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. (shrink)

Peirce was the first logician to define three-valued logical connectives. In 1909, he defined four one-place three-valued connectives and six two-place three-valued connectives, all of which were rediscovered by later logicians. Peirce’s motivation was to accommodate within formal logic a specific, narrow range of propositions he took to be neither true nor false, viz. propositions that predicate of a breach in mathematical or temporal continuity one of the properties that is a boundary-property relative to that breach.

This is a discussion of a three-valued logic in Peirce's writings.

In early 1909, Charles S. Peirce conducted a series of experiments with three-valued logic, anticipating the pioneering work of Jan Łukasiewicz and Emil Post by ten years. These experiments are entirely contained within six or seven pages of Peirce's Logic Notebook. Due to the work of Atwell Turquette, the formalisms contained in those pages are relatively well understood. What is less understood are Peirce's philosophical reasons for conducting those experiments. His explanation of the need for his "triadic" (...) logic is very brief, taking up little more than a single short page in the Notebook. Here he gives us two clues about his motivations, one connected to modal notions and one to his views on continuity. There are two previous accounts of the philosophical motivations behind triadic logic, due to Max Fisch and Turquette, and to Robert Lane. In this paper, I re-evaluate those views and connect the two clues to Peirce's hypothetical cosmology. I argue that in conducting his three-valued experiments, Peirce was trying to create a logic to capture his notion of the evolving universe. (shrink)

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. Charles Sanders Peirce 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. (shrink)

In this collection on the Kyoto School of Philosophy, the author offers the reader Tanabe's religious philosophy, but also, and for the first time, his ...

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. (shrink)

In his Indigenous critique of the field of animal studies, Billy-Ray Belcourt (Driftpile Cree Nation) describes it as having an analytic blind spot when it comes to settler-colonialism, a blind spot that manifests through universalising claims and clumsy arguments about ‘shared’ oppressions, through assumptions that settler colonial political institutions can be a neutral part of the solution, and through a failure to engage with ‘Indigenous studies of other than human life’ (20). In the same article, he calls on decolonial projects (...) to do more to include animality within their purview, to include critiques of animal agriculture and to incorporate critiques of anthropocentrism as ‘a key logic of white supremacy’. Belcourt’s critique of both Animal studies and decolonial projects on the basis of an unequal but mutual marginalisation is an important starting point for research projects like ours that hope to bring Animal studies and Indigenous studies approaches into dialogue about the cultural impacts of introduced animals. Our approach sets out to be ‘triadic’, always involving at least three sides; Settler- Coloniser, Indigene and Animal. (shrink)

Tanabe Hajime, another pole of the so-called Kyoto-School of Philosophy of modern Japan, attempts to construct a dialectical, triadic logic of genus, species and individual as a creative synthesis between Eastern and Western philosophy. Although the formal pattern of his method is influenced by the Hegelian dialectic, the way of his thinking is rather prevailed by Kantian dualism. This makes a sharp contrast to his mentor Nishida Kitaro, whose logic of Topos or Place qua Absolute Nothingness is (...) criticized as all-embracing and static in character by him. The difference between them might be parallel to that of Greek and Latin theology concerning the Trinity. Tanabe never presupposes any preexistent entity as the primordial One in the eternal dimension, but rather maintains the individuality as the free subjective agent in the field of history. The dichotomy between the universal and the individual is overcome in and through the mediation of the third term— the species — as the negatively self-realized, specific form of the genus. The species, however, turns out to be the self-estrangement, when it loses the perpetually negative mediation of the free subjective activity of the individual. (shrink)

By performing an abduction of Bateson and Peirce, we come to understand the logic of the sacred as a semiotic and phenomenological communicative phenomenon. First, I compare and contrast their ideas concerning ontology, epistemology, and logic. Next, I articulate how both theorists construct their epistemologies within a triadic frame of relations that successfully accounts for a communicative logic that activates the integration of body and Mind. Finally, I bring Bateson’s triadic relations of aesthetics,consciousness (mental process), (...) and the sacred in line with Peirce’s existential semiotic categories of Firstness, Secondness, and Thirdness. We discover the sacred is constituted within a communicative logic that is abductive. As such, I argue that Bateson’s epistemology of the sacred becomes more accessible as a postmodern philosophy when viewed as a condition of phenomenological semiosis. (shrink)

