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 tautology (or the complement thereof). Graphs that cannot be simplified beyond a certain point are analogues of the satisfiable formulas of first-order logic. Existential graph 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 – 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 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 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. Characteristica universalis Characteristica universalis, commonly interpreted as universal characteristic, or universal character in English, is a universal and formal language imagined by the German philosopher Gottfried 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 or calculus ratiocinator. Since the characteristica universalis is diagrammatic and employs pictograms (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 1666 De Arte Combinatoria (On the