TY - JOUR

T1 - An analysis of Existential Graphs–part 2

T2 - Beta

AU - Bellucci, Francesco

AU - Pietarinen, Ahti Veikko

N1 - Funding Information:
Open access funding provided by Alma Mater Studiorum - Universitá di Bologna within the CRUI-CARE Agreement. Support received from (A.-V. Pietarinen) the Basic Research Program of the National Research University Higher School of Economics, the Estonian Research Council (PUT 1305, 2016-2018), the Academy of Finland (2016, 2019), and the Chinese National Social Science Fund “The Historical Evolution of Logical Vocabulary and Research on Philosophical Issues” (20& ZD046, 2020).
Publisher Copyright:
© 2021, The Author(s)

PY - 2021/12

Y1 - 2021/12

N2 - This paper provides an analysis of the notational difference between Beta Existential Graphs, the graphical notation for quantificational logic invented by Charles S. Peirce at the end of the 19th century, and the ordinary notation of first-order logic. Peirce thought his graphs to be “more diagrammatic” than equivalently expressive languages (including his own algebras) for quantificational logic. The reason of this, he claimed, is that less room is afforded in Existential Graphs than in equivalently expressive languages for different ways of representing the same fact. The reason of this, in turn, is that Existential Graphs are a non-linear, occurrence-referential notation. As a non-linear notation, each graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic that are obtained by permuting those elements (sentential variables, predicate expressions, and quantifiers) that in the graphs lie in the same area. As an occurrence-referential notation, each Beta graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic in which the identity of reference of two or more variables is asserted. In brief, Peirce’s graphs are more diagrammatic than the linear, type-referential notation of first-order logic because the function that translates the latter to the graphs does not define isomorphism between the two notations.

AB - This paper provides an analysis of the notational difference between Beta Existential Graphs, the graphical notation for quantificational logic invented by Charles S. Peirce at the end of the 19th century, and the ordinary notation of first-order logic. Peirce thought his graphs to be “more diagrammatic” than equivalently expressive languages (including his own algebras) for quantificational logic. The reason of this, he claimed, is that less room is afforded in Existential Graphs than in equivalently expressive languages for different ways of representing the same fact. The reason of this, in turn, is that Existential Graphs are a non-linear, occurrence-referential notation. As a non-linear notation, each graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic that are obtained by permuting those elements (sentential variables, predicate expressions, and quantifiers) that in the graphs lie in the same area. As an occurrence-referential notation, each Beta graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic in which the identity of reference of two or more variables is asserted. In brief, Peirce’s graphs are more diagrammatic than the linear, type-referential notation of first-order logic because the function that translates the latter to the graphs does not define isomorphism between the two notations.

KW - Beta graphs

KW - Existential graphs

KW - Line of identity

KW - Logical notations

KW - Occurrence-referentiality

KW - Peirce

KW - Quantification

UR - http://www.scopus.com/inward/record.url?scp=85105423333&partnerID=8YFLogxK

U2 - 10.1007/s11229-021-03134-3

DO - 10.1007/s11229-021-03134-3

M3 - Article

AN - SCOPUS:85105423333

SN - 0039-7857

VL - 199

SP - 7705

EP - 7726

JO - Synthese

JF - Synthese

IS - 3-4

ER -