This chapter examines María Lugones’s germane and insightful attempt to theorize “intermeshed oppressions,” which, she argues, have been (mis)represented in women of color feminisms by the concepts of “interlocking systems of oppression” and, more recently, “intersectionality.” The latter, intersectionality, introduced by Black feminist legal scholar Kimberlé Williams Crenshaw as a metaphor (1989) and as a “provisional concept” (1991), has become the predominant way of referencing the mutual constitution of what have been theorized as multiple systems of oppression, constructing the multiplicity (...) of social identities. But Lugones’s analysis, which maintains subtle but important distinctions among the concepts of “intermeshed,” “interlocking,” and “intersecting” oppressions, shows that intersectionality theory often conflates fragmentation with multiplicity, and—by reifying “intersectional identities”—reproduces social-ontological fragmentation at the political and perceptual-cognitive levels of representation. Intersectional accounts redeploy unitary categories (for instance, race, gender, class, sexuality, disability) that are defined to the exclusion of each other by privileging the identities of normative group members. Consequently, they remain within what Lugones calls the “logic of purity,” which erases “curdled,” impure, category-transgressive, border-dwelling, mestiza subjects. Although, according to Lugones, intersectionality enables us to discern how the logic of purity produces “women of color” as impossible beings, she argues that the liminal identities of subjects dwelling in categorial interstices can only be made visible by conceptualizing oppressions as fused or “intermeshed.” However, as I interpret her, Lugones is not merely criticizing intersectionality or seeking to transcend its conceptual limitations by proposing an alternate concept. Rather, the concepts of “intermeshed,” “interlocking,” and “intersecting” oppressions do significantly different work in her account and illuminate different aspects of the social ontology, phenomenology, and epistemology of resistance to oppression. First, I situate Lugones with respect to the current conjuncture in intersectionality studies, in which some scholars are calling for a post-intersectional turn. Then, I reconstruct Lugones’s complex account of intermeshed oppressions, interlocking oppressions, and intersectionality. Finally, I discuss the status of intersectionality in the shift in Lugones’s work from “women of color feminisms” to “decolonial feminism.” Intersectionality is now routinely invoked as a representational theory of multiple identities, but Lugones’s heterodox interpretation helps us to see that it is best understood as a critique of representations based on the logic of purity: specifically, of how categorial axes of oppression (mis)represent intermeshed oppressions. Lugones’s triadic distinction (intersecting/interlocking/intermeshed) points toward a provisional usage of intersectionality, namely, to diagnose the fragmentation of social experiences of multiplicity (which, I would argue, is more consistent with the concept’s original aims). In her visionary philosophy, which attempts to theorize resistance against the grain of fragmentation from a conceptual space outside of the “logic of purity,” we find “glimpses” of a non-fragmented account of oppression, and praxical possibilities for liberatory, decolonial feminist coalitions. (shrink)

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. (shrink)

In 1909, Peirce recorded in a few pages of his logic notebook some experiments with matrices for three-valued propositional logic. These notes are today recognized as one of the first attempts to create non-classical formal systems. However, besides the articles published by Turquette in the 1970s and 1980s, very little progress has been made toward a comprehensive understanding of the formal aspects of Peirce's triadic logic (as he called it). This paper aims to propose a new (...) approach to Peirce's matrices for three-valued propositional logic. We suggested that his logical matrices give rise to three different systems, one of them - which we called P3 - is an original and non-explosive logic. Besides that, we will show that the P3 system can easily be transformed into paraconsistent and paracomplete calculi, adding to it, respectively, unary operators of consistency and intuitionistic negation. We conclude with a discussion about philosophical motivations. (shrink)

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. (shrink)

