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- 1.The extremely difficult passage on the construction of the world-soul,
*Timaeus*34 C-36 E, would be even less intelligible without the image of crossed strips with which Plato helps us. One of the more obvious examples of a visual aid in Plato is described in*Republic*VI, 509 E, where a line is divided (in a very peculiar way, it turns out) and each division and subdivision accorded special significance illustrating being or knowing. See, Robert S. Brumbaugh,*Plato’s Mathematical Imagination*(Bloomington: Indiana University Press, 1954). Plato’s “imagination” is not co-extensive with what is merely visible and ordered, but it can be abundantly illuminated by the several score “diagrams” which Brumbaugh finds buried in the text. Plato, by the way, looks upon sight as the sense of greatest benefit to our intellect—*Timaeus*47 A.Google Scholar - 2.Aristotle,
*Prior Analytics*, I, 4, 25 b 31-26 a 1. For important notes on this, see Sir W. D. Ross,*Aristotle’s Prior and Posterior Analytics: A Revised Text with Introduction and Commentary*(Oxford: At the Clarendon Press, 1949,*ad loc*.Google Scholar - 3.See, for example, the scholium to Proposition 15 in Book I of the
*Ethics*. The famous illustration of the proper defining of a circle in the*Improvement of the Understanding*is another case in point.Google Scholar - 4.Of the period of Psellus and shortly after, William Stanley Jevons, in his
*Studies in Deductive Logic: A Manual for Students*(Third edition; London: Macmillan and Company, 1896) says: “During a visit to Italy in 1874, I was much surprised and interested by the multitudes of curious diagrammatic exercises to be found in theological MSS. of the great public libraries of Italy. The abundance of these diagrams shows that rudimentary logical exercises were very popular in the country where, and at the time when, the dawn of modern science began to break. I estiamted that a single MSS. in the Biblioteca Communale … contained at least 800 such diagrams … The MSS. containing these (among others) is assigned in the printed catalogue to the 11th or 12th Century …”—Preface, p. xxii.Google Scholar - 5.For light on this material, consult Clarence I. Lewis,
*A Survey of Symbolic Logic*(Berkeley: University of California Press, 1918),*ad loc.*Leibniz lived during a century when many of the leading mathematicians wished to find visual or even mechanical methods of solving difficult problems—Descartes, Pascal, Napier,*et al.*Google Scholar - 6.
*Letters of Euler to a German Princess on Different Subjects in Physics and Philosophy*, translated from the French by Henry Hunter (London: H. Murray, 1795) 2 vols. See Letters CI–CVIII, written between February 10 and March 7, 1761.Google Scholar - 7.J. H. Lambert, “
*Sechs Versuche einer Zeichenkunst in der Vernunftlehre*” in*Johann Heinrich Lamberts logische und philosophische Abhandlungen*, edited by Johann Bernoulli (Berlin: 1782) Vol. I. A number of diagrammatic devices are found scattered through these papers. For brief accounts, see Lewis,*op. cit.*, and J. Welton,*A Manual of Logic*, 2 vols. (Second edition, London: W. B. Clive, 1901). See also Lambert’s*Neues Organon*(Leipzig: Johann Wendler, 1764).Google Scholar - 8.Sir William Hamilton,
*Lectures on Metaphysics and Logic*, edited by Henry L. Mansel and John Veitch (Edinburgh and London: William Blackwood and Sons, 1874). See especially the Appendices to Vol. IV: Appendix VI (c), (d), (e), (f).Google Scholar - 9.In his
*Studies in Deductive Logic*, Jevons slightly modifies the Euler circles, p. 52. See also his*The Principles of Science: A Treatise on Logic and Scientific Method*(Special American edition; New York: MacMillan and Company, 1874), which contains a description of the logical abacus (p. 119–23) and the logical machine (p. 123–31), the latter being sometimes dubbed the logical piano. See further W. Mays and D. P. Henry, “Jevons and Logic,”*Mind*LXII, 248: p 484ff (October, 1953) for a discussion of this machine in relation to the theory behind it.Google Scholar - 10.John Venn,
