Graphs Theory Research Papers - Academia.edu (original) (raw)

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.