There are at least two politically salient senses of “representation”—acting-for-others and portraying-something-as-something. The difference is not just semantic but also logical: relations of representative agency are dyadic (x represents y), while portrayals are triadic (x represents y as z). I exploit this insight to disambiguate constructivism and to improve our theoretical vocabulary for analyzing political representation. I amend Saward’s claims-based approach on three points, introducing the “characterization” to correctly identify the elements of representational claims; explaining the “referent” in pragmatic, (...) not metaphysical terms; and differentiating multiple forms of representational activity. This enables me to clarify how the represented can be both prior to representation and constituted by it, and to recover Pitkin’s idea that representatives ought to be “responsive” to the represented. These points are pertinent to debates about the role of representatives, the nature of representative democracy, and the dynamics of revolutionary movements. (shrink)

This article explores the usefulness of interdisciplinarity as method of enquiry by proposing an investigation of the concept of information in the light of semiotics. This is because, as Kull, Deacon, Emmeche, Hoffmeyer and Stjernfelt state, information is an implicitly semiotic term (Biological Theory 4(2):167–173, 2009: 169), but the logical relation between semiosis and information has not been sufficiently clarified yet. Across the history of cybernetics, the concept of information undergoes an uneven development; that is, information is an ‘objective’ entity (...) in first order cybernetics, and becomes a ‘subjective’ entity in second order cybernetics. This contradiction relegates the status of information to that of a ‘true’ or ‘false’ formal logic problem. The present study proposes that a solution to this contradiction can be found in Deely’s reconfiguration of Peirce’s ‘object’ (as found in his triadic model of semiosis) into ‘thing’ and ‘object’ (Deely 1981). This ontology allows one to argue that information is neither ‘true’ nor ‘false’, and to suggest that, when considered in light of its workability, information can be both true and false, and as such it constitutes an organism’s purely objective reality (Deely 2009b). It is stated that in the process of building such a reality, information is ‘motivated’ by environmental, physiological, emotional (including past feelings and expectations) constraints which are, in turn, framed by observership. Information is therefore found in the irreducible cybersemiotic process that links at once all these conditions and that is simultaneously constrained by them. The integration of cybernetics’ and semiotics’ understanding of information shows that history is the analytical principle that grants scientific rigour to interdisciplinary investigations. As such, in any attempt to clarify its epistemological stance (e.g. the semiotic aspect of information), it is argued that biosemiotics does not need only to acknowledge semiotics (as it does), but also cybernetics in its interdisciplinary heritage. (shrink)

Charles Peirce incorporates modality into his Existential Graphs by introducing the broken cut for possible falsity. Although it can be adapted to various modern modal logics, Zeman demonstrates that making no other changes results in a version that he calls Gamma-MR, an implementation of Jan Łukasiewicz's four-valued Ł-modal system. It disallows the assertion of necessity, reflecting a denial of determinism, and has theorems involving possibility that seem counterintuitive at first glance. However, the latter is a misconception that arises from overlooking (...) the distinction between the intermediate truth values that are assigned to possibly true propositions as either X-contingent or Y-contingent. Any two propositions having the same ITV are possible together, while any two propositions having different ITVs, including those that are each other's negation, are possible individually yet not possible together. Porte shows that Ł-modal can be translated into classical logic by defining a constant for each ITV such that its implication of another proposition asserts the latter's possibility, while its conjunction with another proposition asserts the latter's necessity. These are expressed in the Alpha part of EG without broken cuts, simplifying derivations and shedding further light on Łukasiewicz's system, as long as graphs including either of the constants are properly interpreted. Ł-modal and Gamma-MR thus capture the two-sided nature of possibility as the limit between truth and falsity in Peirce's triadic conception. (shrink)

