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Dr. Frithjof Dau

Peirce's Existential Graphs --- Readings and Links

This site contains readings and links on the existential graphs, including his work on the logic of relations (relatives, as he called them) of Charles Sanders Peirce (1839-1914). As far as I have read them, I have written some short comments on the articles and books. Of course these comments are flavoured by my very personal and mathematical view on existential graphs (and sometimes much too short), but nonetheless, I hope they can help you select the right readings for you. Moreover, I hope that none of my critical comments will offend any of the authors.

I intend to make this list -in the long run, of course- as complete as possible. So If you know more stuff which should be put on this list, please let me know.

Last, not least: Do not hesitate to inform me about misspellings etc. I do know that my english is far from perfect. And if you find any dead links or other problems, please let me know, too.

2005/11/21: I have created a `small' (1.8 MB) gif with an animated proof (a moving picture of thought, as Peirce would say) of Leibniz' splendid theorem, carried out with Peirce's Alpha graphs. You can find it here: Leibniz' Praeclarum Theorema

Changes on this website:

  • 2007/1/01:
    • Congratulations to Frederik Stjernfelt that his thesis "Diagrammatology: An Investigation on the Borderlines of Phenomenology, Ontology, and Semiotics" has been published by Springer!
  • 2007/01/28:
    • I am pleased to be invited for a special issue on Peircean Diagrammatic Logic for Semiotica, Elsevier, which will be published in 2008. My contribution is about ligatures in existential graphs. It can be found on this site at on my publication list. Moreover, I added another paper on my own on constants and functions in existential graphs.
    • I have two broken links to former sites with scans of Peirce's original manuscripts. If anybody can help me finding the pics which had been on these sites (see below), pleas let me know.
    • I added some remarks to the book of Hammer.
  • 2006/09/12: I uploaded a new version of my own treatise. At the moment, I am very busy and do not have enough time to maintain this site carefully. I hope this will change next year (2007). Nonetheless, I want to draw your attention to three recent papers which had been published in the proceedings of ICCS 2006:
    • Joachim Hereth Correia and Reinhard Pöschel finally found a proof of Peirce's Reduction Thesis in a much more general setting than that of Herzberger or Burch. Their proof is far from being trivial and needs much more sophisticated arguments than Herzberger or Burch need. It is a really great result.
    • Frederik Stjernfelt investigates in "Two Iconicity Notions in Peirce's Diagrammatology", as the title says, the iconicity in Peirce's graphs to an great extent.
    • Last and least :-), I published a paper about interesting proof-theoretic results in Peirces calculus for Alpha graphs.
    All papers offer insights into Peirce graphs which has so far not been unfolded.
  • 2004/08/23
  • 2004/06/20

Books - exclusively on Existential Graphs

  • Markus Arnold:  Einführung in die kontextuelle Relationenlogik.
    Diploma Thesis, Darmstadt 2001 (Advisor: Rudolf Wille)

    This german thesis is based on Burch's book and Wille's paper on the algebra of relations.
    I think if you want to get a copy of this thesis, you should contact Rudolf Wille.

  • Robert W. Burch:  A Peircean Reduction Thesis: The Foundations of Topological Logic.
    Texas Tech University Press, 1991.

    This book is a detailed and precise elaboration of the so-called Peircean Algebraic Logic (PAL). PAL is very closely related to Peirces logic of relatives. I consider the book rather technical and hard to read, but it provides detailed insights into the nature of existential graphs, particularly in the roles of the so-called teridentity and hypostatic abstraction. The main focus of the book is to provide a proof for Peirce's thesis that triadic realtions are neccessary and suffiecient for constructing all relations.
    It is not appropriate for beginners on existential graphs; you should already be familiar with the graphs if you read the book.

  • Frithjof Dau:   Mathematical Logic with Diagrams - Based on the Existential Graphs of Peirce.

