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Exploring the beta quadrant

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The theory of existential graphs, which Peirce ultimately divided into four quadrants (\(\upalpha , \upbeta , \upgamma \) and \(\updelta \)), is a rich method of analysis in the philosophy of logic. Its \(\upbeta \)-part boasts a diagrammatic theory of quantification, which by 1902 Peirce had used in the logical analysis of (i) natural-language expressions such as complex donkey-type anaphora, (ii) quantificational patterns describing new mathematical concepts, and (iii) cognitive information processing. In the \(\upbeta \)-quadrant, he came close to inventing independence-friendly logic, the idea of which he found indispensable in fulfilling the tasks (i)–(iii).

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  1. 1.

    Sylvester, one of the founding fathers of graph theory, published a ground-breaking article on this analogy in the first issue of the American Journal of Mathematics, the series of which he launched in 1878.

  2. 2.

    Peirce’s own account of the history of diagrammatic logics, especially those of Euler diagrams (MS 479, 1903, CP 4.353) follows closely Venn’s Symbolic Logic (1894, p. 477, \(2\mathrm{nd}\) ed., and p. 423), noting that Hamilton misrepresented their origins in Weise’s Nucleus Logicae Weisiane of 1712. (Weise died in 1708 and the book was mostly written by J. C. Lange.) Peirce tells that another Lange, Friedrich Albert, attributes the origins of what we tend to term Euler diagrams to Juan Vives and his tradition (Bellucci 2013). According to Peirce, Euler should be acknowledged, however, as the first logician to have developed the idea into the direction of a calculus, the method of which Venn and Peirce subsequently improved. Peirce also notes how the logic of relations answers why the diagram method works in the way it does and why it is appropriate to the analysis of the meaning of natural-language assertions. MS 479 continues with a detailed study of Venn’s and Peirce’s own improvements.

  3. 3.

    On this Peirce wrote that: “Another fault of ordinary language as an instrument of reasoning is that it is more pictorial, than diagrammatic. It serves the purposes of literature well, but not those of logic” (MS 573, c.1890).

  4. 4.

    As an example Peirce argues for the superiority of maps over oral descriptions: “Well, what I propose to put you into possession of is a way of making a diagram of any fact you please, and to this I shall add a way of writing a description of a fact somewhat resembling an algebraic expression. The diagrammatic method I call the method of Existential Graphs; the other I call my Universal Algebra of Logic” (MS 650, 1910).

  5. 5.

    See e.g. MS 410, “Analysis of Propositions”, a version of Chapter II for the projected book How to Reason (the “Grand Logic”), §74. The Grand Logic is scheduled to appear in Volume 9 of the Chronological Edition of the Writings of Charles S. Peirce.

  6. 6.

    The connecting lines enabled Peirce to surpass preceding developers of diagrammatic notations: “the system would be inferior to the (ill-named) Euler’s diagrams unless extended with lines of identities” (MS 430, p. 7).

  7. 7.

    Much more is relevant to the issue than can be discussed here, beginning with the classic work of Charles H. Kahn from 1973 (see Kahn 2009). A whole generation of analysts has joined the discussions since Kahn’s pioneering exploration of the manifold nuances in the use and meaning of estin in ancient Greek thought.

  8. 8.

    Yet he would procure elaborate studies on the algebra of copula, especially in the 1893 grand logic but already in early 1870s (MSS 411–412, 430, 573–579, 594, 737).

  9. 9.

    Even earlier, and at least since summer 1896, Peirce is using the word ‘cut’ rather meaning affirming the truth of the graph, as a sign pertaining to the entitative, not existential system of logical graphs (MS 481). In less than two years, he would be using a cut as a denial of what the graphs placed on their interiors assert (see e.g. MS 513, May 1898).

  10. 10.

    Note that it is not straightforward to ‘translate’ (3) to the first-order formula, since one needs to introduce additional variables and identities. And even so, it is not clear whether that translation captures well the intended meaning, especially that of the reflexive.

