Existential Graphs Research Articles

Iconic logic: the visual art of drawing the right conclusion.

Most people, evidence suggests, have a hard time thinking straight. Symbolic logic is a tool that can help remedy this problem. Unfortunately, it is highly abstract and uses symbols whose meanings rely on unintuitive arbitrary conventions. Without sacrificing rigor, iconic logic is more concrete and uses icons that resemble what they stand for and whose meanings are thus easier to picture, process, and remember. Here I review and critique iconic existential graphs and concept diagrams-the former link iconic logic to iconic mathematics; the latter expand popular Euler or Venn diagrams and have, to some degree, been empirically investigated for user-friendliness. I lay out how expertise in perception, cognition, and genetics can inform and improve such empirical research to help make iconic logic more ergonomic. After all, logic is a tool, and tools should not only suit their use but also their user.

Enhancing Existential Graphs: Peirce's Late Improvements

Abstract: Charles Peirce developed Existential Graphs as a diagrammatic syntax for representing and reasoning about propositions, with three parts: Alpha for propositional logic, Beta for first-order predicate logic, and Gamma for aspects of modal logic, second-order logic, and metalanguage. He made several adjustments between 1909 and 1911 that merit further consideration: using heavy lines to denote possible states of things in which attached propositions would be true, drawing a red line just inside the edge of a page and writing postulates in the resulting margin, shading oddly enclosed areas instead of drawing thin lines as their boundaries, and using multiple sheets of paper to represent different subjects within the overall universe of discourse. These modifications can be combined to constitute a plausible candidate for Delta, a fourth part that Peirce intended to add but never spelled out. It implements modern formal systems of modal propositional logic in accordance with a version of possible worlds semantics in which the laws for the actual state of things are facts in every possible state of things.

Thinking Scripts

Chinese traditional characters share with Peirce’s existential graphs the fact of being endowed with an object-language that they describe through a nonlinear syntax and in an iconic way. Here iconicity is not restricted to images and perceptive similarity since diagrams and graphic metaphors are iconic too. The graphs are shown to be a borderland between Western traditional logic and Chinese traditional writing and culture, so the écart (Jullien’s concept for cultural distance) between characters and graphs is preserved even though graphs break with the Western prejudice in favor of conventionality at the expense of iconicity in logical systems. The take-home lesson for the study of writing systems is to substitute the orality-writing duality with the interplay among orality, writing, and pictures thus shifting from a linguistic typology to a semiotic one.

Native diagrammatic soundness and completeness proofs for Peirce’s Existential Graphs (Alpha)

Peirce’s diagrammatic system of Existential Graphs (\(EG_{\alpha })\) is a logical proof system corresponding to the Propositional Calculus (PL). Most known proofs of soundness and completeness for \(EG_{\alpha }\) depend upon a translation of Peirce’s diagrammatic syntax into that of a suitable Frege-style system. In this paper, drawing upon standard results but using the native diagrammatic notational framework of the graphs, we present a purely syntactic proof of soundness, and hence consistency, for \(EG_{\alpha }\), along with two separate completeness proofs that are constructive in the sense that we provide an algorithm in each case to construct an \(EG_{\alpha }\) formal proof starting from the empty Sheet of Assertion, given any expression that is in fact a tautology according to the standard semantics of the system.