    As the author of this website, it is not very surprising that I am writing on a treatise about existential graphs on my own ... This treatise is still in preparation. I dare to supply a preliminary version of it which contains the core of my work. As the title says, I focus on a mathematical elaboration of Peirce's Alpha and Beta graphs. Nonetheless, the treatise contains a chapter about my understanding why Peirce considers his EGs to be the luckiest find of his career, some general considerations about the formalization of diagrams, and a chapter with some remarks to the books of Roberts, Shin and Zeman.

    Update: 12. September 2006 I uploaded a new version of the treatise. The content is finished, it will only be subject of further proof-reading. If you find any mistakes, flaws, misprints etc, please let me know.
    compressed pdf-file (3.2 MB)   uncompressed pdf-file (4.5 MB)

  • Kenneth Laine Ketner:   Elements of Logic: An Introduction to Peirce's Existential Graphs.
    Texas Tech University Press, Lubbock, Texas, 1990.

    So far, I don't know this one ...

  • D. D. Roberts:   The Existential Graphs of Charles S. Peirce.
    Mouton, The Hague, Paris, 1973.

    Probadly the most popular secondary literature on existential graphs. This book offers a comprehensive description of the whole system of existential graphs. Particularly, the gamma part of existential graphs is described to a large extent (far more than the broken cuts, which is the main focus of most papers or books which deal with gamma). Obviously, Roberts worked through many, many manusscripts of Peirce, and he justifies his elaboration with a lot of quotations.
    From a mathematical point of view, this book is unfortunately insufficient. Roberts does not provide any (technical or mathematical) definitions, and he relies solely on the graphical representations of graphs. Nonetheless, this is definetely a outstanding work. Together with the books of Shin and and Zeman, this is a "standard reading".

    More detailed critics can be found here.

  • Sun-Joo Shin:   The Iconic Logic of Peirce's Graphs.
    Bradford Book, Massachusetts, 2002.

    I enthusiastically read this book within two days. It is an elaboration of Peirce's alpha- and beta-graphs, including improved translations ("readings") to propositional and first order logic. I liked most the philosophical backgrounds this book offers.
    Similar to the book of Roberts, this treatise lacks mathematical preciseness. Particularly, some of the rules Shin provides are not sound (but these flaws can easily be fixed), and the translation of beta graphs to first oder logic contains a flaw as well (I discussed this with Shin, and she agreed, thus I dare to express this critics).
    Nonetheless, in all other aspects I enjoyed the book very much. It is very good read and can really recommend it. Together with the books of Roberts and and Zeman, this is a "standard reading".

    More detailed critics can be found here.

  • John Stewart :   Theorem Proving Using Existential Graphs
    Master thesis (advisor: Robert Levinson), 1996.

    A computer science approach for an automatic theorem prover on the basis of existential graphs. As far as I know, this is the only work using existential graphs for theorem proving. The underlying structures are abstractions of the diagrams and do not incorporate any diagrammatic or graphical features of existential graphs. Thus, those who are interested especially in problems concerning diagrammatic representations will hardly benefit from this thesis. Moreover, this thesis does not touch any philosophical questions on existential graphs. On the other hand: For those who are intested in deduction procedures, this is of course a good choice.
    My impression is that this thesis sometimes mixes up syntax and semantics. So, in my view, although written by a computer scientist, this thesis lacks sometimes mathematical preciseness.
    As far as I know, this thesis is not published. I think if you want to get a copy of this thesis, you should contact Robert Levinson.

