Euler diagram

An Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is disjoint (has no members in common) with "animals"
"...of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose:-somewhat at random, as they happened to be most accessible :-it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the Eulerian Scheme." (Footnote 1 page 100)
Composite of two pages 115–116 from Venn 1881 showing his example of how to convert a syllogism of three parts into his type of diagram. Venn calls the circles "Eulerian circles" (cf Sandifer 2003, Venn 1881:114 etc) in the "Eulerian scheme" (Venn 1881:100) of "old-fashioned Eulerian diagrams" (Venn 1881:113).

But nevertheless, he contended, "the inapplicability of this scheme for the purposes of a really general Logic" (page 100) and on page 101 observed that, "It fits in but badly even with the four propositions of the common Logic to which it is normally applied." Venn ends his chapter with the observation illustrated in the examples below—that their use is based on practice and intuition, not on a strict algorithmic practice:

“In fact ... those diagrams not only do not fit in with the ordinary scheme of propositions which they are employed to illustrate, but do not seem to have any recognized scheme of propositions to which they could be consistently affiliated.” (pp. 124–125)
in the case of the particular affirmative and negative, for the same diagram is commonly employed to stand for them both, which it does indifferently well

(Sandifer 2003 reports that Euler makes such observations too; Euler reports that his figure 45 (a simple intersection of two circles) has 4 different interpretations). Whatever the case, armed with these observations and criticisms, Venn then demonstrates (pp. 100–125) how he derived what has become known as his Venn diagrams from the "...old-fashioned Euler diagrams." In particular he gives an example, shown on the left.

"VENN'S method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need only strike out (by shading) those which are made to vanish by the data of the problem." (italics added p. 73)

Given the Venn's assignments, then, the unshaded areas inside the circles can be summed to yield the following equation for Venn's example:

"No Y is Z and ALL X is Y: therefore No X is Z" has the equation x'yz' + xyz' + x'y'z for the unshaded area inside the circles (but this is not entirely correct; see the next paragraph).

In Venn the 0th term, x'y'z', i.e. the background surrounding the circles, does not appear. Nowhere is it discussed or labeled, but Couturat corrects this in his drawing. The correct equation must include this unshaded area shown in boldface:

"No Y is Z and ALL X is Y: therefore No X is Z" has the equation x'yz' + xyz' + x'y'z + x'y'z' .

In modern usage the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or the domain of discourse.

Couturat now observes that, in a direct algorithmic (formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "No X is Z". Couturat concluded that the process "has ... serious inconveniences as a method for solving logical problems":

"It does not show how the data are exhibited by canceling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i. e., "throwing of the problem into an equation" and the transformation of the premises, nor with the subsequent steps, i. e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form."(p. 75)
"For more than three variables, the basic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6." (p. 64)

In Chapter 6, section 6.4 "Karnaugh Map Representation of Boolean Functions" they begin with:

Often a set of well-formedness conditions are imposed; these are topological or geometric constraints imposed on the structure of the diagram. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs.

Given the example above, the formula for the Euler and Venn diagrams is: