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Encyclopedia > 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. Sample flowchart diagram A diagram is a simplified and structured visual representation of concepts, ideas, constructions, relations, statistical data, anatomy etc used in all aspects of human activities to visualize and clarify the topic. ... Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ... A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ... 1882 (MDCCCLXXXII) was a common year starting on Sunday (see link for calendar) of the Gregorian calendar or a common year starting on Tuesday of the 12-day slower Julian calendar. ... 1914 (MCMXIV) was a common year starting on Thursday. ...

Contents

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. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... -1... It has been suggested that Predicate calculus be merged into this article or section. ... In logic, normal modal logic is a set L of modal formulas such that L contains all propositional tautologies, Kripkes schema: , and L is closed under substitution, detachment rule: from A and A→B infer B, necessitation rule: from A infer . ...


Alpha

Alpha Graphs
Alpha Graphs

The syntax is: Image File history File links PeirceAlphaGraphs. ... Image File history File links PeirceAlphaGraphs. ... For other uses, see Syntax (disambiguation). ...

  • The blank page;
  • Single letters or phrases written anywhere on the page;
  • Objects (subgraphs) enclosed by a simple closed curve called a cut or sep. A cut can be empty. Cuts can be nested and concatenated at will but must never intersect.

Any well-formed part of a graph is a subgraph. In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...


The semantics are: Semantics (Greek semantikos, giving signs, significant, symptomatic, from sema, sign) refers to the aspects of meaning that are expressed in a language, code, or other form of representation. ...

  • The blank page denotes Truth;
  • Letters, phrases, subgraphs, and entire graphs can be True or False;
  • To surround objects with a cut is equivalent to logical negation or Boolean complementation. Hence an empty cut denotes False;
  • All objects 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. Negation (i. ... A complementation test is used in genetics to decide if two recessive mutant phenotypes are determined by mutations in the same gene or two different genes. ... Astronomical conjunction Grammatical conjunction Logical conjunction This is a disambiguation page, a list of pages that otherwise might share the same title. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... -1... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...


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. Likewise, any subgraph P in node n may be erased if there exists a copy of P in some node ancestral to n (i.e., some node on which n depends). For an equivalent rule in an algebraic context, see C2 in Laws of form.

A proof manipulates a graph by a series of steps, with each step justified by one of the above 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). A tautologically true formula is one whose graph simplifies to the blank page. Graphs that cannot be simplified beyond a certain point are analogues of the satisfiable formulas of first order logic. A tree structure is a way of representing the hierarchical nature of a structure in a graphical form. ... Node may mean: Node (botany), the place on a plant stem where a leaf is attached Node (physics), a spatial locus along a standing wave where the wave has minimal amplitude Node (networking), a device connected to a network, such as a computer or router Node (computer science), a basic... A tree structure is a way of representing the hierarchical nature of a structure in a graphical form. ... The book Laws of Form (hereinafter abbreviated LoF), by G. Spencer-Brown, describes three distinct logical systems: The primary arithmetic (described in Chapter 4), which can be interpreted as Boolean arithmetic; The primary algebra (chapter 6), an algebraic structure that is a provocative and economical notation for the two-element... Look up proof in Wiktionary, the free dictionary. ... Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... In mathematics and in the sciences, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ...


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. In mathematics, a predicate is a relation. ... The domain of discourse, sometimes called the universe of discourse, is an analytic tool used in deductive logic, especially predicate logic. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ...


