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)

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)

Recent philosophical analyses of the epistemic dimension of images in the sciences show a certain trend in acknowledging potential roles of these images beyond their merely decorative or pedagogical functions. We argue, however, that this new debate has yet paid little attention to a special type of pictures, we call ‘visual metaphor’, and its versatile heuristic potential in organizing data, supporting communication, and guiding research, modeling, and theory formation. Based on a case study of Conrad Hal Waddington’s epigenetic landscape images (...) in biology, we develop a descriptive framework applicable to heuristic roles of various visual metaphors in the sciences. (shrink)

Recent philosophical analyses of the epistemic dimension of images in the sciences show a certain trend in acknowledging potential roles of these images beyond their merely decorative or pedagogical functions. We argue, however, that this new debate has yet paid little attention to a special type of pictures, we call ‘visual metaphor’, and its versatile heuristic potential in organizing data, supporting communication, and guiding research, modeling, and theory formation. Based on a case study of Conrad Hal Waddington’s epigenetic landscape images (...) in biology, we develop a descriptive framework applicable to heuristic roles of various visual metaphors in the sciences. (shrink)

Reasoning with diagrams is considered to be a peculiar form of reasoning. Diagrams are often associated with imagistic representations conveyed by spatial arrangements of lines, points, figures, or letters that can be manipulated to obtain knowledge on a subject matter. Reasoning with diagrams is not just ‘peculiar’ because reasoners use spatially arranged characters to obtain knowledge – diagrams apparently have cognitive surplus: they enable a quasi-intuitive form of knowledge. The present paper analyses the issue of diagrams’ cognitive value by enquiring (...) into the tradition of symbolic cognition developed by Leibniz, Lambert, and Kant. The proposal resulting from this enquiry is to question the idea that the cognitive value of diagrams lies solely in allowing evidence for inferences. The imaginative dimension of diagrams connects reasoning to doxastic attitudes of meditation and enquiry. (shrink)

In order to effectively communicate information, the choice of representation is important. Ideally, a chosen representation will aid readers in making desired inferences. In this paper, we develop the theory of observation: what it means for one statement to be observable from another. Using observability, we give a formal characterization of the observational advantages of one representation of information over another. By considering observational advantages, people will be able to make better informed choices of representations of information. To demonstrate the (...) benefit of observation and observational advantages, we apply these concepts to set theory and Euler diagrams. In particular, we can show that Euler diagrams have significant observational advantages over set theory. This formally justifies Larkin and Simon’s claim that “a diagram is worth ten thousand words”. (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.

Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid's reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the received (...) view, this essay provides a contrary analysis by introducing a formal account of Euclid's proofs, termed Eu. Eu solves the puzzle of generality surrounding Euclid's arguments. It specifies what diagrams Euclid's diagrams are, in a precise formal sense, and defines generality-preserving proof rules in terms of them. After the central principles behind the formalization are laid out, its implications with respect to the question of what does and does not constitute a genuine picture proof are explored. (shrink)

En este artículo recuperamos la lógica diagramática lineal de Leibniz para la silogística y descubrimos sus propiedades lógicas y computacionales a través de una aproximación formal en términos metalógicos, lo cual es algo que, hasta donde sabemos, aún falta por hacerse. Así, en esta contribución buscamos, respectivamente, dos metas, una histórica y una lógica: i) prestar más atención a los aspectos algorítmicos del sistema diagramático lineal de Leibniz para la silogística, de los cuales creemos que han sido desdeñados por un (...) prejuicio general en contra del razonamiento diagramático; y ii) probar propiedades metalógicas del sistema para argumentar que es un sistema lógico bona fide. (shrink)

Theories of diagrams and diagrammatic reasoning typically seek to account for either the formal semantics of diagrams, or for the advantages which diagrammatic representations hold for the reasoner over other forms of representation. Regrettably, almost no theory exists which accounts for both of these issues together, nor how they affect one another. We do not attempt to provide such an account here. We do, however, seek to lay out larger context than is generally used for examining the processes of using (...) diagrams in reasoning or communication. A context in which detailed studies of sub-problems, such as the formal semantics or cognitive impact of specific diagrammatic systems, may be embedded.Accounts of the embedding of sentential logics in the computational processes of reasoners and communicators are relatively well developed from several decades of research in AI. Analogies between the sentential and the graphical cases are quite revealing about both similarities and differences. To provide a structure for the 'grand context' of diagrammatic representation and reasoning, and to clarify the relations between its component problems, we examine carefully these analogies and the decomposition they provide of subproblems for analysing diagrammatic reasoning. (shrink)

El primer objetivo de este artículo es mostrar que explicaciones genuinamente geométricas/matemáticas e intrínsecamente diagramáticas de fenómenos físicos no solo son posibles en la práctica científica, sino que además comportan un potencial epistémico que sus contrapartes simbólico-verbales carecen. Como ejemplo representativo utilizaremos la metodología geométrica de John Wheeler (1963) para calcular cantidades físicas en una reacción nuclear. Como segundo objetivo pretendemos analizar, desde un marco inferencial, la garantía epistémica de este tipo de explicaciones en términos de dependencia sintáctica y semántica (...) del contenido lo inferido en las premisas, lo cual denominaremos criterio de validez inferencial (CVI). (shrink)