Existential Graphs Research Articles

A New Analytic/Synthetic/Horotic Paradigm

We study a contemporary need to complement analytic philosophy with pendular, synthetic approaches. We provide new definitions of the dyad analytics/synthetics and complete it with a natural third, horotics. Some historical trends to support a synthetic/horotic paradigm are studied: (i) Peirce’s ideas around his logic of continuity – non Cantorian continuum and existential graphs – emphasizing the importance of mathematical gestures, (ii) Gödel’s understanding of intuitionism as a synthetic counterpart of classical logic, along with a new horotic approach to his work, (iii) Contemporary mathematical achievements (1950-2000), difficult to understand from analytical philosophy perspectives. Finally, we indicate some main features that a systematical synthetic/horotic reasoning should enforce, in order to fulfill its historic sequence.

Existential Graphs: What a Diagrammatic Logic of Cognition Might Look Like

This paper examines the contemporary philosophical and cognitive relevance of Charles Peirce's diagrammatic logic of existential graphs (EGs), the ‘moving pictures of thought’. The first part brings to the fore some hitherto unknown details about the reception of EGs in the early 1900s that took place amidst the emergence of modern conceptions of symbolic logic. In the second part, philosophical aspects of EGs and their contributions to contemporary logical theory are pointed out, including the relationship between iconic logic and images, the problem of the meaning of logical constants, the cognitive economy of iconic logic, the failure of the Frege–Russell thesis, and the failure of the Language of Thought hypothesis.

Peirce's Search for a Graphical Modal Logic (Propositional Part)

This paper deals with modality in Peirce's existential graphs, as expressed in his gamma and tinctured systems. We aim at showing that there were two philosophically motivated decisions of Peirce's that, in the end, hindered him from producing a modern, conclusive system of modal logic. Finally, we propose emendations and modifications to Peirce's modal graphical tinctured systems and to their underlying ideas that will produce modern modal systems.

The Hardness of the Iconic Must: Can Peirce's Existential Graphs Assist Modal Epistemology?

Charles Peirce’s diagrammatic logic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf’s famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not merely to individual things but to how those things are related in larger structures, such that certain aspects of these relations force certain other aspects to be a certain way.

The Hardness of the Iconic Must: Can Peirce's Existential Graphs Assist Modal Epistemology?

The Hardness of the Iconic Must: Can Peirce's Existential Graphs Assist Modal Epistemology?

Charles Sanders Peirce, A Mastermind of (Legal) Arguments

In this article, we try to trace the relationship between semiotics and theory of legal reasoning using Peirce’s idea that all reasoning must be necessarily in signs: every act of reasoning/argumentation is a sign process, leading to “the growth of knowledge. The broad scope and universal character of Peirce’s sign theory of reasoning allows us to look for new conciliatory paradigms, which must be presented in terms of possible synthesis between the traditional approaches to argumentation. These traditional approaches are strongly affected by either the dialectical (logical) perspective or the rhetorical perspective on argumentation, while Peirce’s approach tends to reconcile the rhetorical and methodological aspects of reasoning. This reconcilation is best illustrated by Peircean analysis of argument’s logical and rhetorical structure; while the diagrammatic (iconic) analysis of arguments is performed in the system of Existential Graphs (which is Peirce’s major methodological system, designed for the expressions of propositions in point of their relational structure). Obviously, Peirce’s original division of argument parts offered only the characterisation of the sign activity (involved in the process of reasoning), and thus left much to be desired in terms of practical explication.

Peirce's tutorial on existential graphs

In his formal papers on existential graphs (EGs), Peirce tended to obscure the simplicity of EGs with distracting digressions. In MS 514, however, he presented his simplest introduction to the EG syntax, semantics, and rules of inference. This article reproduces Peirce's original words and diagrams with further commentary, explanations, and examples. Unlike the syntax-based approach of most current textbooks, Peirce's method addresses the semantic issues of logic in a way that can be transferred to any notation. The concluding section shows that his rules of inference can clarify the foundations of proof theory and relate diverse methods, such as resolution and natural deduction. To relate EGs to other notations for logic, this article uses the Existential Graph Interchange Format (EGIF), which is a subset of the CGIF dialect of Common Logic. EGIF is a linear notation that can be mapped to and from the Alpha, Beta, and Gamma variants of EGs. It can also be translated to or from other formalisms, algebraic or geometrical.

