Peirce considered the principal business of logic to be the analysis of reasoning. He argued that the diagrammatic system of Existential Graphs, which he had invented in 1896, carries the logical analysis of reasoning to the furthest point possible. The present paper investigates the analytic virtues of the Alpha part of the system, which corresponds to the sentential calculus. We examine Peirce’s proposal that the relation of illation is the primitive relation of logic and defend the view that (...) this idea constitutes the fundamental motive of philosophy of notation both in algebraic and graphical logic. We explain how in his algebras and graphs Peirce arrived at a unifying notation for logical constants that represent both truth-function and scope. Finally, we show that Shin’s argument for multiple readings of Alpha graphs is circular. (shrink)

1 INTRODUCTION Above the other titles he might justly have claimed, Charles S. Peirce prized the title 'logician'. He expressed in several places his ...

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. (shrink)

In a number of recent philosophical works, the concept of ontography has been raised to involve a revaluation of figurative and visual thinking against logico-conceptual thinking— i. e. a revaluation of a philosophical practice that supplements or departs from the traditional site of philosophy, language. This paper investigates the ontographic dimensions of Charles S. Peirce’s diagrammatology by focusing on his system of »existential graphs« as ontography avant la lettre.

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 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 that supports a “lattice of partial proofs” of pragmaticism. (shrink)

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 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 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. (shrink)

The following two articles comprise two sets of Charles Peirce’s manuscripts, “Recent Developments of Existential Graphs and their Consequences for Logic” (MS 498, MS 499, MS 490 & S-36, 1906) and “Assurance through Reasoning” (MS 669 & MS 670, 1911), written for the National Academy of Sciences meetings in 1906 and 1911. The papers are deposited at Houghton Library, Harvard University. Only some parts of MS 470 have been published before, and in somewhat defective form. Although “Assurance” follows (...) “Recent Developments” chronologically, given the expository style of the former it is recommended to be read before “Recent Developments”. As the title indicates, in the latter Peirce goes on to describe his latest discoveries concerning the method and the logic of existential graphs. The transcription reproduces all significant deletions that appear in the original sheets. Editorial comments and additions are given in brackets. [Alt.:] means the beginning of an alternative sequence. [Del. (shrink)

Lines of Identity 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. (shrink)

This paper examines Charles Peirce's graphical notation for first-order logic with identity. The notation forms a part of his system of "existential graphs," which Peirce considered to be his best work in logic. In this paper a Tarskian semantics is provided for the graphical system.

In his formal papers on existential graphs , 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 , 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. (shrink)

Teachers of critical thinking courses are justified in teaching some amount of propositional logic, especially logical equivalence and formal proofs for validity, but the otherwise informal nature of most critical thinking courses makes it difficult to decide how much propositional logic should enter a course. Most instructors use truth tables to teach the above two topics but they are too off-putting to be useful to most critical thinking students (as are derivations, the common alternative to truth tables). This paper presents (...) an accessible alternative to truth tables, namely, Peirce’s Existential Graphs. Detailing only what one needs to know about Existential Graphs in order to test propositions for logical equivalence and deductive argument forms for validity, this paper describes how they work, their terminology, their inference rules, and several examples which may aid in demonstrating their use to students. (shrink)

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, certain aspects of which relations force certain other aspects to be a certain way. (shrink)

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.

Charles S. Peirce’s pragmatist theory of logic teaches us to take the context of utterances as an indispensable logical notion without which there is no meaning. This is not a spat against compositionality per se , since it is possible to posit extra arguments to the meaning function that composes complex meaning. However, that method would be inappropriate for a realistic notion of the meaning of assertions. To accomplish a realistic notion of meaning (as opposed e.g. to algebraic meaning), Sperber (...) and Wilson’s Relevance Theory (RT) may be applied in the spirit of Peirce’s Pragmatic Maxim (PM): the weighing of information depends on (i) the practical consequences of accommodating the chosen piece of information introduced in communication, and (ii) what will ensue in actually using that piece in further cycles of discourse. Peirce’s unpublished papers suggest a relevance-like approach to meaning. Contextual features influenced his logic of Existential Graphs (EG). Arguments are presented pro and con the view in which EGs endorse non-compositionality of meaning. (shrink)