LOGIC OF THE FUTURE: WRITINGS ON EXISTENTIAL GRAPHS. VOLUME 1: HISTORY AND APPLICATIONS, EDITED BY AHTI PIETARINEN

Reviewed by: Logic of the Future: Writings on Existential Graphs. Volume 1: History and Applications ed. by Ahti Pietarinen Frederik Stjernfelt Edited by Ahti Pietarinen Logic of the Future: Writings on Existential Graphs. Volume 1: History and Applications Berlin/Boston: Walter de Gruyter, 2020, 664 pp. including name and keyword indexes To Peirce scholars and other aficionados of logic, semiotics, and pragmatism, 2017 brought the great news of Bellucci’s Speculative Grammar book, providing the eye-opening first detailed chronological overview over Peirce’s career-length developing of his semiotics. Now, the first volume of Ahti Pietarinen’s long-awaited three-volume publication of the totality of Peirce’s writings on his mature logic representation system known as Existential Graphs (EGs) not only gives us a plethora of hitherto unpublished Peirce papers but also a new and in many ways surprising view into the origin and development of this important fruit of Peirce’s last, creative philosophical burst taking its beginning around 1896, famously to peak in his annus mirabilis of 1903. Ahti Pietarinen has spent an admirable effort not only to find, collect and transcribe the enormous amount of Peirce’s EG writings, but also to frame and comment the results in a planned three-volume, 2000-pages annotated edition named Logic of the Future (LoF). The book is simultaneously the first volume of a new de Gruyter series “Peirceana”, edited by Francesco Bellucci and Ahti Pietarinen. The title has been chosen from a 1909 letter to William James in which Peirce describes the EG system and exclaims: “This ought to be The Logic of the Future”. Now, the first volume of this gargantuan work has appeared, subtitled “History and Applications”. A brief general introduction presents the layout of the whole planned edition, while another brief intro takes the uninitiated reader into the central issues of Existential Graphs. The tentative principle of selection of texts is ambitious: to be [End Page 114] “comprehensive”, that is, publishing the majority of the some 5000 ms. pages in which Peirce address EGs, including a vast amount of alternate text versions presented as footnotes and appendices to texts presented. A second principle is the overall chronological structure of the three volumes. A third is that texts already published (in the CP, in the NEM, or elsewhere) are left out. The two latter criteria, however, are rules of thumbs not rigorously followed. This appears from the detailed 100-pages introduction to the present volume, which goes through each of the volume’s 28 selected texts and text collections from Peirce’s writings, spanning the period from 1895 to 1911. The selection is not strictly chronological, however, which is partially a result of the overall disposition of all of the three volumes. In a certain sense, the central volume of the whole project is that of volume two, still to appear, to cover Peirce’s intense 1903 developments and presentations of the EGs in the “Logical Tracts” and the “Lowell Lectures”, each of which are planned to fill one subvolume of Logic of the Future volume two. The third and final volume, “Pragmaticism and Correspondence” then, is planned to contain post-1903 EG writings related to Peirce’s Pragmaticism paper series in the Monist, in which Peirce puts the EGs to use in his mature arguments for pragmatism if not decidedly a proof of pragmatism—as well as Peirce’s presentation and discussion of the EGs in lengthy correspondence with colleagues. These selection choices leave for the first volume primarily a batch of central texts documenting the early development of the EGs from 1896–98, forming the epicenter of the present volume. A bit surprisingly, this is flanked by two other batches of texts. First, texts 1–4, a small selection of philosophy-of-logic papers, most of them late, around 1910; Pietarinen’s argument is that these texts present Peirce’s most clarified general view of logic, as a preparation to the in medias res-texts of 1896. It appears as a further selection criterion in addition to the ones quoted, that of serving a novice reader with some easily accessible reading to begin with. They are mainly brief, late (1910...

An analysis of Existential Graphs\u2013part 2: Beta

This paper provides an analysis of the notational difference between Beta Existential Graphs, the graphical notation for quantificational logic invented by Charles S. Peirce at the end of the 19th century, and the ordinary notation of first-order logic. Peirce thought his graphs to be “more diagrammatic” than equivalently expressive languages (including his own algebras) for quantificational logic. The reason of this, he claimed, is that less room is afforded in Existential Graphs than in equivalently expressive languages for different ways of representing the same fact. The reason of this, in turn, is that Existential Graphs are a non-linear, occurrence-referential notation. As a non-linear notation, each graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic that are obtained by permuting those elements (sentential variables, predicate expressions, and quantifiers) that in the graphs lie in the same area. As an occurrence-referential notation, each Beta graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic in which the identity of reference of two or more variables is asserted. In brief, Peirce’s graphs are more diagrammatic than the linear, type-referential notation of first-order logic because the function that translates the latter to the graphs does not define isomorphism between the two notations.

