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Visual Reasoning with Diagrams pp 1–18Cite as

Birkhäuser

What is a Logical Diagram?

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Part of the book series: Studies in Universal Logic ((SUL))

Abstract

Robert Brandom’s expressivism argues that not all semantic content may be made fully explicit. This view connects in interesting ways with recent movements in philosophy of mathematics and logic (e.g. Brown, Shin, Giaquinto) to take diagrams seriously—as more than a mere ‘heuristic aid’ to proof, but either proofs themselves, or irreducible components of such. However what exactly is a diagram in logic? Does this constitute a cleanly definable semiotic kind? The paper will argue that such a kind does exist in Charles Peirce’s conception of iconic signs, but that fully understood, logical diagrams involve a structured array of normative reasoning practices, as well as just a ‘picture on a page’.

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Notes

  1. 1.

    This phrase is taken from Marcus Giaquinto [17, p. 3].

  2. 2.

    This example is nicely discussed in [17, pp. 3–4], and [22, pp. 3–4].

  3. 3.

    Discussed in [17, pp. 4–5].

  4. 4.

    “A body of work emerged in the late 19th century which grounded elementary geometry in abstract axiomatic theories…This development is now universally regarded as a methodological breakthrough. Geometric relations which previously were logically free-floating, because they were understood via diagrams, were given a firm footing with precisely defined primitives and axioms” [22, p. 6]. Non-Euclidean geometries are another key example, and I am grateful to an anonymous referee for pointing this out.

  5. 5.

    Although transposed into a rigidly empiricist setting which truth be told sits oddly with Frege’s thinking—and arguably has caused significant problems in the philosophy of mathematics.

  6. 6.

    (A cat.)

  7. 7.

    Randall Dipert has argued against this that it no more follows that resemblance is ‘entirely independent of’ representation because the former relation is symmetric and the latter is not, than that the brother relation is ‘entirely independent of’ the uncle relation as the former is symmetric and the latter is not [15].

  8. 8.

    See also, from a more philological perspective, the work of Reviel Netz, e.g. [23].

  9. 9.

    [5, 32].

  10. 10.

    Price suggests Rorty approaches a global non-cognitivism in [29].

  11. 11.

    This term derives from [6].

  12. 12.

    The term was coined by Simon Blackburn, see in particular [4]. Its links with pragmatism are explored in [28], though [21] argues that the two views share important similarities and differences.

  13. 13.

    I have argued this previously elsewhere: [19, 20].

  14. 14.

    Thus for instance Tractatus 6.42 states, “…there can be no ethical propositions…” [37].

  15. 15.

    Pace the recent structuralist movement in philosophy of science.

  16. 16.

    An example is a shadow-clock, which is iconic insofar as it represents the 24 hour structure of our day, indexical insofar as it relies on the sun physically casting a shadow to tell the time, and symbolic insofar as the numerals on the clock-face have meanings which must be learned.

  17. 17.

    This might have something to do with the fact that key researchers in semantics and logic in the 1960s and 70s also worked in artificial intelligence.

  18. 18.

    Shin also calls this the “multiple carving principle” [35, p. 77].

  19. 19.

    This is not to say that ordinary reasoners would necessarily recognize them as such. This is theoretical not applied logic (what Peirce called logica docens, opposing it to logica utens, a distinction that medieval logicians drew).

  20. 20.

    Although this is counterintuitive to some, the fact that the experience is only possible once one has grasped the proper interpretation of the EG rules does not undermine its directness once those rules have been grasped.

  21. 21.

    It is worth noting that this logic articulated the first version of the Russell-Peano notation for quantifiers which is standard today. Thanks are due to an anonymous referee for suggesting this.

  22. 22.

    One might object that this is incorrect, since a diagram such as a map may be understood to posit the shape of a real-world country. However considered purely qua diagram, a map does not yet have that semiotic function. To interpret it as saying something about a country is: (i) to peg it to a real-world object (thereby rendering it also an index), (ii) to claim something general about that object’s shape (rendering it also a symbol). These are further signs.

  23. 23.

    Thus for instance, Chrisman sums up much recent metaethics by writing, “The realism debate has been pursued (mostly) by investigating the appropriate semantic account of ethical statements” [14, p. 334].

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  1. Philosophy Programme, University of Waikato, Private Bag 3105, Hamilton, 3240, New Zealand

    Catherine Legg

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  1. Catherine Legg
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  1. IRIST, Université de Strasbourg, Rue de l'Université 7, Strasbourg, 67000, France

    Amirouche Moktefi

  2. Department of Philosophy, Yale University, Hillhouse Avenue 38, New Haven, 06511, Connecticut, USA

    Sun-Joo Shin

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Legg, C. (2013). What is a Logical Diagram?. In: Moktefi, A., Shin, SJ. (eds) Visual Reasoning with Diagrams. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0600-8_1

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