  • Frederik Stjernfelt:    Diagrammatology: An Investigation on the Borderlines of Phenomenology, Ontology, and Semiotics
    Synthese Library , Vol. 336. Springer-Verlag, Heidelberg-Berlin, 2007.
    507 p., Hardcover ISBN: 978-1-4020-5651-2 139,95 Euro

    So far, I did not read this book, so I provide a summary from Springers site:

    Diagrammatology investigates the role of diagrams for thought and knowledge. Based on the general doctrine of diagrams in Charles Peirce's mature work, Diagrammatology claims diagrams to constitute a centerpiece of epistemology. The book reflects Peirce's work on the issue in Husserl's contemporanous doctrine of "categorial intuition" and charts the many unnoticed similarities between Peircean semiotics and early Husserlian phenomenology. Diagrams, on a Peircean account, allow for observation and experimentation with ideal structures and objects and thus furnish the access to the synthetic a priori of the regional and formal ontology of the Husserlian tradition. The second part of the book focusses on three regional branches of semiotics: biosemiotics, picture analysis, and the theory of literature. Based on diagrammatology, these domains appear as accessible for a diagrammatological approach which leaves the traditional relativism and culturalism of semiotics behind and hence constitutes a realist semiotics

    Diagrams will never be the same. A fascinating and challenging tour through phenomenology, biology, Peirce's theory of signs and Ingarden's ontology of literature, all neatly tied together through the guiding thread of the diagrammatical. A veritable tour de force.

    Barry Smith, SUNY at Buffalo, U.S.A.

    With his meticulous scholarship, Frederik Stjernfelt shows that Peirce and Husserl were cultivating a broad and fertile common ground, which was largely neglected by both the analytic and the continental philosophers during the 20th century and which promises to be an exciting area of research in the 21st.

    John F. Sowa, Croton-on-Hudson, U.S.A.

  • J. J. Zeman:   The Graphical Logic of C. S. Peirce
    You can find a html-version of this book at the homepage of Zeman.

    This book offers a mathematical elaboration of Peirce's alpha-graphs, Peirce's beta-graphs, and the part of Peirce's gamma-graphs which extend the beta-part by adding the so-called "broken cut" (this part corresponds to modal logic). Similar to Roberts, he justifies his elaboration with a lot of quotations, and he gives some philosophical background, too.
    The only critics I have are: Like Shin and Roberts, Zeman does not provide an extensional semantics of EGs, `only' translations from graphs to propositional (for alpha), first order (for beta) and modal (for gamm) logic. These translatation are correct, but in my eyes a little bit clumsy (you can find a discussion of this in Shin's book as well). Zeman's definition is an abstract, mathematical definition (here I have to withdraw a former statement of mine on this website: sorry for that), but he does not discuss the relationship of his definition and the graphical representations of EGs (this relationship is fairly clear to him).
    For reading this book, one should be little bit familiar with existential graphs: It is not as readable like the book of Shin. On the other hand, this book is in my opinion -from a mathematical point of view- the best book on existential graphs so far.
    Together with the books of Roberts and Shin, this is a "standard reading".

    More detailed critics can be found here.

Books - containing Existential Graphs

  • Frithjof Dau:   The Logic System of Concept Graphs with Negations (And its Relationship to Predicate Logic).
    PhD-Thesis. An extended version will appear in Springer Lecture Notes on Computer Science (LNCS 2892).
    If you have access to springer-link, you can download this thesis at springer-link.

    Ok. This is advertisement for my thesis. Nonetheless, you will find a short introduction into existential graph in this treatise.

  • E. M. Hammer:   Logic and Visual Information.
    CSLI Publications, Stanford, California, 1995.

    An elaboration of different diagrammatic systems, like Venn diagrams, Euler circles, or higraphs, containing an chapter on Peirce's alpha graphs.

    I used this book as a textbook for a lecture on logic with diagrams (the part on venn diagrams). I have to say that from a mathematical point of view, this book is not precise enough, and it contains some (minor) flaws.

  • Derik Hawley:   Logic in Pictures: An Examination of Diagrammatic Representations, Graph Theory and Logic.
    Thesis requirement for the degree of Master of Arts in Philosophy, Waterloo, Canada, 1994.

