It's often hypothesized that the structure of mental representation is map-like rather than language-like. The possibility arises as a counterexample to the argument from the best explanation of productivity and systematicity to the language of thought hypothesis—the hypothesis that mental structure is compositional and recursive. In this paper, I argue that the analogy with maps does not undermine the argument, because maps and language have the same kind of compositional and recursive structure.

Most of us create and use a panoply of non-sentential representations throughout our ordinary lives: we regularly use maps to navigate, charts to keep track of complex patterns of data, and diagrams to visualize logical and causal relations among states of affairs. But philosophers typically pay little attention to such representations, focusing almost exclusively on language instead. In particular, when theorizing about the mind, many philosophers assume that there is a very tight mapping between language and thought. Some analyze utterances (...) as the outer vocalizations of inner thoughts (e.g. Grice 1957, Devitt 2005), while others treat thought as a form of inner speech (e.g. Sellars 1956/1997, Carruthers 2002). But even philosophers who take no stand on the relative priority of language and thought still tend to individuate mental states in terms of the sentences we use to ascribe them. Indeed, Dummett (1993) claims that it is constitutive of analytic philosophy that it approaches the mind by way of language. In many ways, this linguistic model is salutary. Our thoughts are often intimately intertwined with their linguistic expression, and public language does provide a comparatively tractable proxy for, and a window into, the messier realm of thought. However, an exclusive focus on thought as it is expressed in language threatens to leave other sorts of thought unexplained, or even to blind us to their possibility. In particular, many cognitive ethologists and psychologists find it useful to talk about humans, chimpanzees, birds, rats, and even bees as employing cognitive maps. We need to make sense of this way of talking about minds as well as more familiar sentential descriptions. In what follows, I investigate the theoretical and practical possibility of non-sentential thought. Ultimately, I am most interested in the contours of distinctively human thought: what forms does human thought take, and how do those different forms interact? How does human thought compare with that of other animals? In this essay, however, I focus on a narrower and more basic theoretical question: could thought occur in maps? Many philosophers are convinced that in some important sense, thought per se must be language-like.. (shrink)

Responding to the paradox of inference and the related problems in philosophy of logic, this paper argues for the necessity of distinguishing between two different objects of justification: logica utens and logica docens. Then, equipped with Peirce’s critical common-sensist conception of logica utens and his classification of sciences, I propose a diagnosis of and a solution to the problem of justification of logic. I argue that this alternative approach successfully avoids circularity in which most attempts in philosophy of logic have (...) been inescapably trapped. I also anticipate and respond to possible objections to taking a Peircean turn in justifying logic. (shrink)

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

Deduction is decisive but nonetheless mysterious, as I argue in the introduction. I identify the mystery of deduction as surprise-effect and demonstration-difficulty. The first section delves into how the mystery of deduction is connected with the representation of information and lays the groundwork for our further discussions of various kinds of representation. The second and third sections, respectively, present a case study for the comparison between symbolic and diagrammatic representation systems in terms of how two aspects of the mystery of (...) deduction – surprise-effect and demonstration-difficulty – are handled. The fourth section illustrates several well-known examples to show how diagrammatic representation suggests more clues to the mystery of deduction than symbolic representation and suggests some conjectures and further work. (shrink)

Parallelism has been drawn between modes of representation and problem-sloving processes: Diagrams are more useful for brainstorming while symbolic representation is more welcomed in a formal proof. The paper gets to the root of this clear-cut dualistic picture and argues that the strength of diagrammatic reasoning in the brainstorming process does not have to be abandoned at the stage of proof, but instead should be appreciated and could be preserved in mathematical proofs.

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)

We present a reading of the traditional syllogistics in a fragment of the propositional intuitionistic multiplicative linear logic and prove that with respect to a diagrammatic logical calculus that we introduced in a previous paper, a syllogism is provable in such a fragment if and only if it is diagrammatically provable. We extend this result to syllogistics with complemented terms à la De Morgan, with respect to a suitable extension of the diagrammatic reasoning system for the traditional case and a (...) corresponding reading of such De Morgan style syllogistics in the previously referred to fragment of linear logic. (shrink)

Throughout its long history, mathematics has involved the use ofsystems of written signs, most notably, diagrams in Euclidean geometry and formulae in the symbolic language of arithmetic and algebra in the mathematics of Descartes, Euler, and others. Such systems of signs, I argue, enable one to embody chains of mathematical reasoning. I then show that, properly understood, Frege’s Begriffsschrift or concept-script similarly enables one to write mathematical reasoning. Much as a demonstration in Euclid or in early modern algebra does, a (...) proof in Frege’s concept-script shows how it goes. (shrink)

In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...) localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)

This paper systematically outlines Wittgenstein's ab-notation. The purpose of this notation is to provide a proof procedure in which ordinary logical formulas are converted into ideal symbols that identify the logical properties of the initial formulas. The general ideas underlying this procedure are in opposition to a traditional conception of axiomatic proof and are related to Peirce's iconic logic. Based on Wittgenstein's scanty remarks concerning his ab-notation, which almost all apply to propositional logic, this paper explains how to extend his (...) method to a subset of first-order formulas, namely, formulas that do not contain dyadic sentential connectives within the scope of any quantifier. (shrink)

In contrast to Hintikka’s enormously complex distributive normal forms of first- order logic, this paper shows how to generate minimized disjunctive normal forms of first-order logic. An effective algorithm for this purpose is outlined, and the benefits of using minimized disjunctive normal forms to explain the truth conditions of propo- sitions expressible within pure first-order logic are presented.