  11. 11.

    This fragment is from an unpublished variant draft of Sect. 2, entitled “Specimens of Mathematical Reasoning. Subsection A. The Simplest Possible Mathematical System” (68 ms pages). It belongs to Chapter III of his planned book Minute Logic. The drafts of his entire Minute Logic project consist of over 2500 ms pages.

  12. 12.

    Despite stating five Peirce presented four roles. I will complete the account by adding two more.

  13. 13.

    In addition, his argument closely tracks the behavior of the underlying linguistic forms, which he presents and analyzes both in English and in Latin.

  14. 14.

    Cf. Peirce’s remark “after collateral observation has disclosed what single universe is meant”.

  15. 15.

    This refutes Hodges (2001), who claimed that Peirce did not distinguish the opposite roles of the players and thus did not have GTSs. Textual evidence is abundant on the opposite roles that the “make-believe” players were assumed to have in Peirce’s semantic games: the speaker is a “person interested in sustaining the truth of the proposition” and the listener “a person who might be sceptical or hostile to the proposition” (MS 503). Pietarinen (2013) marries Peirce’s endoporeutic semantics with the modern GTS approach in the wider context of games in logic and language.

  16. 16.

    From his “Monist Scheme” notes “Z” (1907). Far from confined to modeling structures of organic chemistry, diagrammatic syntax consists of “Graphs modified by Cuts” (MS 669). The same point—the necessity of giving new interpretations to the lines of separation—equally holds for non-visual logical diagrams, such as auditory or tactile diagrams (see Pietarinen 2010).

  17. 17.

    The developments were so fast that Peirce later speculated on the possibility that the reason for Paul Carus’s refusal to publish more of his works during those hectic years of 1896–1898 had to do with the uncanny pace in which he was developing his systems of logical graphs (CSP to PC, 1896, 1908).

  18. 18.

    Testimony for this is in MS 498 (pp. 1–2): “The system of expressing propositions which is called EGs was invented by me late in the year 1896, as an improvement upon another system published in the Monist for January 1897. But it is curious that 14 years previously, I had, but for one easy step, entered upon the system of existential graphs, reaching its threshold by a more direct way. The current of my investigations at that time swept me past the portal of this rich treasury of ideas. I must have seen that such a system of expression was possible, but I failed to appreciate its merits”. He also credits Mitchell for the discovery of the “quantifiers” and for being “probably” the first to use these existential (\(\Sigma \), sum) and universal (\(\Pi \), product) quantifiers as strings of dependent quantifiers, thus being able to give the system “considerable power” in his “epoch-making paper” of the 1883 Studies in Logic (Dictionary of Philosophy and Psychology, Vol. II, entry “Symbolic Logic”, p. 650, 1902).

  19. 19.

    Peirce explains this in MS 300 (1908) by denying that negation or denial is a “simple concept”, or that consequence would be a “composite of two negations”, so that to say that “If in the actual state of things A is true, then B is true” would be “correctly analyzed as the assertion”: “It is false to say that A is true while B is false”. Endoporeutic interpretation is thus profoundly present in the very idea of the scroll notation.

  20. 20.

    In mid-1890s Peirce investigated the mathematics of curves of various kinds (MS 105, 163, 261, 264). Such forms included the scroll which he appropriated for logical purposes. Now mathematical curves are imaginary objects whose properties do not rely on visual features.

  21. 21.

    Hence, those diagrammatic forms that we encounter in logical graphs are really examples of the kinds of mixed signs in which the quality of the iconic, though perhaps predominant, is coupled with indexical and symbolic ones, just as rhemas are coupled with dicisigns (propositions) to yield arguments.

  22. 22.

