Existential graph

An existential graph is a type of diagrammatic or visual notation for logical expressions, invented 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.

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 than even Peirce could envisage.
 * alpha, isomorphic to the propositional logic and the two-element Boolean algebra;
 * beta, isomorphic to first-order logic with identity, with all formulae closed;
 * gamma, (nearly) isomorphic to normal modal logic.

Alpha


All existential graphs begin with the blank page, denoting Truth. A simple closed curve is called a cut or sep, and denotes negation/complementation. A fundamental syntactical rule asserts that cuts can be nested and concatenated at will, but must never intersect. Concatenated objects are implicitly conjoined. Hence the alpha graphs are a minimalist notation for sentential logic, one grounded in the expressive adequacy of And and Not. The alpha graphs constitute a radical simplification of two-element Boolean algebra and the truth functors.

Rules of inference:
 * Insertion - Any subgraph can be inserted at an odd level.
 * Erasure - Any subgraph can be deleted at an even level.

Rules of equivalence:
 * Double cut - A nested pair of cuts may be added or removed from around any subgraph.
 * Iteration/Deiteration - Any subgraph P can be removed or added anywhere, providing an equivalent subgraph is a subgraph of an ancestor of P.

Beta
Peirce notated predicates using intuitive English phrases; capital Latin letters, the standard usage of contemporary logic, may also be employed. A dot asserts the existence of an individual included 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. A line of identity connecting two or more predicates indicates that the predicates share a common variable. The beta graphs can be read as employing variables that are implicitly quantified. The depth of an object is the number of cuts that enclose it. If the "shallowest" instance of a variable has even (odd) depth, the variable is tacitly existentially (universally) quantified. The beta graphs appear to streamline first order logic with identity, but the secondary literature is not fully clear on this point.

Gamma
Add to alpha a second kind of cut, written using a dashed rather than a solid line. A simple closed curve written using dashed lines can be read as the primitive unary operator of modal logic.

Zeman (1964) was the first to note that:
 * The beta graphs are isomorphic to the predicate calculus;
 * Straightforward emendations of the gamma graphs 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.

Peirce's role
The existential graphs are a curious offspring of the marriage of Peirce the logician/ mathematician with Peirce the founder of a major strand of semiotics. 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, and extended De Morgan's relation algebra, stopping short of the metatheory (about which even Principia Mathematica was innocent). But his evolving semiotic theory led him to doubt the value of logic formulated using conventional linear typography, and to believe that logic and mathematics are best captured by a notation embedded in two (or even three) dimensions. Frege's 1879 Begriffschrifft independently reached the same conclusion, but employing a notation 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 existential graphs, called the entitative graphs. He immediately abandoned this formalism in favor of the existential graphs.

Peirce's graphical logic is but one of his manyfold accomplishments in logic and mathematics (on which see Charles Peirce). Unnoticed during his lifetime, his graphical logic was invariably denigrated or ignored after his death, until the Ph.D. theses by Roberts (1963) and Zeman (1964).

Primary literature

 * 1931-35. The Collected Papers of C.S. Peirce. Pp 320-470 of vol. 4 constitute the locus citandum for the existential graphs. Available online as 4.372-417 and 4.418-529.
 * 1992. Reasoning and the Logic of Things. Ketner, K.L. and Putnam, H., eds.. Harvard University Press.
 * 2001. Semiotic and Significs: The Correspondence between C.S. Peirce and Victoria Lady Welby. Hardwick, C.S., ed. Texas Tech University Press.
 * A transcription of Peirce's MS 514, edited with commentary by John Sowa.

As of this writing, the chronological critical edition of Peirce's works, the Writings, extends only to 1890. Much of Peirce's work on graphical logic 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 25 volumes of the chronological edition appear.

Secondary literature

 * Hammer, Eric M., 1998, "Semantics for Existential Graphs," Journal of Philosophical Logic 27: 489 - 503.
 * Roberts, Don D., 1973. The Existential Graphs of C.S. Peirce. John Benjamins. The definitive version of his 1963 thesis.
 * Shin, Sun-Joo, 2002. The Iconic Logic of Peirce's Graphs. MIT Press.
 * Zeman's 1964 thesis.