Although a number of authors have used diagrams extensively in their studies of Navya-Nyāya, they have done so to explain and illustrate concepts, not with the goal of reasoning with the diagrams themselves. Adherents of diagrammatic... more
Although a number of authors have used diagrams extensively in their studies of Navya-Nyāya, they have done so to explain and illustrate concepts, not with the goal of reasoning with the diagrams themselves. Adherents of diagrammatic reasoning have made claims for its potential by pointing to key structural correspondences between diagrams and logical concepts, arguably lacking in sentential representations, and describing these relations using concepts such as "well matchedness" and "iconicity". A canonical example of this iconicity is the use of Euler diagrams to depict categorical syllogisms. Since the meaning of expressions in Indian logic differs in so many important ways from logic in the Western tradition , the use or adaptation of diagrams developed in the latter would seem to preclude iconicity. Thus, the development of diagrams which reflect the nature of inference in Navya-Nyāya, which centres on the anumāna inference schema, is motivated. In this paper we extend Ganeri's method of depicting the Vaiśeṡika ontology with graphs to include syntax intended to expose the nature of anumāna. The diagrams are given a formal basis: i.e. abstract syntax, inference rules defined abstractly and a graph-theoretic semantics. These are the first formalised logical diagrams that aim to reflect the nature of the anumāna inference. This paper lays the way for further work in extending the formalism to cover more of Navya-Nyāya, and in exploring a dialogue between properties of the formalism and of Navya-Nyāya.
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Symbols represent by codes like conventions, whereas icons represent by similarity (Couturat 1901; Dascal 1978; Gensini 1991; Serfati 2001). Until recently, much of the literature in philosophy of notation tended to follow Leibniz in... more
Symbols represent by codes like conventions, whereas icons represent by similarity (Couturat 1901; Dascal 1978; Gensini 1991; Serfati 2001). Until recently, much of the literature in philosophy of notation tended to follow Leibniz in assuming that we always or most often think in symbols. However, current debates have increasingly been driven by the recognition that any system of representation depends, to some extent, on iconicity, such that the purported line of demarcation is rather a continuum (Stjernfelt 2014). This slow but resolute turn toward iconic signs has begun to change how we see logic. With different emphases, linguists (Simone 1995; Van Langendonck 2007), logicians (Burch 1991; Shin 1994, 2002; Hammer 1995; Allein & Barwise 1996; Pietarinen 2006, 2010, 2011, 2012), historians of mathematics (Nets 2004; Mancosu, Jorgensen & Pedersen 2005; Giaquinto 2007), semioticians (Stjernfelt 2007, 2014; Bordron 2011; Dondero & Fontanille 2012), philosophers of mind (Champagne 2014) and cognitive scientists (Glasgow, Nara y Anan & Chandrasekaran 1995; Hoffmann 2010a, 2010b, 2011; Magnani 2011; Nakatsu 2010) have all recognized that a better understanding of reasoning by icons, specifically diagrams, is crucial to understanding problem-solving, inference-drawing, and hypothesis-making. Arguably, no one has explored the potential of iconic notations more systematically than C. S. Peirce. It is commonly believed that the birth of new formal logic(s) by Frege (1884), Russell (1903) and Couturat (1904) rendered Kant’s appeal to intuition redundant (cf. Kneale & Kneale 1962, VII-VIII; Coffa 1991; Carson & Huber 2006). However, an alternate line of development is clearly discernable in the work of Peirce. For Peirce, the notion of intuition is by no means redundant, being instead the faculty which allows necessary reasoning to yield informative truths (Peirce 1931-1958; Peirce 2010; cf. Hintikka 1980; Hookway 1985; Ketner 1985; Shin 1997; Pietarinen 2006; Stjernfelt 2007; Bellucci 2012). The diagram, in this Peircean paradigm, transforms intuition into a visual commodity amenable to careful public scrutiny. Indeed, one of the most striking features of Peirce’s diagrammatic notation is its depiction of inferences as transformations. Inference rules, being answerable to the self-same nature of images, are motivated in a way that makes them less rule-like. For example, enclosures, a common device used by Peirce, place distinct limits on what counts as included/inside, excluded/outside, or both. One can attempt to transgress these limits, but the iconic sign-vehicles at hand simply repel erroneous interpretations. While Venn exploited this to prove categorical syllogisms, Peirce shows how to generalize the method, thereby giving a novel justification for the normative force of logic. Peirce’s unpublished technical work in logic has thus far influenced debates mainly through the intermediary of specialists (Shin 2002; Sowa 2011), but the upcoming publication of Peirce’s full Existential Graphs (Logic of the Future, edited by A.-V. Pietarinen) promises to augment this rate of influence, by making available a 1000-page buffet of diagrammatic and iconic logical systems. Participants to this symposium are thus invited to reflect on how Peirce-inspired diagrammatic approaches to notation can positively reshape issues pertaining to logic, cognition, and reason-giving practices generally.