    There is a preceding argument in MS 430 according to which in applying iteration to


    we get


    and further


    Notice that in applying iteration, the identities of graphs between the iterated and the original graph-instances will have to be preserved (so that the iteration does not introduce new variables that could change the meaning). The rule of iteration can be applied to the loose end of the lines resting on positive areas to retract them across the cuts inside out, but not retracting them across the cuts outside in.

  23. 23.

    The non-equivalence is also seen form the graphs, since in the lower sub-graph the rule of erasure has been applied to remove an instance of ‘is born of’, and unlike the rules of iteration and deiteration, the rule of erasure is not reversible.

  24. 24.

    For example, LEM fails in IF logic, the usual explanation having been non-determinacy of the associated semantic games of imperfect information. Note also that bringing ovals into contact is not unusual as such, because it already takes place in the scrolls (the signs of consequences). But the connection of boundary lines in the scrolls will have to have a different interpretation, given the fact that it is a contact created by two nested, and not by two co-located, ovals.

  25. 25.

    The Necker Cube effect is unavoidable but inconsequential.

  26. 26.

    There are also cases in which the identity needs to be denoted by the continuity of two-dimensional planes and not merely by one-dimensional lines as on two-dimensional sheets. If these planes of identity branch or cross the boundaries of the oval, we call them planar ligatures.

  27. 27.

    MS 599 is entitled Reason’s Rules, c. 1906, but the statement is from the pages only extant in the original folders at Houghton library and are missing from the 1966 microfilm edition of his Nachlass.

  28. 28.

    See Pietarinen (2006), Chapter 5. Their movement, then, can, but need not be limited to, concern new rules of transformation that one needs to invent for the system given the semantic incompleteness of the resulting logic.

  29. 29.

    Already during his earliest explorations of logical and EGs he had remarked that “there is no reason but the convenience of writing why the graphs should be constructed upon a surface rather than have three dimensions. This limitation is not founded in their nature. In three dimensions, any number of regions can all touch one another, and any two graphs could be directly connected, without extending beyond their common enclosure, however complicated might be the connections of the different graphs” (MS 482, 1897).

  30. 30.

    The reason was that his extensive 1902 Carnegie Application to complete and publish the works was turned down.

  31. 31.

    This is not to deny the import of the critical tone Peirce levied on Weierstrass. But those remarks simply testify how closely Peirce studied him, and in how good company he was when writing the review: his father Benjamin had worked on linear associative algebras which Charles had improved upon, his brother James Mills had lectured on Cauchy and the topic of the theory of functions of a complex variable at Harvard, and Charles himself was just in the process of inventing the iconic notation of ovals to capture, among others, the ordering of the lines of identity—all this surrounded by the Weierstrassian ambiance and around the time of the posthumous publication of his lecture notes (Weierstrass 1895).

  32. 32.

    Forsyth’s 1893 book refers to Weierstrass’s mimeograph of 1886 which I have been unable to locate.

  33. 33.

    For example, Peirce states in his review that “it would be unfair to convey the idea that Forsyth is quite impeccable in his expressions. This is far from being true. Thus, at the beginning of chapter iii., in enunciating Cauchy’s fundamental theorem on the expansion of a holomorphic function, the important words ‘unconditionally and uniformly,’ as describing the mode of convergence, are omitted, as they are overlooked in the proof given” (Peirce 1894, p. 47). The difference between convergence and uniform convergence is not observed anywhere else in the book, either.

  34. 34.

    Peirce’s Puzzle is also an independence phenomenon in which choices for logical connectives (the conditional) are independent of choices for quantifies. That is, choices of what the values of quantifiers can be may depend on which construction branch in the semantic tree we are modeling. A way to capture connective independence is to have graphs modeled by layers of sheets with their respective domains (Pietarinen 2006, Chapter 5).

  35. 35.

    Aside from the vision of developing stereoscopic diagrams which he did not live to carry out, Peirce recognized that “the system of graphs needs to be remodeled to make [higher-order relations]” “the regular morphology of graphs” (MS 504, p. 6). Beginning in 1903, Peirce indeed develops systems of higher-order graphs and the logic of potentials (MS 478; Pietarinen 2015b).

