Diagrammatic reasoning isreasoning by means ofvisualrepresentations. The study ofdiagrammatic reasoning is about the understanding of concepts and ideas, visualized with the use ofdiagrams andimagery instead of bylinguistic oralgebraic means.
Adiagram is a 2D geometric symbolicrepresentation ofinformation according to somevisualization technique. Sometimes, the technique uses a3D visualization which is thenprojected onto the 2D surface. The term diagram in common sense can have two meanings:
In science the term is used in both ways. For example, Anderson (1997) stated more general "diagrams are pictorial, yet abstract, representations of information, andmaps,line graphs,bar charts,engineeringblueprints, andarchitects'sketches are all examples of diagrams, whereas photographs and video are not".[2] On the other hand, Lowe (1993) defined diagrams as specifically "abstract graphic portrayals of the subject matter they represent".[3]
In the specific sense diagrams and charts contrastcomputer graphics, technical illustrations,infographics, maps, andtechnical drawings, by showing "abstract rather thanliteral representations of information".[1] The essences of a diagram can be seen as:[1]
Or as Bert S. Hall wrote, "diagrams are simplified figures, caricatures in a way, intended to convey essential meaning".[4] According toJan V. White (1984) "the characteristics of a good diagram are elegance, clarity, ease, pattern, simplicity, and validity".[1] Elegance for White means that what you are seeing in the diagram is "the simplest and most fitting solution to a problem".[5]
Alogical graph is a special type ofgraph-theoretic structure in any one of several systems of graphicalsyntax thatCharles Sanders Peirce developed forlogic.
In his papers onqualitative logic,entitative graphs, andexistential graphs, Peirce developed several versions of a graphicalformalism, or a graph-theoreticformal language, designed to be interpreted for logic.
In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures.
Aconceptual graph (CG) is a notation for logic based on theexistential graphs ofCharles Sanders Peirce and thesemantic networks ofartificial intelligence. In the first published paper on conceptual graphs,John F. Sowa used them to represent theconceptual schemas used in database systems. His first book[6] applied them to a wide range of topics in artificial intelligence, computer science, and cognitive science. A linear notation, called theConceptual Graph Interchange Format (CGIF), has been standardized in the ISO standard forCommon Logic.
The diagram on the right is an example of thedisplay form for a conceptual graph. Each box is called aconcept node, and each oval is called arelation node. In CGIF, this CG would be represented by the following statement:
In CGIF, brackets enclose the information inside the concept nodes, and parentheses enclose the information inside the relation nodes. The letters x and y, which are calledcoreference labels, show how the concept and relation nodes are connected. In theCommon Logic Interchange Format (CLIF), those letters are mapped to variables, as in the following statement:
As this example shows, the asterisks on the coreference labels *x and *y in CGIF map to existentially quantified variables in CLIF, and the question marks on ?x and ?y map to bound variables in CLIF. A universal quantifier, represented@every*z in CGIF, would be representedforall (z) in CLIF.
Anentitative graph is an element of thegraphicalsyntax forlogic thatCharles Sanders Peirce developed under the name ofqualitative logic beginning in the 1880s, taking the coverage of theformalism only as far as thepropositional or sentential aspects of logic are concerned.[7]
Thesyntax is:
Thesemantics are:
A "proof" manipulates a graph, using a short list of rules, until the graph is reduced to an empty cut or the blank page. A graph that can be so reduced is what is now called atautology (or the complement thereof). Graphs that cannot be simplified beyond a certain point are analogues of thesatisfiableformulas offirst-order logic.
Anexistential graph is a type ofdiagrammatic or visual notation for logical expressions, proposed byCharles Sanders Peirce, who wrote his first paper ongraphical logic in 1882 and continued to develop the method until his death in 1914. Peirce proposed three systems of existential graphs:
Alpha nests inbeta andgamma.Beta does not nest ingamma, quantified modal logic being more than even Peirce could envisage.
Inalpha thesyntax is:
Any well-formed part of a graph is asubgraph.
Thesemantics are:
Hence thealpha graphs are a minimalist notation forsentential logic, grounded in the expressive adequacy ofAnd andNot. Thealpha graphs constitute a radical simplification of thetwo-element Boolean algebra and thetruth functors.
Characteristica universalis, commonly interpreted asuniversal characteristic, oruniversal character in English, is a universal and formal language imagined by the German philosopherGottfried Leibniz able to express mathematical, scientific, and metaphysical concepts. Leibniz thus hoped to create a language usable within the framework of a universal logical calculation orcalculus ratiocinator.
Since thecharacteristica universalis is diagrammatic and employspictograms (below left), the diagrams in Leibniz's work warrant close study. On at least two occasions, Leibniz illustrated his philosophical reasoning with diagrams. One diagram, the frontispiece to his 1666De Arte Combinatoria (On the 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.[8]
In the early 1990sSun-Joo Shin presented an extension of Existential Graphs called Venn-II.[9] Syntax and semantics are given formally, together with a set ofRules 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.
