Euler diagram

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. 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. (see descriptions, left) The small text to the left erroneously says: "The first employment of circular diagrams in logic improperly ascribed to Euler. To be found in Christian Weise", a book which was actually written by Johann Christian Lange. was not Euler but rather Weise; however the latter book was actually written by Johann Christian Lange, rather than Weise. In Hamilton's illustration of the four categorical propositions 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" Given the Venn's assignments, then, the unshaded areas inside the circles can be summed to yield the following equation for Venn's example: : " is and is : therefore is " has the equation 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", Nowhere is it discussed or labeled, but Couturat corrects this in his drawing. 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." and , in turn referenced (among other authors of logic texts) . 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. == Relation between Euler and Venn diagrams ==