Peirce, Bierstadt, and the Topographic Imagination in Nineteenth-Century North America

A passage in a letter serves as a starting point for an exploration of the relationship between the philosopher Charles Sanders Peirce and the painter Albert Bierstadt, taking into account their extended friendship, business partnerships, and respective practices of visual representation. The letter contains Peirce’s description of a painting by Bierstadt that he possessed, which sets the stage for an investigation of his systematic philosophy and a comparison of his existential graphs with the oft-noted “effects” of Bierstadt’s grand canvases. The latter, finally, help to unpack the ethical role of visual art within Peirce’s thought.

Charles Peirce and Bertrand Russell on Euclid

Both Charles Sanders Peirce (1839–1914) and Bertrand Russell (1872–1970) held that Euclid’s proofs in geometry were fundamentally flawed, and based largely on mathematical intuition rather than on sound deductive reasoning. They differed, however, as to the role which diagramming played in Euclid’s emonstrations. Specifically, whereas Russell attributed the failures on Euclid’s proofs to his reasoning from diagrams, Peirce held that diagrammatic reasoning could be rendered as logically rigorous and formal. In 1906, in his manuscript “Phaneroscopy” of 1906, he described his existential graphs, his highly iconic, graphical system of logic, as a moving picture of thought, “rendering literally visible before one’s very eyes the operation of thinking in actu”, and as a “generalized diagram of the Mind” (Peirce 1906; 1933, 4.582). More generally, Peirce personally found it more natural for him to reason diagrammatically, rather than algebraically. Rather, his concern with Euclid’s demonstrations was with its absence of explicit explanations, based upon the laws of logic, of how to proceed from one line of the “proof” to the next. This is the aspect of his criticism of Euclid that he shared with Russell; that Euclid’s demonstrations drew from mathematical intuition, rather than from strict formal deduction.

Logical Principles of a Topological Explanation

AbstractWe aim to demonstrate the applicability of Peirce’s iconic logic in the context of current topological explanations in the philosophy of science. We hold that the logical system of Existential Graphs is similar to contemporary topological approaches, thereby recognizing Peirce’s iconic logic (Beta Graphs) as a valid method of scientific representation. We base our thesis on the nexus between iconic logic and the so-called NonReduction Theorem. We illustrate our assumptions with examples derived from biology (protein folding).

Computing Truth Values in the Topos of Infinite Peirce’s α-Existential Graphs

• We consider infinite-depth Peirce’s α -Existential Graphs. • A generic-figure approach is used to represent α -Existential Graphs in a categorical framework. • We apply the notion of grossone in order to establish well-defined valuations in the category of α -Existential Graphs. We present here an approach to the analysis of the truth values of Peirce’s α -graphs without the restriction of finite number of elements (cuts and characters) on the Sheet of Assertion. We show that the ensuing structure in which such graphs are objects constitutes a topos . While the computation of the truth value of a graph in the topos can be an infinite process, we show that using the concept of grossone (①) the subobject classifier of the topos allows to determine a truth value for each graph.

On the Logical Philosophy of Assertive Graphs

The logic of assertive graphs (AGs) is a modification of Peirce’s logic of existential graphs (EGs), which is intuitionistic and which takes assertions as its explicit object of study. In this paper we extend AGs into a classical graphical logic of assertions (ClAG) whose internal logic is classical. The characteristic feature is that both AGs and ClAG retain deep-inference rules of transformation. Unlike classical EGs, both AGs and ClAG can do so without explicitly introducing polarities of areas in their language. We then compare advantages of these two graphical approaches to the logic of assertions with a reference to a number of topics in philosophy of logic and to their deep-inferential nature of proofs.

