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This document delves into Charles Sanders Peirce's existential graphs, an innovative approach to logical representation. We explore the fundamental concepts, including the Alpha notation for symbolization, the Sheet of Assertion where statements are visually mapped, and the established rules of inference such as Double Cut, Insertion, and Erasure. Additionally, we discuss the implementation in Java and the multiple readings arising from these systems. This integrated understanding serves as a foundation for advanced topics in logic and computational theory, facilitating the application of Peirce’s ideas in modern contexts.
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Peirce’s Existential Graphs Bram van Heuveln Minds and Machines Lab, RPI Summer 2001
Today’s Topics • Alpha • symbolization • rules of inference • some theory • Java implementation
AlphaSheet of Assertion • To assert some statement in EG, you put the symbolization of that statement on a sheet of paper, called the ‘Sheet of Assertion’ (SA). The location of the statement on the SA does not matter, i.e: states the same as:
AlphaSymbolization Symbolization in EG Expression in PL P P ~
AlphaInference Rules Double Cut (De)Iteration Erasure 2k 1 2k 1 Insertion 2k+1 1 2k+1 1
AlphaMultiple Readings Possible Readings P Q P Q or Q P ~(P Q) or ~P ~Q P Q ~(P ~Q) or ~P Q or P Q P Q ~(~P ~Q) or P Q or ~P Q or ~(~Q ~P) or Q P or ~Q P P Q
(i) f() = T (ii) f([]) = F (iii) f(P) = P (iv) f([P]) = ~P (v) f([D]) = ~f(D) (vi) f(D1 D2) = f(D1) f(D2) (vii) f([D1 D2]) = f([D1]) f([D2]) (viii) f([D1 [D2 D3]]) = f(D1) f(D2) f([D1 [D3]]) (ix) f([[D]]) = f(D) (PROLOG Project: Given subset, generate all readings) AlphaMultiple Readings Formalized (see Shin)
AlphaRecursive Conditional Reading Q R S P This graph can be read as: P (Q (R S)), i.e. P Q & P (R S) or P Q & (P R) S
AlphaRelative Recursive Conditional Reading 0 1 2k-1 2k The Recursive Conditional Reading (RCR) relative to any subgraph 2k-1 or 2k is: 0 & 1 2 & (1 3 2k-1) 2k
AlphaSoundness of Insertion 0 & 1 2 & (1 3 2k-1) 2k 0 1 2k-1 2k Strengthening the Antecedent IN 0 & 1 2 & (1 3 2k-1 )2k 0 1 2k-1 2k
AlphaSoundness of Erasure 0 & 1 2 & (1 3 2k-1) (2k ) 0 1 2k-1 2k Weakening the Consequent E 0 & 1 2 & (1 3 2k-1)2k 0 1 2k-1 2k
AlphaSoundness of Iteration/Deiteration Case 1 & 1 2 & (1 3 2k-1) 2k 1 2k-1 2k p, q r p, q (r p) IT/DE & 1 2 & (1 3 2k-1)(2k ) 1 2k-1 2k
AlphaSoundness of Iteration/Deiteration Case 2 & 1 2 & (1 3 2k-1) 2k 1 2k-1 2k p, q r p, (q p) r IT/DE & 1 2 & (1 3 2k-1 )2k 1 2k-1 2k