Since Sun-Joo Shin's groundbreaking study (2002), Peirce's existential graphs have attracted much attention as a way of writing logic that seems profoundly different from our usual logical calculi. In particular, Shin argued that existential graphs enjoy a distinctive property that marks them out as "diagrammatic": they are "multiply readable," in the sense that there are several di erent, equally legitimate ways to translate one and the same graph into a standard logical language. Stenning (2000) and Bellucci (...) and Pietarinen (2016) have retorted that similar phenomena of multiple readability can arise for sentential notations as well. Focusing on the simplest kinds of existential graphs, called alpha graphs (AGs), this paper argues that multiple readability does point to important features of AGs, but that both Shin and her critics have misdiagnosed its source. As a preliminary, and because the existing literature often glosses over such issues, we show that despite their non-linearity, AGs are uniquely parsable and allow for inductive de nitions. Extending earlier discussions, we then show that that in principle, all propositional calculi are multiply readable, just like AGs: contrary to what has been suggested in the literature, multiple readability is linked neither to non-linearity nor to AGs' dearth of connectives. However, we argue that in practice, AGs are more amenable to multiple readability than our usual notations, because the patterns that one needs to recognize to multiply translate an AG form what we call complex symbols, whose structural properties make it easy to perceive and process them as units. Nevertheless, we show that such complex symbols, though largely absent from our usual notations, are not inherently diagrammatic and can be found in seemingly sentential languages. Hence, while ultimately vindicating Shin's idea of multiple readability, our analysis traces it to a di erent source and thus severs its link with diagrammaticity. (shrink)

In this paper, a game-theoretical semantics is developed for the so-called alpha part of Charles S. Peirce's System of Existential Graphs of 1896. This alpha part is that portion of Peirce's graphs that corresponds to propositional logic. The paper both expounds a game-theoretical semantics for the graphs that seems close to Peirce's own intentions and proves for the alpha part of the graphs that this semantics is adequate.

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ 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. (shrink)

This first volume of the Logic of the Future edition collects Peirce's writings on the historical development, theory and application of his graphical method and diagrammatic reasoning. Its 28 selections of texts and extensive general and volume int.

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic (Peirce’s Existential Graphs α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}). The straightforward construction of a presheaf category EGα∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_{\alpha ^{*}}$$\end{document} of cuts-only Existential Graphs (equivalent to the well-studied category of finite forests) provides a basis for the further construction of the category EGα\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_\alpha $$\end{document} 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. (shrink)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ 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. (shrink)

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. (shrink)

The article presents two gestures corresponding to two profound new understandings of Peirce's Continuum and Peirce's Existential Graphs. Vargas and Oostra have revolutionized Peirce's mathematical studies, thanks to a first complete model for Peirce's continuum provided by Vargas, and thanks to the emergence of intuitionistic existential graphs provided by Oostra. The article aims at showing how these careful mathematical constructions can be encrypted in very simple gestures.

A. N. Prior very much admired the logic and philosophy of C.S. Peirce. In the spring of 1962 Prior went to Chicago to study Peirce's ideas. One of the topics that caught his attention was Peirce's existential graphs. This interest continued when he returned to England. In this paper Prior's grappling with the existential graphs will be discussed.

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 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... (shrink)

Peirce’s diagrammatic system of Existential Graphs (EGα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha })$$\end{document} is a logical proof system corresponding to the Propositional Calculus (PL). Most known proofs of soundness and completeness for EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document} 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 (...) class='Hi'>graphs, we present a purely syntactic proof of soundness, and hence consistency, for EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document}, along with two separate completeness proofs that are constructive in the sense that we provide an algorithm in each case to construct an EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document} 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. (shrink)

A framework for “improvisational” social acts and communication is introduced by referring to the idea of “relationalism” such as natural farming, permaculture and deep ecology. Based on this conception, the notion of Existential Graph by C. S. Peirce is introduced. The notion of extended self in deep ecology is substantiated based on the Roy Adaptation Model in Nursing Theory and Narrative approaches. By focusing on Leibnizian notions of space and time and by introducing Petri net, a spatio-temporal model of (...) improvisation is constructed. This model is expected to substantiate the interesting notion of “Ba” proposed by H. Shimizu reflecting Japanese culture. (shrink)

In 2001, Le Bars proved that there exists an existential monadic second-order sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence.

