Triadic Logic Research Articles

Deleuze’s zeroness and Peirce’s pure zero regarding the expansion of semiotics’ categorial frame

Abstract Deleuze (1925–1995), in the early 1980s, adopts Peirce’s (1839–1914) semiotics in order to classify the signs that the images of the cinema display. Aiming at insufflating the Peircean principles with the movement that animates the images of cinema, he provides Peirce’s triadic logic with a new category – Zeroness – which stands for the semiotic movement of cinematic images. Deleuze’s new category has impacts on the main domains of Peirce’s philosophy. Accordingly, our inquiry will focus on the irradiation of Zeroness over the (a) system of categories, the (b) Peircean phenomenology (phaneroscopy), the (c) generative categorial role, the (d) semiotic effectiveness, the (e) doctrine of continuity, and the (f) logic of relatives. This article will show that for Deleuze nearly like more recent Peircean scholarship: (a) zeroth state holds a categorial status, (b) some phenomenon instantiates zeroth state, (c) zeroth state plays a generative role regarding Peirce's categories, (d) zeroth state as a phenomenological category is semiotically effective, (e) zeroth state and the doctrine of continuity, and (f) zeroth state is a “medad.” In order to assess these topics, we recover Deleuze’s advances on Peirce’s philosophy and confront them with the Peircean studies on the zeroth state. Finally, we will see that Deleuze, far from being a Peircean scholar, developed the importance of the zeroth state for Peirce’s semiotics in advance to the subsequent Peircean scholarship. Reciprocally, most of Deleuzian scholars understimate the importance of Peirce for Deleuze’s semiotics.

Harold Macmillan, Margaret Thatcher and British Conservatism’s Politics of Negation

A distinction is often made between two traditions of British Conservative thought: the paternalist “one nation/middle way” tradition, and the more libertarian one that is associated with Margaret Thatcher. Typically, this distinction is mapped onto the ideas that Conservatives hold about equality. While “one nation” Conservatives condone some forms of redistribution, their libertarian counterparts, it is argued, do not. This article reassesses this distinction by comparing the political thinking of two representatives of the two traditions: Harold Macmillan and Margaret Thatcher. Comparing Macmillan and Thatcher’s views on equality, it is argued, is inappropriate as a way of distinguishing between their respective ideologies. Instead, we should draw a distinction between the way that Macmillan and Thatcher arrived at their views about equality. On his part, Macmillan adopted a triadic logic. By conceiving of a “middle way”, he defined his conservatism in relation to the dangerous “extremes” of free market capitalism and state socialism. Thatcher, by contrast, adopted a dyadic logic. Instead of locating herself between two “extremes”, she employed a series of binary distinctions to establish her conception of conservatism. Freedom was preferred to equality; the market was preferred to the state, and so on. The article explores the implications of these differing logics, and it concludes if we are to identify different variants of conservatism, we should be just as concerned with their modes of negation as we are with the affirmative claims they make about concepts like equality.

Untying the Gordian Knot: Process, Reality, and Context

Untying the Gordian Knot: Process, Reality, and Context

Tableau method of proof for Peirce’s three-valued propositional logic

Peirce’s triadic logic has been under discussion since its discovery in the 1960s by Fisch and Turquette. The experiments with matrices of three-valued logic are recorded in a few pages of unpublished manuscripts dated 1909, a decade before similar systems have been developed by logicians. The purposes of Peirce’s work on such logic, as well as semantical aspects of his system, are disputable. In the most extensive work about it, Turquette suggested that the matrices are related in dual pairs of axiomatic Hilbert-style systems. In this paper, we present a simple tableau proof for a fragment of Peirce three-valued logic, called P3, based on similar approaches in many-valued literature. We demonstrated that this proof is sound and complete. Besides that, taking the false as the only undesignated value and adding non-classical negations to the calculus, we can explore paraconsistent and paracompleteness theories into P3. Keywords: Charles S. Peirce, many-valued logics, theory of proof, tableau method.

