Volume 2011, Issue 186 - Diagrammatical reasoning and Peircean logic representations

Charles S. Peirce's epistemological theory of mental diagrams forms the theoretical basis of his attempt to analyze diagrammatic reasoning. Two examples, one from science and another from art, are examined to test the scope of this theory. While the first example shows how scientific diagrams form part of translation processes, similar processes are demonstrated in how paintings are received. The article attempts to connect Peirce and A. J. Greimas's theory of narrative. Relating the two proves useful in allowing Peirce's theory of the connection between the three normative sciences (Logic, Ethics, and Esthetics) to be discussed on a new basis.

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.

That diagrams are analog, i.e., homomorphic, representations of some kind, and sentential representations are not, is a generally held intuition. In this paper, we develop a formal framework in which the claim can be stated and examined, and certain puzzles resolved. We start by asking how physical things can represent information in some target domain. We lay a basis for investigating possible homomorphisms by modeling both the physical medium and the target domain as sets of variables, each with a constraint structure. When a homomorphism exists, the causality of the physical medium can provide a “free ride,” automatically determining the information that is needed to solve a problem. The modeling technique enables us to show how in sentential representations the structure of the 2-D space is used non-homomorphically to define tokens, and to provide a minimal homomorphism with respect to the ordering of the arguments for n-ary functions, predicates, and operators, but otherwise there is no homomorphism with the target domain. Our treatment also characterizes diagrams as a type of a more general class of physical representations in which physical causality plays a role in providing the information needed to solve a problem.

Lines of Identity (LoI) are important elements in Existential Graphs. They can be assembled to whole networks called “ligatures.” They are not straightforwardly understandable: for example, in constrast to LoI, ligatures may stand for more than one object. This article elaborates the handling of ligatures. It is precisely investigated how ligatures are dealt with in the calculus and how they can be modified without changing the meaning of a graph. Finally, a sufficient criterion for reading a ligature similar to a LoI is provided, as well as an algorithm which grasps the correct meaning of an existential graph even if it contains arbitrarily complex ligatures.

I will sustain in this article that Peirce can be seen as the last great representative of that inconspicuous but persistent tradition that, from the thirteenth to the seventeenth century, spent its energies on discovering a universal language. His project of Existential Graphs is in fact grounded on the isomorphism among a Sheet of Assertion, in which Graphs-signs are drawn, a Mind, with its thoughts-signs, and the Universe, with its facts-signs. In the same sense, Leibniz worked on his Characteristica Universalis, seen as a general Encyclopedia or alphabet of human thoughts, and a pictum mundi amphitheatrum.

A linear notation for Charles S. Peirce's alpha and beta diagrammatic systems of existential graphs is presented. These two systems are equivalent to propositional and first-order logic. Some differences between the linear and graphical notation are analyzed, revealing some of the strengths and weaknesses of Peirce's system.

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.

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.

In the first part of this paper, I delineate Peirce's general concept of diagrammatic reasoning from other usages of the term that focus either on diagrammatic systems as developed in logic and AI or on reasoning with mental models. The main function of Peirce's form of diagrammatic reasoning is to facilitate individual or social thinking processes in situations that are too complex to be coped with exclusively by internal cognitive means. I provide a diagrammatic definition of diagrammatic reasoning that emphasizes the construction of, and experimentation with, external representations based on the rules and conventions of a chosen representation system. The second part starts with a summary of empirical research regarding cognitive effects of working with diagrams and a critique of approaches that use “mental models” to explain those effects. The main focus of this section is, however, to elaborate the idea that diagrammatic reasoning should be conceptualized as a case of “distributed cognition.” Using the mathematics lesson described by Plato in his Meno, I analyze those cognitive conditions of diagrammatic reasoning that are relevant in this case.

Our brains make up a series of signs and are engaged in making or manifesting or reacting to a series of signs: through this semiotic activity they are at the same time engaged in “being minds.” An important effect of this semiotic activity of brains is a continuous process of “externalization of the mind” that exhibits a new cognitive perspective on the mechanisms underlying the semiotic emergence of abductive processes of meaning formation. I consider this process of externalization interplay critical in analyzing the relation between meaningful semiotic internal resources and devices and their dynamical interactions with the externalized semiotic materiality suitably stocked in the environment. The last part of the paper will describe some aspects of this interplay between external diagrammatization and internal recapitulation showing the specificity of their co-evolution.

The production of mathematical formalism on state of the art computers is quite different than by pen and paper. In this paper, I examine the question of how different media influence mathematical writing. The examination is based on an investigation of professional mathematicians' use of various media for their writing. A model for describing mathematical writing through turn-takings is proposed. The model is applied to the ways mathematicians use computers for writing, and especially it is used to understand how interaction with the computer system LaTeX is different in the case of formulas than in the case of diagrams.

Some very good arguments can be given in favor of the Augustinean wisdom, according to which it is impossible to provide a satisfactory definition of the concept of time. However, even in the absence of a proper definition, it is possible to deal with conceptual problems regarding time. It can be done in terms of analogies and metaphors. In particular, it is attractive to make use of Peirce's diagrams by means of which various kinds of conceptual experimentation can be carried out. This paper investigates how Peircean diagrams can be used within the study of time. In particular, we discuss 1) the topological properties of time, 2) the implicative structure in tense logic, 3) the notions of open future and branching time models, and finally 4) tenselogical alternatives to branching time models.

In the development of pure mathematics during the nineteenth and twentieth centuries, two very different movements had prevailed. The so-called rigor movement of arithmetization, which turned into set theoretical foundationalism, on the one hand, and the axiomatic movement, which originated in Poncelet's or Peirce's emphasis on the continuity principle, on the other hand. Axiomatical mathematics or mathematics as diagrammatic reasoning represents a genetic perspective aiming at generalization, whereas mathematics as arithmetic or set theory is mainly concerned with foundation and separation. We may thus conclude when Peirce defines mathematics in terms of diagrammatic reasoning, that this implies some very profound distinctions in the epistemology of mathematics.

In this article, the role of abductive reasoning within Peirce's diagrammatic reasoning is discussed. Both abduction and diagrammatic reasoning bring in elements of discovery but it is not clear if abduction should be a part of a fully developed diagrammatic system or not for Peirce. This relates to Peirce's way of interpreting abduction in his later writings. Iconicity and perceptual elements as a basis for discoveries are analyzed, both in deductive and abductive reasoning. At the end, the role of modern ideas of distributed cognition applied to the Peircean scheme is shortly delineated.

Peirce believed that his pragmaticism can be conclusively proven. Beginning in 1903, he drafted several attempts, ending by 1908 with a semeiotic proof. Around 1905, he exposes the proof using the theory of Existential Graphs (EGs). This paper modernizes the semantics Peirce proposed for EGs in terms of game-theoretic semantics (GTS). Peirce's 1905 proof is then reconstructed in three parts, by (1) relating pragmaticism to the GTS conception of meaning, (2) showing that Peirce's proof is an argument for a relational structure of the meaning of intellectual signs that our interpretative and strategic practices give rise to, and (3) bringing out the key links between EGs and pragmaticism.

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.

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.

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.

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.