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Topic: fom - 02 - logical constants
Replies: 1   Last Post: Dec 11, 2012 9:26 PM

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fom

Posts: 1,969
Registered: 12/4/12
fom - 02 - logical constants
Posted: Dec 7, 2012 2:08 AM
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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-

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 21-point 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}




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