Diagrammatic reasoning

Information mapping
Topics and fields
Business decision mapping Data visualization Graphic communication Infographics Information design Knowledge visualization Mental model Morphological analysis Ontology (information science) Schema (psychology) Visual analytics Visual language
Node–link approaches
Argument map Cladistics Cognitive map Concept lattice Concept map Conceptual graph Decision tree Dendrogram Graph drawing Hyperbolic tree Hypertext Issue map Issue tree Layered graph drawing Mind map Object-role modeling Organizational chart Pathfinder network Radial tree Semantic network Sociogram Timeline Topic map Tree structure ZigZag
See also
Design rationale Diagrammatic reasoning Entity–relationship model Geovisualization List of concept- and mind-mapping software Olog Ontology (philosophy) Problem structuring methods Semantic Web Treemapping Wicked problem
v t e

Diagrammatic reasoning is

• visual information device: Like the term "illustration" the diagram is used as a collective term standing for the whole class of technical genres, including graphs, technical drawings and tables.^[1]
• specific kind of visual display: This is only the genre, that shows qualitative data with shapes that are connected by lines, arrows, or other visual links.

In science the term is used in both ways. For example, Anderson (1997) stated more general "diagrams are pictorial, yet abstract, representations of information, and maps, line graphs, bar charts, engineering blueprints, and architects' 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 contrast

• a form of visual formatting devices
• a display that does not show
  quantitative data
  , but rather relationships and abstract information
• with building blocks such as geometrical shapes that are connected by lines, arrows, or other visual links.

Or as Bert S. Hall wrote, "diagrams are simplified figures, caricatures in a way, intended to convey essential meaning".^[4] According to Jan 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]

Logical graph

A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic.

In his papers on

A conceptual graph (CG) is a notation for logic based on the existential graphs of Charles Sanders Peirce and the semantic networks of artificial intelligence. In the first published paper on conceptual graphs, John F. Sowa used them to represent the conceptual 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 the Conceptual Graph Interchange Format (CGIF), has been standardized in the ISO standard for Common Logic.

The diagram on the right is an example of the display form for a conceptual graph. Each box is called a concept node, and each oval is called a relation node. In CGIF, this CG would be represented by the following statement:

[Cat Elsie] [Sitting *x] [Mat *y] (agent ?x Elsie) (location ?x ?y)

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 called coreference labels, show how the concept and relation nodes are connected. In the Common Logic Interchange Format (CLIF), those letters are mapped to variables, as in the following statement:

(exists ((x Sitting) (y Mat)) (and (Cat Elsie) (agent x Elsie) (location x y)))

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 represented forall (z) in CLIF.

Entitative graph

An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic beginning in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned.^[7]

The syntax is:

• The blank page;
• Single letters, phrases;
• Objects (subgraphs) enclosed by a
  simple closed curve
  called a cut. A cut can be empty.

The semantics are:

• The blank page denotes False;
• Letters, phrases, subgraphs, and entire graphs can be True' or False;
• To surround objects with a cut is equivalent to Boolean
  complementation
  . Hence an empty cut denotes Truth;
• All objects within a given cut are tacitly joined by
  disjunction
  .

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 a

An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. Peirce proposed three systems of existential graphs:

• alpha –
  sentential logic and the two-element Boolean algebra
  ;
• beta – isomorphic to first-order logic with identity, with all formulas closed;
• gamma – (nearly) isomorphic to normal modal logic.

Alpha nests in beta and gamma. Beta does not nest in gamma, quantified modal logic being more than even Peirce could envisage.

In alpha the syntax is:

• The blank page;
• Single letters or phrases written anywhere on the page;
• Any graph may be enclosed by a
  simple closed curve
  called a cut or sep. A cut can be empty. Cuts can nest and concatenate at will, but must never intersect.

Any well-formed part of a graph is a subgraph.

The semantics are:

• The blank page denotes Truth;
• Letters, phrases, subgraphs, and entire graphs may be True or False;
• To enclose a subgraph with a cut is equivalent to logical
  complementation
  . Hence an empty cut denotes False;
• All subgraphs within a given cut are tacitly
  conjoined
  .

Hence the alpha graphs are a minimalist notation for

Characteristica universalis

Since the characteristica universalis is diagrammatic and employs

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]

The Venn-II reasoning system

In the early 1990s Sun-Joo Shin presented an extension of Existential Graphs called Venn-II.^[9] 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

References