fom
Posts:
1,969
Registered:
12/4/12


fom  02  logical constants
Posted:
Dec 7, 2012 2:08 AM


What follows it the presentation of logical constants in the form of a projective geometry on 21names.
The namespace conforms with a typical presentation of finite projective geometries using difference sets.
There are three collineations which shall be named:
Negation:
axis
THIS
line elements
THIS SOME OTHERS NO ALL
Contraposition:
axis
THIS
line elements
THIS LEQ XOR NTRU TRU
Conjugation:
axis
THIS
line elements
THIS 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 21point projective plane has 21 lines:
NTRU: {THIS, SOME, OTHER, NO, ALL}
THIS: {THIS, NTRU, LEQ, XOR, TRU} LEQ: {THIS, IF, NIMP, IMP, NIF} XOR: {THIS, OR, NAND, AND, NOR} TRU: {THIS, FLIP, LET, DENY, FIX}
SOME: {SOME, NTRU, IMP, NAND, FIX} IMP: {SOME, LEQ, NIF, OR, DENY} NAND: {SOME, XOR, IF, NOR, LET} FIX: {SOME, TRU, NIMP, AND, FLIP}
OTHER: {OTHER, NTRU, IF, OR, FLIP} IF: {OTHER, LEQ, NIMP, NAND, LET} OR: {OTHER, XOR, IMP, AND, DENY} FLIP: {OTHER, TRU, NIF, NOR, FIX}
NO: {NO, NTRU, NIF, AND, LET} NIF: {NO, LEQ, IMP, NOR, FLIP} AND: {NO, XOR, NIMP, OR, FIX} LET: {NO, TRU, IF, NAND, DENY}
ALL: {ALL, NTRU, NIMP, NOR, DENY} NIMP: {ALL, LEQ, IF, AND, FIX} NOR: {ALL, XOR, NIF, NAND, FLIP} DENY: {ALL, TRU, IMP, OR, LET}
===========================================
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}

