Art of Combinations), represents the Aristotelian theory of how all material things are formed from combinations of the elements earth, water, air, and fire. These four elements make up the four corners of a diamond (see picture to right). Opposing pairs of these are joined by a bar labeled "contraries" (earth-air, fire-water). At the four corners of the superimposed square are the four qualities defining the elements. Each adjacent pair of these is joined by a bar labeled "possible combination"; the diagonals joining them are labeled "impossible combination." Starting from the top, fire is formed from the combination of dryness and heat; air from wetness and heat; water from coldness and wetness; earth from coldness and dryness. The Venn-II reasoning system In the early 1990s Sun-Joo Shin presented an extension of Existential Graphs called Venn-II. Syntax and semantics are given formally, together with a set of Rules of Transformation which are shown to be sound and complete. Proofs proceed by applying the rules (which remove or add syntactic elements to or from diagrams) sequentially. Venn-II is equivalent in expressive power to a first-order monadic language. See also Heuristics How to Solve It by George Pólya Natural deduction Propositional calculus Spatial-temporal reasoning Trikonic Visual reasoning References Further reading Gerard Allwein and Jon Barwise (ed.) (1996). Logical Reasoning with Diagrams. Oxford University Press. Michael Anderson, Peter Cheng, Volker Haarslev (Eds.) (2000). Theory and Application of Diagrams: First International Conference, Diagrams 2000. Edinburgh, Scotland, UK, September 1–3, 2000. Proceedings. Micheal Anderson and R. McCartney (2003). Diagram Processing: Computing with Diagrams. In: Artificial Intelligence, Volume 145, Issue 1–2, April, 2003. James Robert Brown (1999). Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge. James Franklin (2000). Diagrammatic reasoning and modelling in the imagination: the secret weapons of the Scientific Revolution, in 1543 and All That: Image and Word, Change and Continuity in the Proto-Scientific Revolution, ed. G. Freeland & A. Corones (Kluwer, Dordrecht), pp. 53-115. Janice Glasgow, N. Hari Narayanan, and B. Chandrasekaran (ed) (1995). Diagrammatic Reasoning: Cognitive and Computational Perspectives. AAAI Press. Kulpa, Zenon. "Diagrammatic representation and reasoning." Machine GRAPHICS & VISION 3 (1/2. 1994. Gem Stapleton A Survey of Reasoning Systems Based on Euler Diagrams. Electronic Notes in Theoretical Computer Science. 2005. External links Diagrammatic Reasoning Site from the University of Hartford, Connecticut, USA Lecture about Universal Algebra and Diagrammatic Reasoning'' by John Baez, 3 Feb 2006. Homepage