    A master of arts in philosphy thesis. This easy-to-read thesis (50 pages) provides an abstract definition of Peirce's beta graphs by means of mathematical graph theory. This definition is similar to the definition of concept graphs with cuts (see my thesis), and very similar to Pollandts definition of relation graphs (see her ICCS-paper). Thus this approach has my full sympathies. But the argumentation does not go into details, some (in my view important) aspects are discussed only perfunctorily. The philosophical dimension of existential graphs is nearly discussed not at all. But this thesis is the only one I found so far which provides an abstract mathematical definition for existential graphs (which prescinds from the graphical properties of the diagrams), so it is definetely worth reading it.

  • Nathan Houser, Don D. Roberts, James Van Evra (Eds):   Studies in the Logic of Charles Sanders Peirce.
    Indiana University Press; 1977.

  • H. Pape:   Charles S. Peirce: Phänomen und Logik der Zeichen.
    German translation of Peirce's   Syllabus of Certain Topics of Logic. Suhrkamp Verlag Wissenschaft, 1983.

    Unfortunately, this syllabus, which gives valuable insight into Peirce's understanding of his graphs, is published only partly in the collected papers.

  • C. S. Peirce:   Reasoning and the Logic of Things. The Cambridge Conferences Lectures of 1898.
    Cambridge, Massachusetts, London, England, Ed. by K. L. Ketner, H. Putnam, Harvard Univ. Press, Cambridge 1992.

    Lecture three is titled "the logic of relatives" and is about existential graphs. Moreover, you find some discussion in the comprehensive comments on the lectures as well.
    A german translation of this lectures have recently been published by Suhrkamp Verlag.

  • C. S. Peirce,    Collected Papers.
    Vol. 1-6 edited by Charles Hartshorne and Paul Weiss, vol 7-8 edited b< Arthur Burks, Cambridge: Harvard, 1931-58.

    Including a comprehensive 100 pages elaboration of EGs in Vol. 4 which is very instructive.


  • Robert W. Burch:   Peirce on the Application of Relations to Relations..
    Published in an anthology edited by Nathan Houser and Don D. Roberts. Indanapolis: Indiana University Press, forthcoming.

    Robert W. Burch:   Valental Aspects of Peircean Algebraic Logic..
    In: Computers Math. Applic. Vol.23, No.6-9, pp.665-677, 1992.

  • Frithjof Dau:   An Embedding of Existential Graphs into Concept Graphs with Negations.
    In: Priss, U.,Corbett, D:, Angelova, G.\ (Eds.): Conceptual Structures: Integration and Interfaces. ICCS 2002, Borovets, Bulgaria, July 15-July 19, 2002, Proceedings. Springer-Verlag, Heidelberg-Berlin, 2002.

    An article on my own which translates Peirce's beta graphs to concept graphs with cuts (see my thesis). Even if you are not familiar with concept graphs, you can find some good hints how and -more important, why- beta graphs have to be read.

  • Frithjof Dau:   Types and Tokens for Logic with Diagrams: A Mathematical Approach.
    See on this site under 'publications'.

    Another article on my own, where I discuss how graphs have to be handled in order to get a rigour formal system. I embedded my argumentation into a case study, namely Peirce's alpha graphs, thus you find a mathematization of these graphs, (syntax, semantics, calculus, including a proof that the calculus is sound and complete) in this paper.

  • Frithjof Dau:   Some Notes on Proofs with Alpha Graphs See on this site under 'publications'.

    There are basically three results in this paper, but I will only refer to two of them. First, it is shown that the erasre rulecan be removed from the calculus, and the calculus is still weakly complete (i.e., all tautologies can be proven). This is nice as the erasure rule is the only rule which does not fulfill the so-called `finite choice property'. Roughly speaking, this is like removing the cut-rule from a sequent calculus (of course, Gentzen's proof for this result is much more sophisticated than mine for removing the erasure rule), i.e. the remaining calculus is as `nice' as a cut-free sequent calculus. The second result then is interesting: There are formulas which can be proven in sequent calculi in polynomial length, but in cut-free sequent calculi, the lengths of the proofs increases exponentially. For alpha graphs, these formulas can be, even in the restricted calculus, in polynomial length. Thus this result shows drastically the fundamental difference between Peirce's transformation rules and contemporary calculi, like sequent calculi.