Peirce introduced Existential Graphs in late 1896, and they were systematically investigated in his 1903 Lowell Lectures. Alpha graphs for classical propositional logic constitute the first part of EGs. The second and the third parts are the beta graphs for first-order logic and the gamma graphs for modal and higher-order logics, among others. As a logical syntax, EGs are two-dimensional graphs, or diagrams, in contrast to the linear algebraic notations. Peirce's theory of EGs is (...) not only a theory of logical syntax but also a proof theory for many logics. A set of transformation rules of graphs defines a proof system for a graphical logic. In this... (shrink)

Logicians have strongly preferred first-order natural deductive systems over Peirce's Beta Graphs even though both are equivalent to each other. One of the main reasons for this preference, I claim, is that inference rules for Beta Graphs are hard to understand, and, therefore, hard to apply for deductions. This paper reformulates the Beta rules to show more fine-grained symmetries built around visual features of the Beta system, which makes the rules more natural and easier to use and understand. (...) Noting that the rules of a natural deductive system are natural in a different sense, this case study shows that the naturalness and the intuitiveness of rules depends on the type of representation system to which they belong. In a diagrammatic system, when visual features are discovered and fully used, we have a more efficacious deductive system. I will also show that this project not only helps us to apply these rules more easily but to understand the validity of the system at a more intuitive level. (shrink)

One of the claims made for C. S. Peirce's existential graphs has been that they are a deductively complete formulation of first-order logic with identity. As Peirce presented them, this is true only for certain versions of first-order logic :those which do not include terms for individuals. I amend Peirce's rules here, showing, in particular, how they are capable of demonstrating that, for instance, ?Jack is in the kitchen? contradicts ?Jack is not in the kitchen?

In this essay, we propose that Peirce’s Existential Graphs can derive the desired uniqueness implication (or in a weaker claim, the definite description readings) of donkey pronouns in conjunctive discourse (A man walks in the park. He whistles), without postulating a separate category of E-type pronouns.

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. (shrink)

Peirce’s beta graphs are roughly equivalent to our first-order predicate logic. However, Bellucci and Pietarinen have recently argued that the beta graphs are not well-equipped to handle asymmetric relative terms. I survey four proposed solutions to the problem and find them all wanting. I offer a fifth solution according to which Peirce’s beta graphs function at three different levels of abstractness from natural language. I diagnose the problem of asymmetric relative terms as arising when we transition from (...) the first to the second level of abstractness. Making grammatical information encoded in natural language explicit at the first level of abstractness and interpreting graphs at the second level of abstractness as shorn of grammatical information resolves the problem. The solution is both well-motivated by Peirce’s own commitments and increases the expressiveness of the beta graphs. (shrink)

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 . This paper modernizes the semantics Peirce proposed for EGs in terms of game-theoretic semantics . Peirce's 1905 proof is then reconstructed in three parts, by relating pragmaticism to the GTS conception of meaning, 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 bringing out the key links between EGs and pragmaticism. (shrink)

The logic of assertive graphs is a modification of Peirce’s logic of existential graphs, 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 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. (shrink)

Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic second-order sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain "built-in" relations, such as the successor relation). The proof makes use of Ehrenfeucht-Fraisse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential (...) monadic second-order sentence. (shrink)

The theory of existential graphs, which Peirce ultimately divided into four quadrants , is a rich method of analysis in the philosophy of logic. Its $$\upbeta $$ β -part boasts a diagrammatic theory of quantification, which by 1902 Peirce had used in the logical analysis of natural-language expressions such as complex donkey-type anaphora, quantificational patterns describing new mathematical concepts, and cognitive information processing. In the $$\upbeta $$ β -quadrant, he came close to inventing independence-friendly logic, the idea of (...) which he found indispensable in fulfilling the tasks –. (shrink)

We propose an original use of techniques from random graph theory to find a Monadic ∑ 1 1 sentence without an asymptotic probability. Our result implies that the 0-1 law fails for the logics ∑ 1 1 and ∑ 1 1 . Therefore we complete the classification of first-order prefix classes with or without equality, according to the existence of the 0-1 law for the corresponding ∑ 1 1 fragment. In addition, our counterexample can be viewed as a single explanation (...) of the failure of the 0-1 law of all the fragments of existential second-order logic for which the failure is already known. (shrink)