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Venn-type diagrams for arguments of n terms

Published online by Cambridge University Press:  12 March 2014

Daniel E. Anderson  and
Frank L. Cleaver
Daniel E. Anderson
Affiliation:
Ohio Wesleyan University
Frank L. Cleaver
Affiliation:
University of South Florida
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Extract

The attempt to find usable diagrams for n terms of the sort devised by John Venn seems to have originated with Venn himself, who published diagrams for up to five classes (the fifth class, however, was shaped like a doughnut, and contained an area outside itself — like the hole in the doughnut). Venn then suggested that “if we wanted to use a diagram for six terms (x, y, z, w, v, u) the best plan would probably be to take two five-term figures, one for the u part and one for the non-u part of all the other combinations …” Such a method would, as Venn admits, be somewhat confusing to the eye, and thus fail to fulfil one of the major purposes of the diagram; nevertheless, it is a method which could be extended to cover n classes.

Venn also suggests (but without illustration) another method for drawing diagrams for n terms: “… the rule of formation would be very simple. We should merely have to begin by drawing any closed figure, and then proceed to draw others subject to the one condition that each intersect once, and once only, all the existing subdivisions produced by those which had gone before.” Then, in a rather surprising footnote, he adds: “It will be found that when we adhere to continuous figures, instead of the discontinuous five-term figure given above, there is a tendency for the resultant outlines thus successively drawn to assume a comb-like shape after the first four or five. If we begin by circles or other rounded figures the teeth are curved, if by parallelograms then they are straight.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 30 , Issue 2 , June 1965 , pp. 113 - 118
Copyright
Copyright © Association for Symbolic Logic 1965

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References

1 Symbolic logic, London (1881)Google Scholar. The attempt to diagram logical arguments is of course much older, originating, perhaps among the ancients. Hamilton, William (Discussions on philosophy and literature (1866))Google Scholar reproduces diagrams of the syllogism tha t date from the fifth century. Venn reproduces diagrams of class inclusion attributed to Ludovicus Vives, and to Maass. The last chapter of Venn's book contains a fairly thorough review of the early literature. A more up to date review is given in Logic machines and diagrams, Gardner, Martin, New York (1958), ch. 2Google Scholar. Gardner reproduces many of the proposed diagrams which otherwise may only be found scattered through numerous journals — many of which are now difficult to obtain.

2 Ibid., n. 1, p. 108.

3 Ibid., p. 108.

4 n. 2, pp. 108–9.

5 More recently a method for constructing diagrams for n terms was devised by More, Trenchard Jr. (On the construction of Venn diagrams, this Journal, vol. 24 (1959))Google Scholar. In November of 1953, M. Karnaugh (Map method for synthesis of combinational logic circuits, Transactions of the American Institute of Electrical Engineers, Communications and electronics Pt. I, No. 9) refined a chart originally designed by Veitch, E. W. (A chart method for simplifying truth functions, Association for Computing Machinery. Proceedings, 05 2, 3 (1952))Google Scholar to accommodate up to six terms, but for more than six terms the regions become discontinuous. More's method (cited above) allows for the construction of diagrams for n terms in which the regions for all terms are continuous; but the adaptation of the diagram to allow for shading to indicate emptiness and asterisks to indicate non-emptiness is confusing to the eye and, for pedagogical purposes at least, not very useful.

6 Two extensions of the use of graphs in elementary logic, Hocking, William Ernest; University of California publications in philosophy, Vol. 2 (1909)Google Scholar.

7 Cf. Lewis, C. I., A survey of symbolic logic, Berkeley, (1918)Google Scholar; Quine, W. V., Methods of logic, New York, (1950)Google Scholar; Lee, H. N., Symbolic logic, New York (1961)Google Scholar.

8 It is worth noting that this diagram fits exactly the description given in Venn's n. 2, pp. 108–9, quoted p. 113 above.