  • Frithjof Dau:   Ligatures in Peirce's Existential Graphs. To appear in a special issue `Peircean diagrammatical logic' of Semiotika, Elsevier. See on this site under 'publications'.

    This paper describes a deeper understanding of ligatures -i.e., networks of lines of identity- in Peirce's existential graphs. In constrast to lines of lines of identity, ligatures do not neccessarily denote a single object. This makes them in some cases harder to read and understand. The paper introduces a special case of ligatures, so-called single-object-ligatures, which can basically be understood like lines of identity, and based on this notion, a simplified reading of existential graphs is introduced. The paper is based on some results of my treatise on EGs, but provides the results of it in a more informal manner.

  • Frithjof Dau:   Constants and Functions in Peirce's Existential Graphs. Submitted to ICCS 2007. See on this site under 'publications'.

    Peirce's existential graphs describe a logic ofrelations, i.e., the Beta part of existential graphs corresponds to first order logic with identity, but without constants or functions. This paper investigates how the syntax, semantics and particularly the calculus of existential graphs has to be extended in order to capture constants and functions as well. I provide the paper, as it is submitted to ICCS (it is under review).

  • E. M. Hammer:   Semantics for Existential Graphs.
    Journal Philosophical Logic, Vol. 27, 1998, 489--503.

  • J. Hereth Correia, R. Pöschel:   The Teridentity and Peircean Algebraic Logic
    In: Schärfe, Henrik; Hitzler, Pascal; Ohrstrom, Peter (Eds.): Conceptual Structures: Inspiration and Application. 14th International Conference on Conceptual Structures, ICCS 2006, Aalborg, Denmark, July 16-21, 2006, Proceedings. Springer, LNAI, Vol. 4068

    A sophisticated mathematical proof for Peirce's Reduction Thesis in a very general setting. The proof is ugly (many case distinctions), but correct (yes, I read it). The result is simply great!

  • J. Hintikka:   The Place of C  S. Peirce in the History of Logical Theory.
    In: Brunning and Forster (1997), 13--33.

  • Mary Keeler:  The Philosophical Context of Peirce's Existential Graphs.

    As the title of the paper says, it consideres existential graphs from a philosophical point of view (and does not contain a single diagram ;-)). Not easy to read (at least not for me), but it cointains some good and important points (e.g., Peirce's purpose in developping existential graphs).


  • Ralf Müller:    Peirce's Existential Graphs: First Inquiries towards a Proper Understanding.
    In: Jaap van Brakel, Michael van Heerden (Eds.): C.S. Peirce, categories to constantinople, proceedings of the international symposium of Peirce, Leuven University Press, 1997.

  • P. Øhrstrøm:  Some Peircean Problems Regarding Graphs for Time and Modality
    Second International Conference on Conceptual Structures,University of Maryland, Proceedings, 1994, p.78-92.
    Together with Jan Schmidt and Harmen van den Berg.

  • P. Øhrstrøm:  Logic and Existential Graphs
    Topics in Cognitive Science and HCI 6, (redigeret af Inger Lytje), 1995, p. 137-154.

  • P. Øhrstrøm:  Existential Graphs and Tense Logic
    in: Conceptual Structures: Knowledge Representation as Interlingua - Lecture Notes in Artificial Intelligence 115.
    Peter W. Eklund, Gerard Ellis, Graham Mann (Eds) (44th International Conference on Conceptual Structures, ICCS '96, Sydney, Australia, August 1996, Proceedings), Springer, Berlin. pp. 203-217 ISBN: 3-540-61534-2.

  • P. Øhrstrøm:  C. S. Peirce and the Quest for Gamma Graphs
    Conceptual Structures: Fulfilling Peirce's Dream, Lecture Notes in Artificial Intelligence, Springer Verlag 1997, p.357-370.