Peirce’s Triadic Logic and Its (Overlooked) Connexive Expansion

In this paper, we present two variants of Peirce’s Triadic Logic within a language containing only conjunction, disjunction, and negation. The peculiarity of our systems is that conjunction and disjunction are interpreted by means of Peirce’s mysterious binary operations Ψ and Φ from his ‘Logical Notebook’. We show that semantic conditions that can be extracted from the definitions of Ψ and Φ agree (in some sense) with the traditional view on the semantic conditions of conjunction and disjunction. Thus, we support the conjecture that Peirce’s special interest in these operations is due to the fact that he interpreted them as conjunction and disjunction, respectively. We also show that one of our systems may serve as a suitable base for an interesting implicative expansion, namely the connexive three-valued logic by Cooper. Sound and complete natural deduction calculi are presented for all systems examined in this paper.

Peirce's Triadic Logic: Modality and Continuity

Peirce's Triadic Logic: Modality and Continuity

Signos, metáforas y educación física

Sólo podemos pensar mediante signos como las metáforas: procesos semióticos que vinculan varios objetos con ayuda de una evocación basada en la semejanza. “La acción motriz, punta de lanza de la educación física” es una de las metáforas favoritas de Pierre Parlebas, y no le falta razón. Su exploración mediante la lógica triádica de Peirce permite avanzar en el conocimiento de los tres símbolos fundamentales de nuestro ámbitos: la educación física, entendida como práctica de intervención motriz dirigida por legítimas aspiraciones transformadoras; la acción motriz, entendida como punto de encuentro de las conductas motrices que se quieren modificar y las intenciones educativas que les dan valor; y la praxiología motriz, entendida como respuesta a la necesidad que los motricistas tenemos de anticipar y organizar correctamente los efectos esperados de las practicas motrices y los efectos deseados en los proyectos educativos. Pragmatismo y praxiología motriz tienen mucho que aportar a la educación física, aunque sólo sea la excusa para conectar a dos pensadores brillantes que cruzaron los límites marcados por sus respectivos siglos para ir más allá de las herencias recibidas.

“Fantastic Tricks before High Heaven,” Measure for Measure and Performing Triads

Reading Measure for Measure through the logic of substitution has been a long-standing critical tradition; the play seems to invite topical, political, and religious parallels at every turn. What if the logic of substitution in the play goes beyond exchange and seeks out a triadic logic instead? This insistent searching for the triad appears most notably in the performance of Measure for Measure by Cheek by Jowl (2013–2019). Cheek By Jowl’s strategies of touring, simplicity, movement, and liberation create a dynamic and ever-evolving performance. This article puts Cheek by Jowl’s performance of Measure for Measure in conversation with C.S. Peirce’s (and subsequent theorists) explorations of triadic logic with Puttenham’s rhetoric of traductio (repetition with variation, and "tranlacing"), in addition to critical work on substitutions in the play. Tracing the superfluity of substitutions in rhetoric and performance of the play allows us to see how the play refuses binaries, and energizes triadic logic as a means to liveness in performance. Both Shakespeare’s play and the Cheek By Jowl production use a triadic structure which suggests the Trinity, foregrounding the body as a site of mediation and liveness.

Rethinking approaches to building a harmonious food market and modelling its development