Mathematics is a struggle for many. To make it more accessible, behavioral and educational scientists are redesigning how it is taught. To a similar end, a few rogue mathematicians and computer scientists are doing something more radical: they are redesigning mathematics itself, improving its ergonomic features. Charles Peirce, an important contributor to ordinary symbolic logic, also introduced a rigorous but non-symbolic, graphical alternative to it that is easier to picture. In the spirit of this iconic logic, George Spencer-Brown founded iconic (...) mathematics. Performing iconic arithmetic, algebra, and even trigonometry, resembles doing calculations on an abacus, which is still popular in education today, has aided humanity for millennia, helps even when it is merely imagined, and ameliorates severe disability in basic computation. Interestingly, whereas some intellectually disabled individuals excel in very complex numerical tasks, others of normal intelligence fail even in very simple ones. A comparison of their wider psychological profiles suggests that iconic mathematics ought to suit the very people traditional mathematics leaves behind. (shrink)

Technology is not only an object of philosophical reflection but also something that can change this reflection. This paper discusses the potential of computer-supported argument visualization tools for coping with the complexity of philosophical arguments. I will show, in particular, how the interactive and web-based argument mapping software “AGORA-net” can change the practice of philosophical reflection, communication, and collaboration. AGORA-net allows the graphical representation of complex argumentations in logical form and the synchronous and asynchronous collaboration on those “argument maps” on (...) the internet. Web-based argument mapping can overcome limits of space, time, and access, and it can empower users from all over the world to clarify their reasoning and to participate in deliberation and debate. Collaborative and web-based argument mapping tools such as AGORA-net can change the practice of arguing in two dimensions. First, arguing on web-based argument maps in both collaborative and adversarial form can lead to a fundamental shift in the way arguments are produced and debated. It can provide an alternative to the traditional four-step process of writing, publishing, debating, and responding in new writing with its clear distinction between individual and social activities by a process in which these four steps happen virtually simultaneously, and individual and social activities become more closely intertwined. Second, by replacing the linear form of arguments through graphical representations of networks of inferential relations which can grow over time in an infinite space, these tools do not only allow a clear visualization of structures and relations, but also forms of collaboration in which, for example, participants work on different “construction zones” of larger argument maps, or debates are performed at specific points of disagreement on those maps. I introduce the term synergetic logosymphysis to describe a practice that combines these two dimensions of collaborative- and web-based argument mapping. (shrink)

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.

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

Following the guiding thread of Peirce’s use of diagrammatic syntax in his system of existential graphs , which depends crucially on the role of the Sheet of Assertion, we introduce the notion of Sheet of Indication as the basis for a general diagrammatic semantics applicable to a wide range of diagrams. We then show how Peirce’s EG-alpha graphs may be understood as instances of SIs and how logically coherent models of the graphs are represented in the SI semantics.

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

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and (...) Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains. (shrink)

While the topic of assertion has recently received a fresh wave of interest from Peirce scholars, to this point no systematic account of Peirce’s view of assertion has been attempted. We think that this is a lacuna that ought to be filled. Doing so will help make better sense of Peirce’s pragmatism; further, what is hidden amongst various fragments is a robust pragmatist theory of assertion with unique characteristics that may have significant contemporary value. Here we aim to uncover this (...) theory, and to show that assertion for Peirce is not a mere corollary of pragmatic conceptions of truth, judgement, and belief, but is rather a central aspect of his philosophy. (shrink)

I develop an account of scientific representations building on Charles S. Peirce's rich, and still underexplored, notion of iconicity. Iconic representations occupy a central place in Peirce's philosophy, in his innovative approach to logic and in his practice as a scientist. Starting from a discussion of Peirce's approach to diagrams, I claim that Peirce's own representations are in line with his formulation of iconicity, and that they are more broadly connected to the pragmatist philosophy he developed in parallel with his (...) scientific work. I then defend the contemporary relevance of Peirce's approach to iconic representations, and specifically argue that Peirce offers a useful ‘third way’ between what Mauricio Suárez has recently described as the ‘analytical’ and ‘practical’ inquiries into the concept of representation. As a philosophically minded scientist and an experimentally inclined philosopher, Peirce never divorced the practice of representing from questions about what counts as a representation... (shrink)

