Euler diagram

The Swiss mathematician Leonhard Euler (1707–1783) is one of the most important authors in the history of this type of diagram, but he is only the namesake, not the inventor. Euler diagrams were first developed for logic, especially syllogistics, and only later transferred to set theory. In the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s. Since then, they have also been adopted by other curriculum fields such as reading as well as organizations and businesses.

Euler diagrams consist of simple closed shapes in a two-dimensional plane that each depict a set or category. How or whether these shapes overlap demonstrates the relationships between the sets. Each curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. Curves which do not overlap represent disjoint sets, which have no elements in common. Two curves that overlap represent sets that intersect, that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve completely within the interior of another is a subset of it.

History

Diagrams reminiscent of Euler diagrams and with similar functions seem to have existed for a long time. However, exact dates for these diagrams can only be determined historically after the invention of printing press.

Before Euler

The first authors to print an Euler-esque diagram and briefly discuss it in their texts were Juan Luis Vives (in 1531), Nicolaus Reimers (in 1589), Bartholomäus Keckermann (in 1601) and Johann Heinrich Alsted (in 1614). The first detailed elaboration of these diagrams can be traced back to Erhard Weigel (1625–1699), who called this type of diagram a 'logometrum' (a measuring instrument for logic). Weigel was the first to prove all valid syllogisms with the aid of shapes in a two-dimensional plane. In the case of generally affirmative judgements (all-sentences), the geometric shape for the subject should lie completely within the shape for the predicate. In the case of negative judgements (no-sentences), it should lie completely outside. In the case of particular judgements (sentences with 'some', 'some...not'), the geometric shapes should partially overlap and not overlap. To prove a syllogism, one must first draw all possible figures for the premises and then see whether one can also read the conclusion from them. If this is the case, the syllogism is valid; otherwise, it is invalid.

Erhard Weigel used initial letters to represent the diagrams, whereas his students, such as Johann Christoph Sturm (1635–1703) and Gottfried Wilhelm Leibniz (1646–1716), used circles or lines. Another tradition can be traced back to Christian Weise (1642–1708), who is said to have used these diagrams in his teaching. This is reported by his students Samuel Großer and Johann Christian Lange. Lange in particular went beyond syllogistics with these diagrams and worked with quantified predicates, for example.

Euler and the time after

In his Letters to a German Princess, Euler focused solely on traditional syllogistics. He further developed Weigel's approach and not only tested the validity of syllogisms, but also developed a method for drawing conclusions from premises. At the same time as Euler, Gottfried Ploucquet and Johann Heinrich Lambert also used similar diagrams. However, the diagrams only became widely known in the 1790s through Immanuel Kant (1724–1804), who used them in his lectures on logic and his students then spread knowledge of the diagrams throughout Europe. In the 19th century, Euler diagrams became the most widely used form of representation in logic, esp. by 'Kantians' such as Arthur Schopenhauer, Karl Christian Friedrich Krause or Sir William Hamilton.

Since the history of the diagrams was only partially researched in the 19th century, most logicians attributed the diagrams to Euler, leading to numerous misunderstandings, some of which persist to this day. As shown in the illustration to the right, Sir William Hamilton erroneously asserted that the original use of the circles to "sensualize... the abstractions of logic" was not Euler but rather Weise; however the latter book was actually written by Johann Christian Lange, rather than Weise. He references Euler's Letters to a German Princess.

In Hamilton's illustration of the four categorical propositions which can occur in a syllogism as symbolized by the drawings A, E, I, and O are:

A: The Universal Affirmative

Example: All metals are elements.

E: The Universal Negative

Example: No metals are compound substances.

I: The Particular Affirmative

Example: Some metals are brittle.

O: The Particular Negative

Example: Some metals are not brittle.

Euler diagrams in the era of Venn

John Venn (1834–1923) comments on the remarkable prevalence of the Euler diagram:

"... 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."

But nevertheless, he contended, "the inapplicability of this scheme for the purposes of a really general logic" and then noted 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.”

Finally, in his Venn gets to a crucial criticism (italicized in the quote below); observe in Hamilton's illustration that the O (Particular Negative) and I (Particular Affirmative) are simply rotated:

“We now come to Euler's well-known circles which were first described in his Lettres a une Princesse d'Allemagne (Letters 102–105). The weak point about these consists in the fact that they only illustrate in strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions. ... This defect must have been noticed from the first 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”.[italics added]

Whatever the case, armed with these observations and criticisms, Venn then demonstrates how he derived what has become known as his Venn diagrams from the “... old-fashioned Euler diagrams.” In particular Venn gives an example, shown at the left.

By 1914, Couturat (1868–1914) had labeled the terms as shown on the drawing at the right. Moreover, he had labeled the exterior region (shown as a′b′c′) as well. He succinctly explains how to use the diagram – one must strike out the regions that are to vanish:

"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]

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 background surrounding the circles, does not appear: That is, the term marked "0", x′y′z′ . 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 use, the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or the domain of discourse.

Couturat observed 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."

Thus the matter would rest until 1952 when Maurice Karnaugh (1924–2022) would adapt and expand a method proposed by Edward W. Veitch; this work would rely on the truth table method precisely defined by Emil Post and the application of propositional logic to switching logic by (among others) Shannon, Stibitz, and Turing. For example, Hill & Peterson (1968) present the Venn diagram with shading and all. They give examples of Venn diagrams to solve example switching-circuit problems, but end up with this statement:

"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."

In Chapter 6, section 6.4 "Karnaugh map representation of Boolean functions" they begin with:

"The Karnaugh map¹ [¹Karnaugh 1953] is one of the most powerful tools in the repertory of the logic designer. ... A Karnaugh map may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram."

The history of Karnaugh's development of his "chart" or "map" method is obscure. The chain of citations becomes an academic game of "credit, credit; ¿who's got the credit?": Karnaugh (1953) referenced Veitch (1952), Veitch, referenced Shannon (1938), and Shannon (1938), in turn referenced (among other authors of logic texts) Couturat (1914). In Veitch's method the variables are arranged in a rectangle or square; as described in Karnaugh map, Karnaugh in his method changed the order of the variables to correspond to what has become known as (the vertices of) a hypercube.

Modern use of Euler diagrams

In the 1990s, Euler diagrams were developed as a logical system. The cognitive advantages of the diagrams soon became apparent. The diagrams were therefore not only used as set diagrams, but have since been used in many different ways and functions in computer science including artificial intelligence and software engineering, information technology, bioscience, medicine, economics, statistics and many other fields, and their philosophy and history have been discussed. In 2000, the conference series The Theory and Application on Diagrams: An International Conference Series began, which regularly addresses current research on Euler diagrams, among other topics.