Existential graphs and proofs of pragmaticism

We show how Peirce's architectonics folds on itself and finds local consequences that correspond to the major global hypotheses of the system. In particular, we study how the pragmaticist maxim (i.e., the pragmatic maxim fully modalized, support of Peirce's architectonics) can be technically represented in Peirce's existential graphs, well-suited to reveal an underlying continuity in logical operations, and can provide suggestive philosophical analogies. Further, using the existential graphs, we formalize — and prove one direction of — a “local proof of pragmaticism,” trying thus to explain the prominent place that existential graphs can play in the architectonics of pragmaticism, as Peirce persistently advocated. Finally, we present a web of “continuous iterations” of some key Peircean concepts (maxim, classification, abduction) that supports a “lattice of partial proofs” of pragmaticism.

Remarks on the iconicity and interpretation of existential graphs

In the 1890s, Peirce reformulated quantification theory by expressing it in a language of diagrams, called existential graphs. Peirce thought that the iconicity of his graphs made them suitable for analyzing logical reasoning. Iconic signs can be said to show their meaning, and this paper studies the ways in which graphs do this. Peirce's pragmatic analysis of propositions resembles game-theoretical semantics, and existential graphs show what they mean by displaying the structure of the semantic game for the proposition represented by a graph.

The fine structure of Peircean ligatures and lines of identity

Lines of identity in Peirce's existential graphs (beta) are logically complex structures that comprise both identity and existential quantification. Yet geometrically they are simple: linear continua that cannot have “furcations” or cross “cuts.” By contrast Peirce's “ligatures” are geometrically complex: they can both have furcations and cross cuts. Logically they involve not only identity and existential quantification but also negation. Moreover, Peirce makes clear that ligatures are composed of lines of identity by virtue of the fact that such lines can be “connected” with one another and can “abut upon” one another at a cut. This paper shows in logical detail how ligatures are composed and how they relate to identity, existential quantification, and negation. In so doing, it makes use of Peirce's non-standard account of the linear continuum, according to which, when a linear continuum is separated into two parts, (1) the parts are symmetric rather than (as the standard account of Dedekind holds) asymmetric, and (2) the one point at which separation occurs actually becomes two points, each of which is a Doppelgänger of the other.

Ligatures in Peirce's existential graphs

Ligatures in Peirce's existential graphs

Peirce's alpha graphs and propositional languages

Many do not doubt that Peirce's Existential Graphs are diagrammatic, as opposed to symbolic. However, when we are pressured to draw a distinction between the two different forms of representation, we find ourselves at a loss and our intuition quite vague. In this paper, I locate fundamental differences between two logically equivalent systems, Peirce's Alpha system and propositional languages. Suppose we have only two sentential connectives, ¬ and ̂. In spite of its truth-functional completeness, we don't want to use this language for the translation of English sentences or as a deductive calculus. We would adopt this language only when we intend to develop logical theories. That is, it is convenient to have fewer connectives for a meta-theory, but not for practical use. So, there seems to be a trade-off between a language with fewer connectives and a language with more connectives. This view has been accepted without question. In this paper, I will argue that this trade-off is limited to linear symbolic systems and that we could have a diagrammatic system with fewer operations but no need to suffer from problems like those of a sentential language with fewer connectives. How is that possible? A comparison between Peirce's Alpha system and a propositional language is presented to answer this question. The case study identifies the following unique property of a non-linear diagrammatic system as a main source of the discrepancy between two different types of representation: One and the same diagram can be read off in more than one way by carving it up in many ways, but without ambiguity.

Peircean Algebraic Logic and Peirce's Reduction Thesis

Robert Burch describes Peircean Algebraic Logic (PAL) as a language to express Peirce's “unitary logical vision” (1991: 3), which Peirce tried to formulate using different logical systems. A “correct” formulation of Peirce's vision then should allow a mathematical proof of Peirce's Reduction Thesis, that all relations can be generated from the ensemble of unary, binary, and ternary relations, but that at least some ternary relations cannot be reduced to relations of lower arity. Based on Burch's algebraization, the authors further simplify the mathematical structure of PAL and remove a restriction imposed by Burch, making the resulting system in its expressiveness more similar to Peirce's system of existential graphs. The drawback, however, is that the proof of the Reduction Thesis from Burch (A Peircean reduction thesis: The foundations of topological logic, Texas Tech University Press, 1991) no longer holds. A new proof was introduced in Hereth Correia, and Pöschel (The teridentity and Peircean algebraic logic: 230–247, Springer, 2006) and was published in full detail in Hereth (Relation graphs and contextual logic: Towards mathematical foundations of concept-oriented databases, Technische Universität Dresden dissertation, 2008). In this paper, we provide proof of Peirce's Reduction Thesis using a graph notation similar to Peirce's existential graphs.

