Peirce

1
Peirces Existential Graphs

• Bram van Heuveln
• Minds and Machines Lab, RPI
• Summer 2001

2
Todays Topics

• Alpha
• symbolization
• rules of inference
• some theory
• Java implementation

3
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
?
?
4
AlphaSymbolization
Symbolization in EG
Expression in PL
P
P
?
?
? ? ?
?
?
?
?
? ? ?
?
?
? ? ?
5
AlphaInference Rules
?
?
?
?
?
?
Double Cut
?
?
?
?
?
?
?
?
?
?
?
(De)Iteration
?
?
Erasure
?
?
?
1
2k
1
2k
?
?
?
?
Insertion
?
1
2k1
1
2k1
6
AlphaMultiple Readings
Possible Readings
P ? Q or Q ? P
Q
P
Q
P
(P ? Q) or P ? Q
(P ? Q) or P ? Q or P ? Q
Q
P
(P ? Q) or P ? Q or P ? Q or (Q ?
P) or Q ? P or Q ? P
Q
P
7
AlphaMultiple Readings Formalized (see Shin)

• (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)

8
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
9
AlphaRelative Recursive Conditional Reading
?1
?2k-1
?2k
?
?0
The Recursive Conditional Reading (RCR) relative
to any subgraph ?2k-1 or ?2k is ?0 ?1 ? ?2
? (?1 ? ?3 ? ? ? ?2k-1) ? ?2k
10
AlphaSoundness of Insertion
?0 ?1 ? ?2 ? (?1 ? ?3 ? ? ? ?2k-1) ? ?2k
?1
?2k-1
?2k
?
?0
Strengthening the Antecedent
IN
?0 ?1 ? ?2 ? (?1 ? ?3 ? ? ? ?2k-1 ? ?) ? ?2k
?
?1
?2k-1
?2k
?
?0
11
AlphaSoundness of Erasure
?0 ?1 ? ?2 ? (?1 ? ?3 ? ? ? ?2k-1) ? (?2k ?
?)
?
?1
?2k-1
?2k
?
?0
Weakening the Consequent
E
?0 ?1 ? ?2 ? (?1 ? ?3 ? ? ? ?2k-1) ? ?2k
?1
?2k-1
?2k
?
?0
12
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
?
?
13
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
?
?