What the present work aimed to achieve is an assessment of the origin an d unity of Husserl s Logical Investigations. My approach was to take the history of its development as fundamental for the determination of its basic structure. Therefore, I proceeded to analyse Husserl s development between the Philosophy of Arithmetic and Logical Investigations with re spect to the fundamental issues in the justification of knowledge in mathematics and logic. In Husserl s own words, one of the (...) concerns that set him on the road to phenomenology was the clarification and analysis of the relation b etween the subjectivity of knowing and the objectivity of knowledge. Fro m my investigations it has, I hope, become apparent how this problem ori ginated in his earliest works and, through various influences and exchanges, led to his theory of intentionality in the first editio n of the Logical Investigations. The Logical Investigations is located between his Brentanist phase and t he ultimate development of transcendental phenomenology. It is the sedim ent of Husserl s logical investigations during the 1890s, which is the m ain period I analysed in the present work. Husserl himself often remarke s on the early origin of some of the issues he tried to deal with in the Logical Investigations. Still in his sketches for a new preface to the second edition of 1913, he took pains to point out the developmental his tory of the work from the Philosophy of Arithmetic onward. Regarding the Philosophy of Arithmetic, more often than not, Husserl remarks that it was just and elaboration of his Habilitationsschrift. Hence, following h is lead, we tried to embrace the whole period of his psychological and l ogical analyses from 1886/7 to 1900/01. Of course, I was certainly not the first, nor will I be the last, to be intrigued by Husserl s development in these years. Numerous books and articles have been written long before mine on precisely these issue s, often dealing with specific topics, such as the uneasy relation of psychology and psychologism in Husserl s work or fundamental inʂ 58;uences such as Bolzano s, which I therefore chose to leave aside. Not simply to duplicate or summarise such pre-existing scholarship, I tried to concentrate on the issues to which I felt I could contribute an orig inal insight, advancing the field. I would like to indicate ᤙ 7;ve points here, where I think I attained this goal: 1. The enduring relevance of Husserl s mathematical ba ckground. 2. Support for an early development of a theory of hig her order objects 3. A better evaluation of Frege s influence 4. A contextualisation of Schuhmann s thesis regarding the relevance of Twardowski for Husserl s notion of intentionality 5. The dependence of the unity of the Logical Investig ations on its historical development. Without understanding its developmental history we would misunderstand i ts internal coherence. In the first part we saw Husserl s early struggle to integrate the mathematical and psychological points of view on the philosophy of math ematics, a struggle that lasted from the Habilitationsschrift at least u p to and including his Doppelvortrag. The basic problem is the justi@ 257;cation of knowledge, in the case of mathematics, the justifica tion of knowledge in formal sciences, obtained with symbolic methods. Su ch problems regarding the subjectivity of knowing and the objectivity of knowledge guided the development of his position from Philosophy of Ari thmetic to Logical Investigations. In articulating his solution Husserl analysed the various relations betw een founding and founded layers of knowledge and whether and how one cou ld pass from on to the other. Already at the time of the Philosophy of A rithmetic he tried to account for higher order acts, relations and objec ts, the mature account of which would have to wait until the Logical Inv estigations and later. Among various influences on his progression towards phenomenology, Frege has been often considered to have provided the conversion to an ti-psychologism with his review of the Philosophy of Arithmetic. I think I have demonstrated that the review is almost entirely irrelevant in th is sense and that we should rather look at a much earlier influenc e on the Philosophy of Arithmetic. By showing that Brentano s conception of intentionality was already more sophisticated than until recently assumed, thanks to some recent public ations of unpublished material from Brentano s and Husserl s Nachlass, I think Schuhmann s thesis is right, but should be reinterpreted as being less radical than it would appear. Both Husserl and Twardowski were in& #64258;uenced by Brentano as his students. Following his and Meinong s l ead, they both elaborated a theory on intentionality, in connection with the other theories of intentionality in his school and the Brentano-Bol zano paradox. Furthermore, by providing a more continuous reading of Hus serl s development through the transcendental turn, I hope to have made clear that his later theory of the noema is not a return to a neo-Twardo wskian triadic position, but a reaffirmation, now in transcendental key, of his 1894 take on the matter. The Logical Investigations are uniquely determined by their historical c ontext in the middle of Husserl s development, falling almost exactly be tween the Habilitationsschrift and the Ideas. Due to their troubled hist ory, including changes in Husserl s earliest position around 1890, the f ailure of the second volume of the Philosophy of Arithmetic, various in& #64258;uences around 1894, and the subsequent reorientation and broadeni ng of his theories to a non-psychologistic logic and mathesis universali s, the Logical Investigations might at first seem to fall apart in two fundamentally disconnected books, the second one of which containin g six heterogeneous studies. It is my contention that it is exactly its troubled history that ties the Logical Investigations together, as Husse rl developed his logical investigations in the 1890s on the basis of a h andful thorny problems, including intentionality as centerpiece. From th e Habilitationsschrift to the Ideas it is the relation between subject a nd object, in all respects, ontological as well as epistemological, that guides his research. The Logical Investigations is a intermediate sedim ent of his struggles, containing an elaboration and improvement of his p osition in his early works and also pointing ahead towards the later pos ition of transcendental phenomenology as an implication and continuation of the solutions he proposes. Furthermore, I tried to address the problem that, while Husserl regarded his Logical Investigations as the breakthrough to phenomenology, many f undamental notions of the later transcendental phenomenology seem to be missing in the first edition of the Logical Investigations. The me thod of the reduction, the notion of a pure ego and a full account of th e noema are at most sketched or implied. Hence, I also looked ahead to t he further development of phenomenology after the Logical Investigations. As Husserl s influence has extended beyond the movement he found ed, I felt I could not bypass his significance for the currently d ominant analytical philosophy. Especially with respect to some of the co re topics of the Logical Investigations, but also earlier themes such as Husserl s semiotics, there is an interest in phenomenology from the ana lytic side. Nevertheless, precisely with regard to the unifying theme of intentionality, analytical philosophy often seems to miss the mark, and spectacularly so. I discussed Searle s influential, but in my opi nion, misguided and flawed approach, and I hope it is clear that I have chosen my method and approach in complete antithesis to his. The central contributions I was able to give, were only possible thanks to solid historical groundwork as necessary prerequisite for systematic analyses. Wherever possible, a reference to the original primary sources was provided, allowing the texts to speak for themselves and providing the reader with the critical instruments to evaluate my interpretations. As Husserl stood in a fruitful exchange with many other scholars, as st udent, colleague, friend and teacher, the intellectual context of his th oughts is essential to a correct understanding of his theories and aims. I hope to have found the right balance in the historical and the system atical aspects of my work to suit the interests of the reader and the re quirements of scholarship. (shrink)

I read Stefan Collini’s What are Universities For? last week with very mixed feelings. In the past, I’ve much admired his polemical essays on the REF, “impact”, the Browne Report, etc. in the London Review of Books and elsewhere: they speak to my heart. If you don’t know those essays, you can get some of their flavour from his latest article in the Guardian yesterday. But I found the book a disappointment. Perhaps the trouble is that Collini is too decent, (...) too high-minded, has too unrealistically exalted a view about what actually happens in universities which is too coloured by attitudes appropriate to the traditional humanities. And he is optimistic to the point of fantasy if he thinks that people are so susceptible to “the romance of ideas and the power of beauty” that they will want, or can be brought to want, lots and lots of universities in order to promote the ideas (as if they would suppose that the task of “conserving, understanding, extending and handing on to subsequent generations the intellectual, scientific, and artistic heritage of mankind” was clearly ill-served when there were only forty universities in England, as opposed to a hundred and whatever). (shrink)

A common reaction to Hegel’s suggestion that we collapse Kant’s distinction between form and content is that, since such a move would also deprive us of any way of distinguishing the merely logical from the real possibility of our concepts, it is incoherent and ought to be rejected. It is true that these two distinctions are intimately related in Kant, such that if one goes, the other does as well. But it is less obvious that giving them up as Kant (...) conceives them is as incoherent a proposal as many of Hegel’s critics think. It has been the point of a recent account of Hegel’s idealism to demonstrate that his critique of Kant’s dichotomy between form and content or concept and intuition does not commit him to the view that human cognition is materially creative of its content in the manner of a God-like or intuitive intellect—does not, in other words, signify a return to a pre-Critical metaphysics. This is the interpretative stance I adopt here as well, in hope of giving defenders of Kant reasons for taking Hegel more seriously. (shrink)