The beta graphs can be read as employing implicitly quantified variables, as a system in which all formula are to be taken as closed. If the "shallowest" part of a line of identity has even (odd) depth, the associated variable would be tacitly existentially (universally) quantified. The beta graphs appear to streamline first order logic with identity, but the secondary literature is not in agreement on just how this is accomplished. Peirce's writings do not address this question, because first order logic was first clearly defined only with the 1928 first edition of David Hilbert and Wilhelm Ackermann's Principles of Theoretical Logic. In predicate logic, existential quantification is an attempt to formalize the notion that something (a logical predicate) is true for something, or at least one relevant thing. ... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... David Hilbert (January 23, 1862, Wehlau, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany – December 24, 1962 Lüdenscheid, Germany ) was a German mathematician best known for the Ackermann function, an important example in the theory of computation. ... Principles of Theoretical Logic is the title of the 1950 American translation of the 1938 second edition of David Hilberts and Wilhelm Ackermanns classic text Grundzüge der theoretischen Logik, on elementary mathematical logic. ...


Gamma

Add to alpha a second kind of simple closed curve, written using a dashed rather than a solid line, to alpha. The resulting second style of cut can be read as the primitive unary operator of modal logic. In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, a unary operation is an operation with only one operand. ... A modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ...


Zeman (1964) was the first to note that:

  • The beta graphs are isomorphic to the predicate calculus (Roberts 1973 and Shin 2002 discuss this finding at length);
  • 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.

First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... A modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... S4 can refer to: SATA International S4 algebra - a variety of modal logic, also called Interior algebra the Audi S4 car This is a disambiguation page — a list of pages that otherwise might share the same title. ... S5 stands for Simple Standards-Based Slide Show System and is an XHTML-based file format for defining slideshows. ... In logic, normal modal logic is a set L of modal formulas such that L contains all propositional tautologies, Kripkes schema: , and L is closed under substitution, detachment rule: from A and A→B infer B, necessitation rule: from A infer . ...

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 metalogic (something which eluded even Principia Mathematica). 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. His work went a step further than Euler's diagrams which were revised also by Venn. Frege's 1879 Begriffschrifft independently advanced in the same direction, but employed a notation very different from Peirce's. Semiotics, or semiology, is the study of signs and symbols, both individually and grouped in sign systems. ... -1... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... It has been suggested that Predicate calculus be merged into this article or section. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Look up Relation in Wiktionary, the free dictionary In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation (of set theory and logic) and relational algebra. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... In geometry, the nine point circle is a circle that can be constructed for any given triangle. ... Venn may mean: Venn diagrams, special diagrams in logics John Venn (1834 – 1923), a british logician and the inventor of Venn diagrams John Venn (1586–1650), was Vice-Chancellor of Oxford University This is a disambiguation page: a list of articles associated with the same title. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ...


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 immediately abandoned this formalism in favor of the existential graphs. An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned. ...


Peirce's graphical logic is but one of his many 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). Charles Sanders Peirce (pronounced purse), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ...


References

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 Hilary Putnam, eds.. Harvard University Press.
  • 2001. Semiotic and Significs: The Correspondence between C.S. Peirce and Victoria Lady Welby. Hardwick, C.S., ed. Lubbock TX: 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 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 25 volumes of the chronological edition appear. Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ... The Harvard University Press is a publishing house, a division of Harvard University, that is highly respected in academic publishing. ... Victoria, Lady Welby (also styled the Hon. ... A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ...


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.

See also

Ampheck, from Greek double-edged, is a term coined by Charles Sanders Peirce for either one of the pair of logically dual operators, variously referred to as Peirce arrows, Sheffer strokes, or NAND and NNOR. Either of these logical operators is a sole sufficient operator for deriving or generating all... Elsie the cat is sitting on a mat John F. Sowas conceptual graphs (CGs) are a system of logic based on the existential graphs of Charles Sanders Peirce and the semantic networks of artificial intelligence. ... An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned. ... A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ... Charles Sanders Peirce (pronounced purse), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ...

External links

  • Stanford Encyclopedia of Philosophy: Peirce's Logic by Eric Hammer. Employs parentheses notation.
  • Van Heuveln, Bram, "Existential Graphs." Dept. of Cognitive Science, Rensselaer Polytechnic University. Alpha only.
  • Gottschall, Christian, Proof Builder — Java applet for deriving Alpha graphs.


 
 

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