Diagrams Research

The diagrams research community is interested in all aspects of diagrams, from their formal foundations to exploring how best to support people in understanding information. Major advances have been made in many areas, with those most relevant to this winter school encompassing logic, philosophy and history.

Diagrammatic logics were first seen as a formal alternative to symbolic logics when Sun-Joo Shin produce a highly regarded seminal piece of work in the 1990s. In particular, she formalized a diagrammatic logic based on Venn diagrams and established its soundness and completeness. An example of a so-called Venn-II diagram is shown below. It comprises three parts, nowadays called unitary diagrams. The first unitary diagram, on the left, expresses that there is nothing in A and B, through the use of shading. It also expresses that there is something in either A or B, but not both, through the use of an x-sequence. The middle unitary diagram expresses something is in both A and B and that nothing is in B but not A. The righthand unitary diagram expresses that there is nothing in either A or B that is also in C and that there is something in all of A, B and C; this unitary diagram is thus a contradiction. Lastly, the three unitary diagrams are joined together by horizontal line segments. These lines express disjunctive information: the lefthand or the middle or the righthand unitary diagram is true.

The sessions on diagrammatic logics at the winter school will introduce Shin's seminal work and show how research in this area has evolved to the modern day. An increasingly important aspect of this research direction is the desire to produce usable diagrammatic logics that have measurable benefits over their symbolic counterparts. This usability aspect will be encompassed by the sessions on diagrammatic logics too.

The expressiveness of diagrammatic logics has largely been limited to what is known as monadic first-order logic, sometimes with equality. Monadic first-order logics only allow assertions along the lines of elements belonging to sets, or otherwise. This is seen in the Venn-II diagram above where, for example, the middle diagram says there is an element in both A and B but no elements in B but not in A. Venn-II is a monadic first-order logic, but it does not have any means of expressing equality explicitly. When equality can be asserted, one can also assert distinctness (through the use of negation). Spider diagrams, developed by Gil et al., extend the expressiveness of Venn-II to monadic first-order logic with equality. The example spider diagram on the left below expresses that there are two different (i.e. unequal) elements, one in both A and B and the other in either A or B. The spider diagram on the right expresses that there is an element in A and a different element in B. The information in the righthand diagram can be inferred from the diagram on the left. That is, if it is the case that the diagram on the left is making a true statement, then so too must the diagram on the right. Inference is a core focus of research into diagrammatic logics and will form a major component of the sessions on diagrammatic logics.

Further, it is also deemed useful to include syntax to represent specific individuals (sometimes called constants), beyond just asserting the existence of elements: saying that the individual called c is an element of A or B is not the same as saying that there is an element in A or B. Various diagrammatic logics include individuals, such as spider diagrams and an extension of Venn-II called Venn-i, introduced by Choudhury and Chakraborty. The two diagrams below are both examples of Venn-i diagrams.

The diagram on the left expresses that the individual c is an element of A or B and that there is an element in both A and B. A novel feature of Venn-i, unique amongst diagrammatic logics, is its ability to explicitly express the absence of an individual from a set; this can be seen in the righthand diagram which expresses that c is not outside of A union B (as well as expressing there is an element in both A and B). The use of constants in diagrammatic logics will be discussed during the special sessions on non-classical diagrammatic logics.

Diagrammatic notations, on which diagrammatic logics are built, have a long history. They can be traced back to work by Hull, Euler, Venn, Peirce and Leibniz to name just a few. Hull was probably the first person to draw what we now call Euler diagrams, since their introduction is generally attributed to Euler. Venn identified potential weaknesses of Euler diagrams and introduced what we now call Venn diagrams. In fact, Euler diagrams are widely confused with Venn diagrams and what we see today is essentially a blend of the two notations. Peirce introduced the core syntax of Shin's Venn-II system and he is also responsible for the introduction of existential graphs. Further, Leibniz introduced the use of lines to represent sets, unlike the use of closed curves seen in Euler and Venn diagrams. The historical development and philosophical debate around diagrams will be covered during the sessions on historical and philosophical aspects of logical diagrams.