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. The graphs. Peirce proposed three systems of existential graphs: "Alpha" nests in "beta" and "gamma". "Beta" does not nest in "gamma", quantified modal logic being more general than put forth by Peirce. Alpha. The syntax is: Any well-formed part of a graph is a subgraph. The semantics are: 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": "Rules of equivalence": A proof manipulates a graph by a series of steps, with each step justified by one of the above rules. If a graph can be reduced by steps to the blank page or an empty cut, it 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. Beta. Peirce notated predicates using intuitive English phrases; the standard notation of contemporary logic, capital Latin letters, may also be employed. A dot asserts the existence of some individual in the domain of discourse. Multiple instances of the same object are linked by a line, called the "line of identity". There are no literal variables or quantifiers in the sense of first-order logic. A line of identity connecting two or more predicates can be read as asserting that the predicates share a common variable. The  presence of lines of identity requires modifying the "alpha" rules of Equivalence. The beta graphs can be read as a system in which all formula are to be taken as closed, because all variables are implicitly quantified. If the "shallowest" part of a line of identity has even (odd) depth, the associated variable is tacitly existentially (universally) quantified. Zeman (1964) was the first to note that the "beta" graphs are isomorphic to first-order logic with equality (also see Zeman 1967). However, the secondary literature, especially Roberts (1973) and Shin (2002), does not agree on just how this is so . Peirce's writings do not address this question, because first-order logic was first clearly articulated only some years after his death, in the 1928 first edition of David Hilbert and Wilhelm Ackermann's "Principles of Mathematical Logic". Gamma. Add to the syntax of "alpha" a second kind of simple closed curve, written using a dashed rather than a solid line. Peirce proposed rules for this second style of cut, which can be read as the primitive unary operator of modal logic. Zeman (1964) was the first to note that straightforward emendations of the "gamma" graph rules yield the well -known modal logics S4 and S5. Hence the "gamma" graphs can be read as a peculiar form of normal modal logic. This finding of Zeman's has gone unremarked to this day, but is nonetheless included here as a point of interest. Peirce's role. The existential graphs are a curious offspring of Peirce the logician/mathematician with Peirce the founder of a major strand of semiotics. Peirce's graphical logic is but one of his many accomplishments in logic and mathematics. In a series of papers beginning in 1867, and culminating with his classic paper in the 1885 "American  Journal of Mathematics", Peirce developed much of the two-element Boolean algebra, propositional calculus, quantification and the predicate calculus, and some rudimentary set theory. Model theorists consider Peirce the first of their kind. He also extended De Morgan's relation algebra. He stopped short of metalogic (which eluded even "Principia Mathematica"). But Peirce's evolving semiotic theory led him to doubt the value of logic formulated using conventional linear notation, and to prefer that logic and mathematics be notated in two (or even three) dimensions. His work went beyond Euler's diagrams and Venn's 1880 revision thereof. Frege's  1879 "Begriffsschrift" also employed a two-dimensional notation for logic, but one very different from Peirce's. Peirce's first published paper on graphical logic (reprinted in Vol. 3 of his "Collected Papers") proposed a system dual (in effect) to the "alpha" existential graphs, called the entitative graphs. He very soon abandoned this formalism in favor of the existential graphs. In 1911 Victoria, Lady Welby showed the existential graphs to C. K. Ogden who felt they could usefully be combined with Welby's thoughts in a "less abstruse form." Otherwise they attracted little attention during his life and were invariably  denigrated or ignored after his death, until the Ph.D. theses by Roberts (1964) and Zeman (1964). Further reading. Primary literature. Currently, the chronological critical edition of Peirce's works, the "Writings", extends only to 1892. Much of Peirce's work on logical graphs consists of manuscripts written after that date and still unpublished. Hence our understanding of Peirce's graphical logic is likely to change as the remaining 23 volumes of the chronological edition appear.