  36. 36.

    Dictionary of Philosophy and Psychology, ed. J.M. Baldwin, New York: Macmillan, vol. 2, 640–651.


  1. Forsyth, A. R. (1893). Theory of functions of a complex variable. Cambridge: Cambridge University Press.

  2. Beeson, M. J. (1985). Foundations of constructive mathematics: Metamathematical studies. Berlin: Springer-Verlag.

  3. Bellucci, F. (2013). Diagrammatic reasoning. Some notes on Charles S. Peirce and Friedrich A. Lange. History and Philosophy of Logic, 34, 293–305.

  4. Bellucci, F., & Pietarinen, A.-V. (2015). Existential graphs as an instrument for logical analysis. Part 1: Alpha (to appear).

  5. Brown, R., & Porter, T. (2008). Category theory and higher dimensional algebra: Potential descriptive tools in neuroscience. arXiv:math/0306223v2.

  6. Henkin, L. (1961). Some remarks on infinitely long formulas. Infinitistic methods: Proceedings of the symposium on foundations of mathematics, Warsaw, 2–9 Sept 1959 (pp. 167–183). Warsaw: Panstwowe Wydawnictwo Naukowe.

  7. Hilpinen, R. (1982). On C. S. Peirce’s theory of the proposition: Peirce as a precursor of game-theoretical semantics. The Monist, 65, 182–188.

  8. Hintikka, J. (1973). Quantifiers vs. quantification theory. Dialectica, 27, 329–358.

  9. Hintikka, J. (1982). Game-theoretical semantics: Insights and prospects. Notre Dame Journal of Formal Logic, 23, 219–241.

  10. Hintikka, J. (1996). The principles of mathematics revisited. New York: Cambridge University Press.

  11. Hintikka, J. (2004). Analyses of Aristotle. Dordrecht: Springer.

  12. Hodges, W. (2001). Dialogue foundations: A septical look. The Aristotelian Society, Supplementary Volume LXXV, 2001, 17–32.

  13. Houser, N. et al. (Ed.). (1997). Studies in the logic of Charles Peirce. Bloomington: Indiana University Press.

  14. Kahn, C. H. (2009). Essays on being. Oxford: Oxford University Press.

  15. Landman, F. (2000). Event and plurality: The Jerusalem lectures. Dordrecht: Kluwer Academic.

  16. Liu, X., & Pietarinen, A.-V. (2015). Ramsey’s encounters with Peirce’s ‘logic of the future’ (to appear).

  17. von Neumann, J. (1958). The computer and the brain. Yale: Yale University Press.

  18. Pauly, A., & Ziegler, M. (2013). Relative computability and uniform continuity of relations. Journal of Logic & Analysis, 5, 1–39.

  19. Peirce, C. S. (1885). On the algebra of logic: A contribution to the philosophy of notations, in Writings of Charles S. Peirce, 1884–1886: The chronological edition, the Peirce edition project (pp. 162–190). Bloomington: Indiana University Press. (Originally appeared in American Journal of Mathematics 7 (1885), 180–202).

  20. Peirce, C. S. (1894). A review of Forsyth, Harkness and Picard. The Nation, 58(15), 197–199.

  21. Peirce, C. S. (1897). The logic of relatives. The Monist, 7, 161–217.

  22. Peirce, C. S. (1967). Manuscripts, Houghton Library of Harvard University. Richard Robin, annotated catalogue of the papers of Charles S. Peirce, Amherst: University of Massachusetts Press. (1967); and The Peirce papers: A supplementary catalogue. Transactions of the C. S. Peirce Society, 7, 37–57.

  23. Peirce, C. S. (1998). The essential Peirce, volume 2. The Peirce edition project. Bloomington: Indiana University Press.