Purpose. The aim of the article is to identify the possibility of exploring the process of tripartite market interaction of the multiple food provision sphere stakeholders as a holistic unity through the formalization of dynamic change, describing their properties with taking into account the time factor in mind. Methodology of research. The advantages of the triadic logic of conducting scientific research aimed at harmonizing the interaction of multiply stakeholders in the food provision sphere above the dyad are established using of the comparison method. The need to build a harmonious food market requires the synthesis of traditional techniques with new knowledge. The use of synthesis, the essence of which is the imaginary unification of individual parts of the object of scientific research into a single whole, due also to the fact that the scope of food security is a coherent unity. Given the need to visualize the objects of scientific search, modelling has a special role in the process of scientific knowledge of the nature of economic phenomena and processes, as well as finding out the causes of appearance of the problematic moments, it is the modelling itself. Findings. The complexity and multidimensionality of tripartite interaction of the food provision requires a rethinking of traditional approaches to building semi-structured, open systems. As a result of comparative analysis, the need to move from dialectic to conducting a study of harmonization of market interaction in the field of food provision sphere on the basis of trialectics is proved. The continuation of the trialectic logic is the use of synthesis, which made it possible to realize that the interaction of the absolute opposite forms a triune space, which should be regarded as a harmoniously developing integrity. The use of simulation modelling made it possible to visualize the synthetic interaction of the stakeholders of the triad of subject and functional components of the food provision sphere. Originality is to combine vector algebra with spatial geometry, which made it possible to visualize dynamic market interaction in the food provision sphere as in a systematic triad with taking into account the time factor in mind. Practical value. The triple helix as an institutional matrix for the harmonization of the stakeholders' market interaction of the triad of subject and functional components of the domestic food provision sphere in the whole of production, consumer and regulatory aspects should ensure its synergistic stability in the context of globalization of food markets. Key words: food provision sphere; marketing strategy; stakeholder; motion vector; systematic Triad; trilateral market interaction; balancing economic interests; harmonious food market; triune space; moral economy; triple helix model; synergy; disproportions; dyadic paradigm; trialectics.

An Application of Peircean Triadic Logic: Modelling Vagueness

Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence (AI). Charles Sanders Peirce (1839–1914) was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting Peircean semiotics within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic.

God, Incarnation in the Feminine, and the Third Presence

This paper deals with the possibility of an incarnation in the feminine in our age. In the first part, we discuss sexual genealogies in ancient Israel and address the problem of the extreme vulnerability of feminine life in the midst of an ancient sacrificial crisis. The second part opens with an analysis of Feuerbach’s interpretation of the Trinity. The triadic logic, as found within various religious contexts, is also affirmed. Based on our analyses from the first and the second part, in the third part, we address the problems of feminine vulnerability and fragility on one hand, and triadic thinking on the other hand, and relate them to an original proposal for the future matrixial theology of incarnation.

Пирс и парадокс лжеца

The liar paradox is a well-known logical and philosophical paradox, yet it has always been a tough call for professional philosophers, and it played a critical role in the history of philosophy, logic, and even mathematics. examine the works of the founder of pragmatism Charles Sanders Peirce and his approaches to this problem, and try to reconstruct the evolution of his views on this issue. We may approach the proposition This proposition is not from three points of view: its meaning, its truth value and its significance. The earliest Peirce's approach focuses on the meaning and truth value. He argues that the proposition This proposition is not is meaningless and at the same time both true and not true. He introduces this way of reasoning in the 1865 in his Harvard lectures on the logic of science. His second approach was presented in the article Grounds of Validity of the Laws of Logic: Further Consequences of Four Incapacities, published in 1869. He rejected his earlier account and based a novel solution in truth value on a Scholastic philosopher Paul of Venice's statement that every proposition asserts its own truth. Thus, the proposition This proposition is not implies assertion of its own truth and at the same time says that it is not true; ergo it is false inasmuch it is self-contradictory. In 1903 Peirce rejected this approach because for any proposition it is true that it is not equivalent to proposition I assert this proposition. Peirce has never presented a third solution of the liar paradox, but there are three issues closely connected with the topic: Russell-Peirce connection, Peirce mature theory of truth, which may propose a way to solve the liar paradox focusing on its significance, and Peirce's development of triadic logic. discuss them in the second part of the article.

Plasticity and Creativity in the Logic Notebook

Peirce’s architectonics, far from rigid, is bended by many plastic transformations, deriving from the cenopythagorean categories, the pragmaticist (modal) maxim, the logic of abduction, the synechistic hypotheses and the triadic classification of sciences, among many other tools capable of molding knowledge. Plasticity, in turn, points to interlacements between mathematics and art, and shapes some associated conceptual forces in the boundary of the disciplines: variation, modulation and invariance; transformability, continuity and discreteness; creative emergence. In this article we focus on this third aspect, through bounded, well defined case studies in the Logic Notebook. The first section describes the manuscript and its interest for a study of creativity, leading to a short speculation on “creative reason” and “plastic imagination” in Peirce. The second section studies five precise cases of creative emergence in the Logic Notebook: differential relatives, existential graphs, sequence diagrams, triadic logic, translatability. Some major surprises occur in those detailed studies.

