«1. Peirce’s Method for Teaching Logic Peirce’s existential graphs (EGs) are the simplest, most elegant, and easiest-to-learn system of logic ever ...»
A.4 Extensions for Gamma Graphs Throughout his long career, Peirce experimented with a variety of notations for logic and a wide range of semantic extensions that went far beyond ordinary first-order logic. In his article of 1885, in which he presented his most complete version of the algebraic notation, he used the terms first-intentional logic for quantifiers that range over simple individuals and second-intentional logic for quantifiers that range over relations. In that article, he used second intentional logic to define equality x=y by a statement that for every relation R, R(x) if and only if R(y). Ernst Schröder translated Peirce’s terms to erste Ordnung and zweite Ordnung, which Bertrand Russell translated back to English as first order and second order. Peirce also introduced notations for three-valued logic, modal logic, and metalanguage about logic. Roberts (1973) summarizes the various graphical and algebraic notations and cites the publications and manuscripts in which Peirce discussed them.
Peirce used the term Gamma graphs for the many variations of EGs that went beyond first-order (or firstintentional) logic. As early as 1898, he used the following example of a metalevel statement in EGs:
The sentence inside the oval could be expressed in EGIF as a proposition or medad with a name enclosed in double quotes. The line of identity and phrase outside the oval could be expressed by a defining node and a monadic relation with an enclosed name. But neither EGIF nor CGIF as defined by ISO/IEC 24707 can represent a line of identity linked to an oval. A proposed extension to Common Logic called IKL (Hayes and Menzel 2006) can support such constructs. An extension to the CLIF dialect with the IKL semantics uses an
operator that followed by a CLIF sentence to denote the proposition stated by the sentence:
("is much to be wished" (that ("You are a good girl")))
An equivalent extension to CGIF or EGIF would use the following notation:
[*x ("You are a good girl")] ("is much to be wished" ?x) Either the EG or its translation to CLIF, CGIF, or EGIF could be read That you are a good girl is much to be wished. A syntactic extension to EGIF for such expressions could be represented with an optional EG in the
rule for DefiningNode:
DefiningNode = '[', DefiningLabel, [EG] ']';
When a line of identity represents a proposition, the bound label prefixed with # could be used in the type position of a medad to assert the proposition: (#?x). It could then be negated in the usual way, ~[(#?x)], to say that it is false that you are a good girl. In some writings, Peirce used ovals with colors or dotted boundaries to represent modality. EGIF can use identifiers such as Possible or Necessary for relations applied to propositions. The first line of the following EGIF defines x as the proposition that you are a good girl and y as its negation. The second line says that if x is necessary, then y is not possible. With the double negation erased by rule 3e, that line would say it’s false that x is necessary and y is possible.
[*x ("You are a good girl")] [*y ~[(#?x)]] ~[(Necessary ?x) ~[ ~[(Possible ?y)] ]] The semantics of EGIF is formally defined by the model theory of Common Logic or the IKL extensions. For Alpha and Beta graphs, the EGIF semantics seems to be consistent with what Peirce had intended.
Determining exactly what he had intended for his many variations of Gamma graphs is still a research project, for which EGIF can be a useful tool.
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