fom
Posts:
1,035
Registered:
12/4/12
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Re: fom - 02 - CORRECTED - logical constants
Posted:
Dec 11, 2012 9:26 PM
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What follows it the presentation of
logical constants in the form of a
projective geometry on 21-names.
The namespace conforms with a typical
presentation of finite projective
geometries using difference sets.
There are three collineations which
shall be named:
Negation:
axis-
NOT
line elements-
NOT OTHER NO SOME ALL
Contraposition:
axis-
NOT
line elements-
NOT LEQ XOR NTRU TRU
Conjugation:
axis-
NOT
line elements-
NOT FLIP LET DENY FIX
All admissible projectivities for the dual plane
shall fix these lines.
The list that follows takes the form:
Line Name: {list of line elements}
A 21-point projective plane has 21 lines:
NTRU: {NOT, NO, OTHER, ALL, SOME}
NOT: {NOT, NTRU, LEQ, XOR, TRU}
LEQ: {NOT, IF, NIMP, IMP, NIF}
XOR: {NOT, OR, NAND, AND, NOR}
TRU: {NOT, FLIP, LET, DENY, FIX}
NO: {NO, NTRU, IF, OR, FLIP}
IF: {NO, LEQ, NIMP, NAND, LET}
OR: {NO, XOR, IMP, AND, DENY}
FLIP: {NO, TRU, NIF, NOR, FIX}
OTHER: {OTHER, NTRU, IMP, NAND, FIX}
IMP: {OTHER, LEQ, NIF, OR, DENY}
NAND: {OTHER, XOR, IF, NOR, LET}
FIX: {OTHER, TRU, NIMP, AND, FLIP}
ALL: {ALL, NTRU, NIMP, NOR, DENY}
NIMP: {ALL, LEQ, IF, AND, FIX}
NOR: {ALL, XOR, NIF, NAND, FLIP}
DENY: {ALL, TRU, IMP, OR, LET}
SOME: {SOME, NTRU, NIF, AND, LET}
NIF: {SOME, LEQ, IMP, NOR, FLIP}
AND: {SOME, XOR, NIMP, OR, FIX}
LET: {SOME, TRU, IF, NAND, DENY}
===========================================
A difference set presentation is given by:
18: {0, 9, 11, 4, 3}
0: {0, 18, 8, 6, 1}
8: {0, 10, 19, 13, 14}
6: {0, 16, 12, 2, 15}
1: {0, 7, 17, 5, 20}
9: {9, 18, 13, 12, 20}
13: {9, 8, 14, 16, 5}
12: {9, 6, 10, 15, 17}
20: {9, 1, 19, 2, 7}
11: {11, 18, 10, 16, 7}
10: {11, 8, 19, 12, 17}
16: {11, 6, 13, 2, 5}
7: {11, 1, 14, 15, 20}
4: {4, 18, 14, 2, 17}
14: {4, 8, 13, 15, 7}
2: {4, 6, 19, 16, 20}
17: {4, 1, 10, 12, 5}
3: {3, 18, 19, 15, 5}
19: {3, 8, 10, 2, 20}
15: {3, 6, 14, 12, 7}
5: {3, 1, 13, 16, 17}
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