A Generic Figures Reconstruction of Peirce’s Existential Graphs (Alpha)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic (Peirce’s Existential Graphs $$\alpha $$). The straightforward construction of a presheaf category $${{\mathcal {E}}}{{\mathcal {G}}}_{\alpha ^{*}}$$ of cuts-only Existential Graphs (equivalent to the well-studied category of finite forests) provides a basis for the further construction of the category $${{\mathcal {E}}}{{\mathcal {G}}}_\alpha $$ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the formal investigation of relations between syntax and semantics in such diagrammatic systems.

Natural logic is diagrammatic reasoning about mental models

Abstract Diagrammatic reasoning, based on visualization and analogy, is the foundation for reasoning in ordinary language and the most esoteric theories of mathematics. Long before they write formal proof, mathematicians develop their ideas with diagrams, visualize novel patterns, and discover creative analogies. For over two millennia, Euclid’s diagrammatic methods set the standard for mathematical rigor. But the abstract algebra of the 19th century led many mathematicians to claim that all formal reasoning must be algebraic. Yet C. S. Peirce and George Polya recognized that Euclid’s diagrammatic reasoning is better match to human thought patterns than the algebraic rules and notations. A combination of Peirce’s graph logic, Polya’s heuristics, and Euclid’s diagrams is better candidate for natural logic than any algebraic formalism. Psychologically, it supports Peirce’s claim that his existential graphs (EGs) provide a moving picture of the action of the mind in thought. Logically, EGs have formal mapping to and from the ISO standard for Common Logic. Computationally, algorithms for Cognitive Memory (CM) and virtual reality (VR) can support cross-modal analogies between language-like and image-like representations.

“Logic of the Future” as C.S. Peirce Understood It (First Volumes of Peirceana)

Finally, the first book started Peirceana. Peirceana is expected as a new series that provides access to both Peirce’s mostly unpublished late works and secondary papers, in which ideas of this American philosopher are developed. This edition is opened with three volumes on Peirce’s manuscripts on “Logic of the Future.” The thinker gave this definition to his theory of existential graphs, i.e., a diagrammatical logical project that includes three sections. The sections can roughly correspond to propositional logic, first-order logic and modal logic. The theory of existential graphs is a particular conception. It predicts ideas of proof theory, mathematics of continuity, and cognitive sciences. Besides, it has an impressive philosophical foundation and perfectly emphasizes the importance of diagrammatic notations. Since many writings are published here for the first time, the edition per se is a significant event for the history of logic. However, Peirceana can do more than this. It urges to think of how the logic of 20‒21st centuries would have looked like if Peirce’s intentions were discovered earlier. At the same time, it clarifies the ground for contemporary investigations. Today we have a bunch of papers about various non-classical modifications, extensions, restrictions, and applications of the original theory. Peirceana is projected as the publication that can gather specialists (logicians, philosophers, as well as IT and cognitive science scholars) who are interested in the development of Peirce’s approach. It lets the readers reconsider our ordinary logical conceptions (e.g., reasoning, logical and language analysis, and others) as well as the conventional understanding of the evolution of modern logic. The current paper has two objectives: firstly, it estimates this diplomatic partly critical edition with its several thoroughly elaborated introductions, and secondly, it briefly introduces the existential graphs theory. A set of examples demonstrates the basic principles of this system.

Existential Graphs on nonplanar surfaces

Existential graphs on the plane constitute a two-dimensional representation of classical logic, in which a Jordan curve stands for the negation of its inside. In this paper we propose a program to develop existential Alpha graphs, which correspond to propositional logic, on various surfaces. The geometry of each manifold determines the possible Jordan curves on it, leading to diverse interpretations of negation. This may open a way for appointing a "natural" logic to any surface.

Как сделать тавтологии ясными?