*Symbolic Logic*(London: MacMillan and Company, 1881). This is one of the most important books to consider spatial representation of logical forms: its contributions are chiefly contained in chapters 5 and 11. There is an interesting historical summary given in chapter 20; here Venn lists no fewer than 25 ways in which “No S is P” can be indicated symbolically. We must remember that Venn was writing three-quarters of a century ago; the additions to this list are no doubt overwhelming by now. One notes that Venn has a difficulty in separating symbolism from geometrical representations, a difficulty quite similar to one that perhaps ought to be raised in the present essay. But in the case at hand there would be little purpose served in making the separation: both symbols and designs can be shown clearly on the feltboard.Google Scholar - 11.Gottlob Freger§ rather notorious diagrams were first expounded in his
*Begriffschrifft, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens*(Halle, 1879). The first chapter of this is rendered into English in Peter Geach and Max Black,*Translations from the Philosophical Writings of Gottlob Frege*(Oxford: Basil Blackwell, 1952), p. 1–20. There is a brief account in Joergen Joergensen,*A Treatise of Formal Logic*(Copenhagen and London: 1931) 3 vols.Google Scholar - 12.Giuseppe Peano and Others,
*Formulaire de Math’ematiques*(Turin: 1895–1908).Google Scholar - 13.Some of these details were first explained to me by Donald A. Ingli, director, and Rolland P. Schlieve, assistant director of Audio-Visual Aids at Southern Illinois University, and I wish to thank them here.Google Scholar
- 14.The Wittgenstein truth-tables make their first appearance in his
*Tractatus Logico-Philosophicus*(London: Kegan Paul, Trench, Trubner, and Company, 1922) par. 4.31.Google Scholar - 15.—Welton,
*op. cit.*,, contains a good chapter on the value of these circles, and other diagrams of propositions, and syllogisms.Google Scholar - 16.John Neville Keynes,
*Studies and Exercises in Formal Logic*(Third edition; London: MacMillan and Company, 1894), Part III, chapter 4.Google Scholar - 17.—Venn,
*op. cit.*,, p. 100 n., says: “Until I came to look somewhat closely into the matter I had no idea how prevalent such an appeal had become. Thus of the first 60 logical treatises, published during the last century or so, which were consulted for this purpose … it appeared that 34 appeal to the aid of diagrams, nearly all of these making use of the Eulerian Scheme.”Google Scholar - 18.John Stuart Mill,
*A System of Logic, Ratiocinative and Inductive*(Eighth edition; London: Longmans, Green, Reader, and Ryer, 1872), Book III, chapter 8.Google Scholar - 19.W. E. Johnson,
*Logic*, Volume II (Cambridge: At the University Press, 1922). Chapter 10 is on demonstrative induction, the type most nearly in accord with Mill’s methods, and easily presentable by the same visual means.Google Scholar - 20.Ronald A. Fisher,
*The Design of Experiments*(New York: Hafner-Publishing Company, 1947). See particularly chapter 5.Google Scholar - 21.Raymond L. Wilder,
*Introduction to the Foundations of Mathematics*(New York: John Wiley and Sons, 1952), chapter 1.Google Scholar - 22.Morris R. Cohen and Ernest Nagel,
*An Introduction to Logic and Scientific Method*(New York: Harcourt Brace and Company, 1934).Google Scholar - 23.—Wilder,
*op. cit.*,, chapter 2.Google Scholar - 24.Described in Jevons,
*Studies in Deductive Logic*, chapter 11. A modified form of these cards is given in Martin Gardner, “Logic Machines,”*Scientific American*186:71; March 1952.Google Scholar - 25.Edmund Callis Berkeley,
*Giant Brains, or Machines That Think*(New York: John Wiley and Sons, 1949). See especially chapter 9, which furnishes a short account of the relation of symbolic logic to electric relays. These could well be illustrated on the feltboard, preferably with flocked lightweight cardboard strips.Google Scholar

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