This book presents novel formalizations of three of the most important medieval logical theories: supposition, consequence and obligations. In an additional fourth part, an in-depth analysis of the concept of formalization is presented - a crucial concept in the current logical panorama, which as such receives surprisingly little attention.Although formalizations of medieval logical theories have been proposed earlier in the literature, the formalizations presented here are all based on innovative vantage points: supposition theories as algorithmic hermeneutics, theories of consequence analyzed (...) with tools borrowed from model-theory and two-dimensional semantics, and obligations as logical games. For this reason, this is perhaps the first time that these medieval logical theories are made fully accessible to the modern philosopher and logician who wishes to obtain a better grasp of them, but who has always been held back by the lack of appropriate ‘translations' into modern terms.Moreover, the book offers a reflection on the very nature of logic, a reflection that is prompted by the comparisons between medieval and modern logic, their similarities and dissimilarities. It is thus a contribution not only to the history of logic, but also to the philosophy of logic, the philosophy of language and semantics.The analysis of medieval logic is also relevant for the modern philosopher and logician in that, being the unifying methodology used across all disciplines at that time, logic really provided unity to science. It thus presents a unified model of scientific investigation, where logic plays the aggregating role. (shrink)

This thesis is on the history and philosophy of logic and semantics. Logic can be described as the ‘science of reasoning’, as it deals primarily with correct patterns of reasoning. However, logic as a discipline has undergone dramatic changes in the last two centuries: while for ancient and medieval philosophers it belonged essentially to the realm of language studies, it has currently become a sub-branch of mathematics. This thesis attempts to establish a dialogue between the modern and the medieval traditions (...) in logic, by means of ‘translations’ of the medieval logical theories into the modern framework of symbolic logic, i.e. formalizations. One of its conclusions is that, when properly understood within their own framework, the interest of medieval logical theories for modern investigations go beyond mere historical interest, but that a thorough conceptual analysis of such theories must be undertaken in order to avoid conceptual misprojections. While such translations of medieval into modern logic have been attempted before, the approach presented here is innovative in that attention is paid to the similarities as well as to the dissimilarities between the two traditions, and to what can be learned from the medieval masters for modern investigations in logic and semantics. (shrink)

Endoporeutic method exemplifies the fundamental principle concerning the direction of the flow of interpretation in Peirce’s diagrammatic logic of existential graphs. It boils down to the fact that in EGs, one starts with the outermost graph-instance or the cut and proceeds inwards according to the conventions of EGs. The interpretation is dialogic, which Peirce laid out in his semeiotics in terms of two players, the Interpreter and the Utterer, and in EGs in terms of the Grapheus and the Graphist. Endoporeutic (...) method is thus closely related to semantic games. It is also an early sketch of model theory in logic, and marks an outlook on logic that puts the notion of compositionality under critical scrutiny. (shrink)

The years 1880-1914 were a time of intense experimentation in the visual arts. Representative conventions became variable, and artists deliberately departed from a concept of depiction considered as physical resemblance or photographic similarity. Visual representations progressed toward a conceptualization of figures and objects that transcended perceptual data, and the rendering of pictorial objects turned into an experiment involving complex visualization processes. This paper explores the interplay between artistic and scientific representative practices between 1880 and 1914. I argue that science and (...) technology acted as substantial challenges upon the concept of resemblance in art and that the rhythm of scientific and technological discoveries at the turn of the 20th century paralleled a shift from a notion of similarity to one of homomorphism in the conceptualization of pictorial representation. Homomorphism denotes representations which dispense with a point-to-point correspondence between depicted objects and perceptual data. I developed the concept from a scholarly study of Charles S. Peirce’s pragmatic account of representation, and in particular his theory of iconicity. Peirce defined iconic signs as “partaking in the character of the object” (CP 4.531), that is, as preserving the same relational structure as their object. A theory of iconicity as homomorphism successfully exemplifies representative relations based on structure preservation. Applied to 20th century representative practice, homomorphism offers a plausible explanation for representations in which a considerable conceptual effort is required, independently of a point-to-point correspondence between depicted objects and perceptual data. Using four case studies - the photographer Alfred Stieglitz and the painters André Derain, Max Weber and Pablo Picasso - I argue that representative practice between 1880 and 1914 was strongly informed by experimental scientific practices and that the shift from figurative to conceptual representation in art was triggered by a more significant theoretical shift involving representation as a general philosophical notion. Rather than a normative quest for the necessary and sufficient conditions for representation, I will propose a pragmatic evaluation of the means and strategies through which artists and scientists devise perspicuous and useful representations of the world. My analysis of the correlations between artistic and scientific representations at the turn of the 20th century aims to fulfill a twofold purpose. From a historical viewpoint, it draws significant parallels between the experimental aspects of representative practices in art and science considered as ways of exploring natural phenomena and intervening upon them. From a philosophical viewpoint, my goal is to propose a novel epistemological framework to assess how the shift in the conceptualization of representations affected subsequent styles of knowing and experimental practices in art and science. Ultimately, by combining the relative merits of historical and philosophical accounts of representation, I will argue for the advantages and desirability of a philosophically informed history of representative practice. (shrink)