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John F. Sowa (VivoMind Research, LLC, USA)

Copyright: © 2014
|Pages: 34

DOI: 10.4018/978-1-4666-6042-7.ch020

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TopExistential graphs and the conceptual graphs based on them are formally defined, but they follow the long tradition of deriving logical patterns from language patterns. For the first formal logic, Aristotle developed a stylized or controlled version of natural language (NL). To clarify the references and reduce ambiguity, he replaced pronouns with letters. For his *Elements of Geometry*, Euclid followed Aristotle’s conventions as far as he could. When he needed more expressive power, Euclid added diagrams and a broader range of language patterns. Controlled Greek was the first CNL, but logicians and mathematicians translated it to controlled Latin, Arabic, and other languages. Over the centuries, they abbreviated words and phrases with various symbols and organized them in diagrams. The plus sign +, for example, is a simplified ampersand &, which abbreviated a hand-written *et* in Latin. The oldest surviving type hierarchy is the Tree of Porphyry from the third century AD.

In the 19th century, George Boole (1854) presented his *laws of thought* as an algebra for propositional logic: 1 for truth; 0 for falsehood; + for *or*; × for *and*; and − for *not*. Thirty years later, Gottlob Frege and Charles Sanders Peirce independently developed notations for first-order logic (FOL). Frege (1879) invented tree diagrams for representing FOL, but nobody else adopted his notation. Peirce added *n*-adic relations to Boolean algebra in 1870, introduced quantifiers in 1880, and extended the algebraic notation to both first-order and higher-order logic in 1885. Giuseppe Peano (1889) adopted Peirce’s algebra and changed some of the symbols to create the modern notation for predicate calculus. But in 1896, Peirce invented *existential graphs* (EGs) as a more *diagrammatic* notation for “the atoms and molecules of logic.”

As an example, the English sentence *A cat is on a mat* and a controlled English version would be identical. Since Boolean algebra cannot represent the details of relations and quantifiers, it can only represent the full proposition by a single unanalyzed letter *p*. For his relational algebra of 1870, Peirce invented a notation for the details at the word and phrase level: Cat* _{i}* for a cat

Since Peano wanted to mix mathematical and logical symbols in the same formulas, he invented new symbols by turning letters upside down or backwards. He replaced Boole’s + for *or* with ∨ for the Latin *vel*, and he turned ∨ upside down for *and*. For the existential quantifier, he turned E backwards for ∃. With these symbols, Peano’s version of Peirce’s formula becomes:

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