  24. Pietarinen, A.-V. (2003). Peirce’s game-theoretic ideas in logic. Semiotica, 144, 33–47.

  25. Pietarinen, A.-V. (2006). Signs of logic: Peircean themes in the philosophy of language, games, and communication, Synthese Library 329. Dordrecht: Springer.

  26. Pietarinen, A.-V. (2007). Semantic games and generalised quantifiers. In A.-V. Pietarinen (Ed.), Game theory and linguistic meaning (pp. 183–206). Oxford: Elsevier.

  27. Pietarinen, A.-V. (2010). Is non-visual diagrammatic logic possible? In A. Gerner (Ed.), Diagrammatology and diagram praxis (pp. 73–84). London: College Publications.

  28. Pietarinen, A.-V. (2011a). Moving pictures of thought II: graphs, games, and pragmaticism’s proof. Semiotica, 186, 315–331 (Translated into Portuguese, 2013).

  29. Pietarinen, A.-V. (2011b). Existential graphs: What the diagrammatic logic of cognition might look like. History and Philosophy of Logic, 32(3), 265–281. (Translated into Chinese and Russian, 2013).

  30. Pietarinen, A.-V. (2013). Logical and linguistic games from Peirce to Grice to Hintikka (with comments by J. Hintikka), Teorema XXXIII/2, 121–136.

  31. Pietarinen, A.-V. (2014). Two papers on existential graphs by Charles S. Peirce: 1. Recent developments of existential graphs and their consequences for logic (1906), 2. Assurance through reasoning (1911), Synthese. doi:10.1007/s11229-014-0498-y.

  32. Pietarinen, A.-V. (2015a). Logic of the future: Peirce’s writings on existential graphs (to appear).

  33. Pietarinen, A.-V. (2015b). No entity without substantive possibility (to appear).

  34. Pietarinen, A.-V. (2015c). Peirce’s development of quantification theory (to appear).

  35. Pietarinen, A.-V., & Bellucci, F. (2014). New light on Peirce’s concept of retroduction and scientific reasoning. International Studies in the Philosophy of Science, 28(2).

  36. Pietarinen, A.-V., & Bellucci, F. (2015). The iconic moment. Towards a Peircean theory of diagrammatic imagination (to appear).

  37. Roberts, D. (1973). The existential graphs of Charles S. Peirce. The Hague: Mouton.

  38. Russell, B. (1919). Introduction to mathematical philosophy. London: Georg Allen.

  39. Saarinen, E. (Ed.). (1979). Game-theoretical semantics. Essays on semantics by Hintikka, Carlson, Peacocke, Rantala, and Saarinen. Dordrecht: D. Reidel.

  40. Shin, S.-J., & Hammer, E. (2010). Peirce’s logic, The Stanford encyclopedia of philosophy. In E. N. Zalta (ed.). Retrieved Dec, 2011 from

  41. Tao, T. (2007). Retrieved Sept, 2011 from

  42. Weierstrass, K. T. W. (1895). Mathematische werke (Vol. 2). Berlin: Reimer.

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Supported by DiaMind (The Diagrammatic Mind: Logical and Communicative Aspects of Iconicity, Estonian Research Council PUT267, 2013–2015 and the Academy of Finland 12786, 2013–2017, Principle Investigator A.-V. Pietarinen). Earlier versions were presented in the Peirce and Early Analytic Philosophy Symposium, University of Helsinki, May 2009, and in Logic Now & Then conference in Brussels in December 2011. Writing up of the final version was supported by the 2014 High-End Foreign Experts Program of State Administration of Foreign Experts Affairs, P. R. China. I thank the participants of these events and the reviewers of the present journal for helpful questions and comments.

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Correspondence to Ahti-Veikko Pietarinen.

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Pietarinen, A. Exploring the beta quadrant. Synthese 192, 941–970 (2015).

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  • Existential graphs
  • Beta
  • Quantifiers
  • Independence-friendly logic
  • Linguistic analysis
  • Philosophy of notation