representation.
It is certain that in order to be effectively employed within virtual communities |>*k requires additional principles, notably methodological meta-principles, as well as advanced tools for realizing these principles in the interest of consensus building. Seen from a pragmatic perspective, collaboratories will need participants interested in such inquiry, tool building, tool and principle testing, virtual community building, etc.
And while |>*k might contribute to bringing about diagrammatically necessary and sufficient conditions for trikonic analysis-synthesis of any topic or issue amenable to such analysis, and although trikonic theory/practice may perhaps help generate themes and topics for significant discussion within the community, |>*k itself does not directly bring about agreement concerning the subject of any given analysis. This will require a companion tool permitting the individual-communal observation of changing patterns of the extent of agreement (also significant disagreement), but especially highlighting deep consensus when it does occur, in particular inquiries and experiments. Additionally, the results of the research will need to be documented in group reports summarizing the experimental findings (in some cases including the experimental process) for the purpose of developing action plans and the like.
Promising research from diverse disciplines has begun to converge on the problematic of collaboratory development. For example, building on the work of Peter Skagestad--who first distinguished Artificial Intelligence (AI) from Intelligence Augmentation (IA), showing that Peirce provided a theoretical basis for the latter much as Alan Turing had for the former--Joseph Ransdell  has argued that new applications of IA should perhaps be informed in their design by Peirce’s notion that all thought is dialogical. |>*k means to contribute to and to catalyze exactly that manner of dialogue.
Such meta-theoretical principles as the assurance that the participants in a virtual collaboratory will be involved in specifications relating to all matters appropriate to the role(s) of each participant within the collaboratory (determining the principles underlying the community, its processes, methods, structure, tools considered and chosen together and ensemble, and so forth). One promising application of such meta-theoretical principles to group report writing, the GRASS project , might be suitably adapted for charting progress towards agreement in trikonic analyses. In conjunction with such meta-principles and tools |>*k would hope to play its small part in the furthering of Intelligence Augmentation.
[Note: The author is indebted to Ben Udell for creating the graphics illustrating this paper, for his help in editing the text, and for his "parametric" language in the analysis of the 9-adic sign division.]
[CP] Collected Papers of Charles Sanders Peirce, 8 vols. Edited by Charles Hartshorne, Paul Weiss, and Arthur Burks (Harvard University Press, Cambridge, Massachusetts, 1931-1958).
[EP] The Essential Peirce, 2 vols. Edited by Nathan Houser, Christian Kloesel, and the Peirce Edition Project (Indiana University Press, Bloomington, Indiana, 1992, 1998).
[RLT] Reasoning and the Logic of Things: the Cambridge Conferences Lectures of 1898. Edited by Kenneth Laine Ketner (Harvard University Press, Cambridge, Massachusetts, 1992).
1. R. Burch (1991) A Peircean Reduction Thesis. Texas Tech University Press.
2. A. de Moor (1999) Empowering Communities: A Method for the Legitimate User-Driven Specification of Network Information Systems. Ph.D. thesis, Tilburg University, the Netherlands.
3. A. de Moor. GRASS (Group Report Authoring Support System): Arena for Societal Discourse. http://grass-arena.net/
4. A. de Moor (2002). Making Doug's Dream Come True: Collaboratories in Context. In Proc. of the PORT's Pragmatic Web Workshop, Borovets, Bulgaria, July 15. 12
5. A. de Moor, M. Keeler, and G. Richmond (2002). Towards a Pragmatic Web. In Proc. of the 10th International Conference on Conceptual Structures (ICCS 2002), Borovets, Bulgaria, Lecture Notes in Artificial Intelligence, No. 2393, Springer-Verlag, Berlin.
6. M. Keeler. Hegel in a Strange Costume: Reconsidering Normative Science for Conceptual Structures Research. In Proc. of the 11th International Conference on Conceptual Structures (ICCS 2003), Dresden, Germany Lecture Notes in Artificial Intelligence, No. 2746, Springer-Verlag, Berlin.
7. M. Keeler. Using Brandom’s Framework to Do Peirce’s Normative Science: Pragmatism as the Game of Harmonizing Assertions? In Proc. of the 12th International Conference on Conceptual Structures (ICCS 2004), Huntsville, Alabama, Lecture Notes in Artificial Intelligence, No. 3127, Springer-Verlag, Berlin.
8. J. J. Liszka, A General Introduction to the Semeiotic of Charles Sanders Peirce. Bloomington and Indianapolis. Indiana University Press. 1996.
9. K.A. Parker, The Continuity of Peirce’s Thought. Nashville & London: Vanderbilt University Press, 1998.
10. R. J. Parmentier,. "Signs' Place in Medias Res: Peirce's Concept of Semiotic Mediation." Semiotic Mediation: Sociocultural and Psychological Perspectives. Ed. Mertz, Elizabeth & Parmentier. 1985.
11. G. Richmond (with B. Udell), trikonic, slide show in ppt format of presentation at PORT Workshop, ICCS 2004 Huntsville, Alabama.http://www.cspeirce.com/menu/library/aboutcsp/richmond/trikonicb.ppt
12. J. Sowa and A. K. Majumdar, "Analogical Reasoning" In Proc. of the 11th International Conference on Conceptual Structures (ICCS 2003), Dresden, Germany Lecture Notes in Artificial Intelligence, No. 22746, Springer-Verlag, Berlin.
13. A Thief of Peirce: The Letters of Kenneth Laine Ketner and Walker Percy. Ed. by Patrick Samway, S.J. University Press of Mississippi, 1995.
14. J.Ransdell (2002) The Relevance Of Peircean Semiotic To Computational Intelligence Augmentation.http://www.cspeirce.com/menu/library/aboutcsp/ransdell/ia.htm
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END OF: Gary Richmond, "Outline of trikonic : Diagrammatic Trichotomic "
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