Existential graph

An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote on graphical logic as early as 1882, and continued to develop the method until his death in 1914. Peirce proposed three systems of existential graphs: alpha, isomorphic to 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 general than put forth by Peirce. 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 negation or Boolean 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 sentential logic, grounded in the expressive adequacy of And and Not. The alpha graphs constitute a radical simplification of the two-element Boolean algebra and the truth functors. The depth of an object is the number of cuts that enclose it. Rules of inference: Insertion - Any subgraph may be inserted into an odd numbered depth. Erasure - Any subgraph in an even numbered depth may be erased. Rules of equivalence: Double cut - A pair of cuts with nothing between them may be drawn around any subgraph. Likewise two nested cuts with nothing between them may be erased. This rule is equivalent to Boolean involution. Iteration/Deiteration – To understand this rule, it is best to view a graph as a tree structure having nodes and ancestors. Any subgraph P in node n may be copied into any node depending on n.