  • P. Øhrstrøm:   A Software System for Learning Peircean Graphs
    Published in W.Tepfenhart & W. Cyre (eds.). Conceptual Structures: Standards and Practices, Lecture Notes in Artificial Intelligence 1640, 1999, p. 184-197. Together with Torben Braüner and Claus Donner.

  • Silke Pollandt:    Relation Graphs: A Structure for Representing Relations in Contextual Logic of Relations.
    In G. Angelova, D. Corbett, U: Priss (Eds.): Conceptual Structures: Integration and Interfaces. LNAI 2393, Springer Verlag, Berlin--Heidelberg 2002.

    A paper, based on Burch's book and Wille's paper, which provides a mathematical elaboration of relation graphs (existential graphs with "pending edges" which describe relations).

  • D. D. Roberts:   The Existential Graphs.
    Computers Math.\ Appl.., Vol. 23, No. 6--9, 1992, 639--63.

  • Sun-Joo Shin:   Diagrams and a Theory of Seeing.

    A 3-pages paper on the different ways an existential graphs ca be perceived.

  • Sun-Joo Shin:    Reviving the Iconicity of Beta Graphs.
    In: Theory and Application of Diagrams: First International Conference, Diagrams 2000, Edinburgh, Scotland, UK, September 2000. Proceedings
    Lecture Notes in Computer Science ,Springer-Verlag Heidelberg, 2000.
    The document is downloadable from the springer-site.

    First of all: The results of this paper can be found in Shin'as book as well.
    The paper offers a new translation (Shin uses the term "reading") of existential graphs to first order logic which benefits from the "visually clear facts" in a diagram. I do like her approach, as it unfolds the specific advantages of diagrammatic systems.
    Unfortunately, her reading is not totally correct: see my paper. But this flaw can be corrected. If this is done, the translation is indeed very nice (much better to understand than Zemans translation of EGs (see Zeman's book).

  • Sun-Joo Shin:   Reconstituting Beta Graphs into an Efficacious System.
    In: Journal of Logic, Language and Information, Vol. 8, No. 3, July 1999.

  • Sun-Joo Shin:   Multiple Readings of Alpha Graphs
    In: M.Anderson, B.Meyer, P.Olivier (Eds.): Diagrammatic Representation and Reasoning.
    Springer-Verlag, Heidelberg-Berlin, 2002.

  • C. S. Peirce, John F. Sowa:   Existential Graphs: MS 514 by Charles Sanders Peirce with Commentary by John F. Sowa.

  • J. F. Sowa:   Logic: Graphical and Algebraic.
    Manuskript, Croton-on-Hudson 1997.

  • F. Stjernfelt: Two Iconicity Notions in Peirce's Diagrammatology

    In: Schärfe, Henrik; Hitzler, Pascal; Ohrstrom, Peter (Eds.): Conceptual Structures: Inspiration and Application. 14th International Conference on Conceptual Structures, ICCS 2006, Aalborg, Denmark, July 16-21, 2006, Proceedings. Springer, LNAI, Vol. 4068

    The first paper which introduces two notions of iconicity, an operational and a optimal (imho, the latter is basically the common understanding of iconicity in Peirce's graphs). Pretty interesting.

  • Rudolf Wille:   Lectures on Contextual Logic of Relations.
    A paper from 2000 based on Burch's book (see also Markus Arnold:  Einführung in die kontextuelle Relationenlogik). You have to contact Rudolf Wille to get this paper.

  • J. J. Zeman:   Peirce and Philo


  • J. J. Zeman:   Peirce's Philosophy of Logic


Images of original manusscripts of Peirce

A couple of people, most of them from the PORT-community (PORT: Peirce Online Resource Testbed), offer some images of original manusscripts of Peirce. I have to say that, when I discovered them for the first time, I was touched. Here are some links:

Further Stuff

The following links are not especially on existential graphs.

Dr. Frithjof Dau
dau (at) dr-dau (dot) net

University of Wollongong
Wollongong, 2522 N.S.W., Australia

Letzte Änderung / last change: 2011-05-14