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