Existential graph

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

• alpha, isomorphic to the propositional calculus and the two-element Boolean algebra;
• beta, isomorphic to the predicate calculus with identity;
• gamma, isomorphic (or nearly so) to normal modal logic.

The 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 constitute a minimalist notation for the expressive adequacy of AND and NOT.

Peirce notated predicates using intuitive English phrases; capital Latin letters, as per contemporary logical usage, may also be employed. A dot asserts the existence of an object contained in the domain of discourse. There are no literal variables or quantifiers. Multiple instances of the same variable are denoted by lines (called "lines of identity") linking predicates. The beta graphs can be read as employing variables that are implicitly existentially quantified. If the "shallowest" instance of a variable has even (odd) depth, the variable is tacitly existentially (universally) quantified. Modality is conveyed by simple closed curves written using dashed lines.

It is now evident that the alpha graphs constitute a radical simplification of Boolean algebra and the truth functors. The beta graphs most likely streamline first order logic with identity to a similar degree, but the secondary literature is less than fully clear on this point.

The existential graphs constitute a curious marriage of Peirce the logician/mathematician, and Peirce the founder of a major strand of semiotics. In a series of papers beginning in 1867, and ending with his classic paper in the 1885 American Journal of Mathematics, Peirce developed much of the two-valued Boolean algebra, propositional calculus, quantification and the predicate calculus, and some rudimentary set theory. 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.

In the existential graphs, signs are expressed as vertices, and edges function as 'lines of identity' establishing logical relations among signs. A gamut of visual expressions for various logical relationships was established. For example, shading is used to assert negation.

Primary Literature The main source for the existential graphs are the 100+ pages devoted to them in vol. 4 of the Collected Papers of C.S. Peirce, published in 1933. Also see:

Peirce, C.S. (1992) Reasoning and the Logic of Things. Ketner, K.L. and Putnam, H., eds.. Harvard University Press.

Peirce, C.S. (2001) Semiotic and Significs: The Correspondence between C.S. Peirce and Victoria Lady Welby. Hardwick, C.S., ed. Texas Tech University Press.

The ongoing chronological edition of Peirce's works stops in 1890, and 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 Peirce's graphical logic went unnoticed during his lifetime. Peirce's graphical logic was invariably denigrated or ignored until it became the subject of the Ph.D. theses by Roberts (1963) and Zeman (1964) [1]. Zeman was the first to notice that the beta graphs were isomorphic to the predicate calculus, and that the gamma graphs could be read as a peculiar flavor of normal modal logic. Roberts published a revised version of his thesis as:

Roberts, Don D (1973) The Existential Graphs of C.S. Peirce. John Benjamins.

Also see:

Shin, Sun-Joo (2002) The Iconic Logic of Peirce’s Graphs. MIT Press.

The primary algebra of G. Spencer Brown's Laws of Form can be seen as an algebraic formalism isomorphic to the alpha existential graphs, that fully exploits duality. Peirce had proposed related formalisms in 1886 and 1902, but failed to appreciate the inherence of duality.

See also

• Conceptual graph

External link

• Zeman's 1964 Ph.D. thesis on Peirce's graphical logic
• John Sowa's commentary on Peirce's graphical logic