В статье показывается, каким образом первый раздел теории экзистенциальных графов Ч. Пирса отвечает на вопрос Л. Витгенштейна: «Как должна быть устроена система знаков, чтобы каждая тавтология распознавалась в ней одним и тем же способом?» Теория экзистенциальных графов или теория графов – диаграмматическая логическая система, базовой единицей которой является диаграмма (внешне похожая на диаграммы Эйлера). Ее первый раздел, альфа-графы, примерно соотносится с пропозициональным фрагментом классической логики. Синтаксис теории нагляден, точнее, он иконичен, а потому иконичным оказывается и решение задачи Витгенштейна. Чтобы определить тип формулы, не требуется никаких преобразований. Тавтологии наблюдемы. Возможность усматривать тавтологии объясняется не только диаграмматическими особенностями синтаксиса, но и его минимальностью. Единственным знаком теории (первый раздел) является разрез (контур упомянутой круговой диаграммы): размещение разрезов рядом друг с другом, внутри друг друга порождает не нового вида знаки, а различного вида графы. Разрез выполняет техническую и логическую функции. В этом смысле теория графов оказывается лаконичнее теорий с NAND- или NOR-операторами. В свете рассуждений о тавтологиях в статье затрагивается вопрос эволюции разреза. Разрез, который при самом простом толковании понимается как негация, представляет собой вырожденную импликацию. Именно импликация, а не негация, конъюнкция или дизъюнкция оказывается первичным знаком теории. На первый взгляд такое решение может показаться странным: импликация – самая сложная для понимания логическая операция. Вместе с тем именно импликация подчеркивает фундаментальную роль логического следования, отражает его основные свойства (антисимметричность и транзитивность).

ОБУЧЕНИЕ ГРАФАМИ — ДИАГРАММЫ Ч. С. ПИРСА И ПРЕПОДАВАНИЕ ЛОГИКИ

The paper explains how Peirce’s Existential Graphs theory can be fruitfully used in logic classes. This theory is the last logical project of the American philosopher that allows to observe logical relations. In other words, it provides a new perspective of logical analysis. First, I offer a brief introduction of the theory, which is more philosophical than technical. After that, I cite some opinions of scholars who have applied these ideas in their classes (logic for philosophy or computer science departments). The paper demonstrates both the proper logical and philosophical advantages of the theory. However, the philosophical part concerns the questions of philosophy of logic. Finally, I mention some obviously perplexing features of the approach which point to the limits of its applicability.

A Graphical Deep Inference System for Intuitionistic Logic

A graphical approach to intuitionistic propositional logic is presented. The system GrIn is a deep inference system and is formulated in terms of Peirce’s existential graphs. GrIn is shown to be sound and complete with respect to the class of all Heyting algebras. The system GrIn is shown to be equivalent to the Gentzen sequent calculus G3ip via translations between intuitionistic graphs and intuitionistic formulae. We also discuss the intuitionistic interpretation of logical constants proposed by the graphical approach.

Existential Graphs as Ontographic Media

Existential Graphs as Ontographic Media

Diagrams and alien ways of thinking

The recent wave of data on exoplanets lends support to METI ventures (Messaging to Extra-Terrestrial Intelligence), insofar as the more exoplanets we find, the more likely it is that “exominds” await our messages. Yet, despite these astronomical advances, there are presently no well-confirmed tests against which to check the design of interstellar messages. In the meantime, the best we can do is distance ourselves from terracentric assumptions. There is no reason, for example, to assume that all inferential abilities are language-like. With that in mind, I argue that logical reasoning does not have to be couched in symbolic notation. In diagrammatic reasoning, inferences are underwritten, not by rules, but by transformations of self-same qualitative signs. I use the Existential Graphs of C. S. Peirce to show this. Since diagrams are less dependent on convention and might even be generalized to cover non-visual senses, I argue that METI researchers should add some form of diagrammatic representations to their repertoire. Doing so can shed light, not just on alien minds, but on the deepest structures of reasoning itself.