Pascal and the Persistence of Platonism in Early Modern Thought

AbstractThe following paper argues that Blaise Pascal, in spite of his famous opposition between the God of the Philosophers and the God of “Abraham, Isaac and Jacob” has significant affinities with the tradition of Renaissance Platonism and is in fact a Platonist in his overall outlook. This is shown in three ways. Firstly, it is argued that Pascal’s skeptical fideism has roots in the notion of faith developed in post-Plotinian neo-Platonism. Secondly, it is argued that Pascal makes considerable use of the Platonic notion of an indefinite dyadic principle. Thirdly, it is argued that Pascal’s religious psychology gives a centrality to the body that brings it close to the theurgical standpoint of figures like Iamblichus. Pascal is then contrasted to figures like Cusanus and Pico in that a dyadic principle of opposition is more prominent in his work than a triadic logic of mediation.

Understanding Peircean Semiotics and Its Influence on Linguistics – An Exploration on Triadic Logic System of Categories and Model of Signs

Abstract Peircean Semiotics is inaccessible to most people for its triadic logic system of categories and model of signs. Author of this paper tries to understand it through a historical view of logic systems of categories and interpret Peircean semiotics and its applications in linguistic studies.

Analysis, synthesis, systems thinking and the scientific method: rediscovering the importance of open systems

AbstractThis paper reconsiders the role of systems thinking in science and addresses an issue raised by Peter Checkland some 25 years ago, that of the relationship between systems thinking, analysis and the scientific method. The paper argues that the scientific method is most usefully interpreted as a dialectic between analysis and synthesis supported by the triadic logic of C.S. Peirce, and that the role of systems thinking is to frame this dialectic. Three generic forms of the system concept are identified—closed systems, input‐output systems and open systems. Consequently, systems thinking is shown to play a central role in mainstream science. The paper concludes that a system is best defined as a cognitive construct for making sense of complexity and the organization of knowledge and that contemporary system thinking is best identified as the ethical, scientific pursuit of knowledge using the socio‐ecological (open) systems frame. Copyright © 2007 John Wiley & Sons, Ltd.

Peirce's Semiotics, Subdoxastic Aboutness, and the Paradox of Inquiry

The author suggests that educational philosophy should benefit from addressing questions traditionally asked within discourse in the philosophy of mind, namely: the relation between the mind and world and the problems of intentionality (or aboutness), meaning, and representation. Peirce's semiotics and his category of creative abduction provide a novel conceptual framework for exploring these questions. A model of reasoning and learning, based on Peirce's triadic logic of relations, is analysed. This model, it is argued, is fruitful for overcoming the paradox of new knowledge that was first debated by Socrates in his dialogue with Meno.

Quantification for Peirce's preferred system of triadic logic

Without introducing quantifiers, minimal axiomatic systems have already been constructed for Peirce's triadic logics. The present paper constructs a dual pair of axiomatic systems which can be used to introduce quantifiers into Peirce's preferred system of triadic logic. It is assumed (on the basis of textual evidence) that Peirce would prefer a system which rejects the absurd but tolerates the absolutely undecidable. The systems which are introduced are shown to be absolutely consistent, deductively complete, and minimal. These dual axiomatic systems reveal an interesting elegance, independent of their historical motivation.

Philosophical Links to TA: Hegel and the Concept of the Adult Ego State

The “Hegelian Model” of triadic logic is applied to the PAC triad to explain certain conceptual connections among the three basic ego states. Oftentimes, the Adult is regarded simply as a third alternative to Parent and Child. An argument is developed to support the logical primacy of the Integrated Adult ego state, which is considered to subsume the Parent and Child by integrating them into the full range of Adult behavior. This is in distinction to the more traditional concept of the Adult per se. The Hegelian Model is held to provide theoretical justification for the PAC triad.

Alternative Axioms for Peirce's Triadic Logic

Alternative Axioms for Peirce's Triadic Logic