On operational and optimal iconicity in Peirce's diagrammatology

Two different concepts of iconicity compete in Peirce's diagrammatical logic. One is articulated in his general reflections on the role of diagrams in thought, in what could be termed his diagrammatology — the other is articulated in his construction of Existential Graphs as an iconic system for representing logic. One is operational and defines iconicity in terms of which information may be derived from a given diagram or diagram system — the other has stronger demands on iconicity, adding to the operational criterion a demand for as high a degree of similarity as possible and may be termed optimal iconicity. Peirce himself does not clearly distinguish these two iconicity notions, a fact that has caused some confusion. By isolating them, we get a clearer and more refined conceptual apparatus for analyzing iconic signs, from pictures to logic. This paper investigates the two iconicity notions and addresses some of the problems they involve.

Linear notation for existential graphs

Linear notation for existential graphs

The images of film and the categories of signs: Peirce and Deleuze on media

In his two Cinema books, Gilles Deleuze updates his lifelong philosophical inquiry using semiotic terminology, which explicitly draws on concepts of Peirce. In a direction that is arguably of utmost significance for contemporary semiotics and media studies, Deleuze upgrades his lifelong interest in philosophy and psychoanalysis using a pragmatic semiotics that involves major claims for the non- linguistic status of semiotics, as suitable for his subject matter, film, and for a fundamental distinction between semiotic and logical or linguistic based semiology. Philosophy as theory, and semiotic theory at that, becomes grounded in innumerable case studies of individual films. In the same way, Peirce drew on case studies in science, mathematics, and logic, synthesizing these in proposals for existential graphs that could assist practical reasoning, so the Cinema books are quite distinct writings by a conceptual philosopher like Deleuze, in their sustained attention to enumerated technical subject matter. The Cinema books can be regarded in part and, interpretively, as a whole, as an inspired, albeit intuitive, translation of Peirce for and as media studies, and through extrapolation, new and televisual media. This paper argues that the Cinema books provide a pertinent and foundational basis for an encompassing semiotic model of contemporary media.

A diagrammatic reasoning system for the description logic [formula omitted

Diagrammatic reasoning is a tradition of visual logic that allows sentences that are equivalent to first order logic to be written in a visual or structural form: usually for improved usability. A calculus for the diagram can then be defined that allows well-formed formulas to be derived. This calculus is intended in the analog of logical inference. Description logics (DLs) have become a popular knowledge representation and processing language. DLs correspond to decidable fragments of first order logic; their notation is in the style of symbolic, variable-free formulas. Moreover, DLs are equipped with table au theorem provers that are proven to be sound and complete. Although DLs have roots in diagrammatic languages (such as semantic networks), they are elaborated in a purely symbolic manner. This paper discusses how DLs can be equivalently represented in terms of a diagrammatic reasoning system. First, existing diagrammatic reasoning systems, namely spider- and constraint diagrams, as well as existential and conceptual graphs, are investigated to determine if they are compatible with DLs. It turns out that Peirce's existential graphs are better suited for this purpose than the alternatives we examine. The paper then redevelops the DL ALC , which is the smallest propositional DL, by means of labeled trees, and provides a diagrammatic representation for these trees in the style of Peircean graphs. We provide a calculus based on C.S. Peirce's calculus for existential graphs and prove the soundness and completeness of the calculus. The calculus acts on labeled trees, but can be best understood as a diagrammatic calculus whose rules modify the Peircean-style representation of ALC .

PEIRCEAN APPROACHES TO EMERGENT SYSTEMS IN COGNITIVE SCIENCE AND RELIGION

Abstract.Cognitive science and religion provides perspectives on human cognition and spirituality. Emergent systems theory captures the subatomic, physical, biological, psychological, cultural, and transcendent relationships that constitute the human person. C. S. Peirce's metaphysical categories and existential graphs enrich traditional cognitive science modeling tools to capture emergent phenomena. From this richer perspective, one can reinterpret the traditional doctrine of soul as form of the body in terms of information as the constellation of constitutive relationships that enables real possibility.

COLLECTIVE SELF‐ORGANIZATION IN GENERAL BIOLOGY: GILLES DELEUZE, CHARLES S. PEIRCE, AND STUART KAUFFMAN

Abstract.Stuart Kauffman's proposal in Investigations to ground a “general biology” in the laws of self‐organization governing systems of autonomous agents runs up against the methodological problem of how to integrate formal mathematical with semantic and semiotic approaches to the study of evolutionary development. Gilles Deleuze's concept of the virtual and C. S. Peirce's system of existential graphs provide a theoretical framework and practical art for answering this problem of method by modeling the creative event of collective self‐organization as both represented and practiced in the scientific community.

Using existential graphs to integrate mental logic theory and mental model theory

There are two main theories as to how humans reason deductively. Mental logic theory states that humans reason through the application of formal reasoning schemas of the kind encountered in formal logic. Mental model theory states that humans reason through the construction, combination, and elimination of models compatible with the given information, and inspection of the models that remain. I propose that there exists a natural compromise between these two competing theories of reasoning.