[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 20

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Since multiplication by a 2-adic relative term
is a logical analogue of matrix multiplication
in linear algebra, all of the products that we
computed above can be represented in terms of
logical matrices and logical vectors.

Here are the absolute terms again, followed by
their representation as "coefficient tuples",
otherwise thought of as "coordinate vectors".

1  =  B +, C +, D +, E +, I +, J +, O

   =  <1, 1, 1, 1, 1, 1, 1>

b  =  O

   =  <0, 0, 0, 0, 0, 0, 1>

m  =  C +, I +, J +, O

   =  <0, 1, 0, 0, 1, 1, 1>

w  =  B +, D +, E

   =  <1, 0, 1, 1, 0, 0, 0>

Since we are going to be regarding these tuples as "column vectors",
it is convenient to arrange them into a table of the following form:

   | 1 b m w
---o---------
 B | 1 0 0 1
 C | 1 0 1 0
 D | 1 0 0 1
 E | 1 0 0 1
 I | 1 0 1 0
 J | 1 0 1 0
 O | 1 1 1 0

Here are the 2-adic relative terms again, followed by
their representation as coefficient matrices, in this
case bordered by row and column labels to remind us
what the coefficient values are meant to signify.

'l' = B:C +, C:B +, D:O +, E:I +, I:E +, O:D =

'l'| B C D E I J O
---o---------------
 B | 0 1 0 0 0 0 0
 C | 1 0 0 0 0 0 0
 D | 0 0 0 0 0 0 1
 E | 0 0 0 0 1 0 0
 I | 0 0 0 1 0 0 0
 J | 0 0 0 0 0 0 0
 O | 0 0 1 0 0 0 0

's' = C:O +, E:D +, I:O +, J:D +, J:O =

's'| B C D E I J O
---o---------------
 B | 0 0 0 0 0 0 0
 C | 0 0 0 0 0 0 1
 D | 0 0 0 0 0 0 0
 E | 0 0 1 0 0 0 0
 I | 0 0 0 0 0 0 1
 J | 0 0 1 0 0 0 1
 O | 0 0 0 0 0 0 0

Here are the matrix representations of
the products that we calculated before:

'l'1 = "lover of anybody" =

| 0 1 0 0 0 0 0 | | 1 |   | 1 |
| 1 0 0 0 0 0 0 | | 1 |   | 1 |
| 0 0 0 0 0 0 1 | | 1 |   | 1 |
| 0 0 0 0 1 0 0 | | 1 | = | 1 |
| 0 0 0 1 0 0 0 | | 1 |   | 1 |
| 0 0 0 0 0 0 0 | | 1 |   | 0 |
| 0 0 1 0 0 0 0 | | 1 |   | 1 |

'l'b = "lover of a black" =

| 0 1 0 0 0 0 0 | | 0 |   | 0 |
| 1 0 0 0 0 0 0 | | 0 |   | 0 |
| 0 0 0 0 0 0 1 | | 0 |   | 1 |
| 0 0 0 0 1 0 0 | | 0 | = | 0 |
| 0 0 0 1 0 0 0 | | 0 |   | 0 |
| 0 0 0 0 0 0 0 | | 0 |   | 0 |
| 0 0 1 0 0 0 0 | | 1 |   | 0 |

'l'm = "lover of a man" =

| 0 1 0 0 0 0 0 | | 0 |   | 1 |
| 1 0 0 0 0 0 0 | | 1 |   | 0 |
| 0 0 0 0 0 0 1 | | 0 |   | 1 |
| 0 0 0 0 1 0 0 | | 0 | = | 1 |
| 0 0 0 1 0 0 0 | | 1 |   | 0 |
| 0 0 0 0 0 0 0 | | 1 |   | 0 |
| 0 0 1 0 0 0 0 | | 1 |   | 0 |

'l'w = "lover of a woman" =

| 0 1 0 0 0 0 0 | | 1 |   | 0 |
| 1 0 0 0 0 0 0 | | 0 |   | 1 |
| 0 0 0 0 0 0 1 | | 1 |   | 0 |
| 0 0 0 0 1 0 0 | | 1 | = | 0 |
| 0 0 0 1 0 0 0 | | 0 |   | 1 |
| 0 0 0 0 0 0 0 | | 0 |   | 0 |
| 0 0 1 0 0 0 0 | | 0 |   | 1 |

's'1 = "servant of anybody" =

| 0 0 0 0 0 0 0 | | 1 |   | 0 |
| 0 0 0 0 0 0 1 | | 1 |   | 1 |
| 0 0 0 0 0 0 0 | | 1 |   | 0 |
| 0 0 1 0 0 0 0 | | 1 | = | 1 |
| 0 0 0 0 0 0 1 | | 1 |   | 1 |
| 0 0 1 0 0 0 1 | | 1 |   | 1 |
| 0 0 0 0 0 0 0 | | 1 |   | 0 |

's'b = "servant of a black" =

| 0 0 0 0 0 0 0 | | 0 |   | 0 |
| 0 0 0 0 0 0 1 | | 0 |   | 1 |
| 0 0 0 0 0 0 0 | | 0 |   | 0 |
| 0 0 1 0 0 0 0 | | 0 | = | 0 |
| 0 0 0 0 0 0 1 | | 0 |   | 1 |
| 0 0 1 0 0 0 1 | | 0 |   | 1 |
| 0 0 0 0 0 0 0 | | 1 |   | 0 |

's'm = "servant of a man" =

| 0 0 0 0 0 0 0 | | 0 |   | 0 |
| 0 0 0 0 0 0 1 | | 1 |   | 1 |
| 0 0 0 0 0 0 0 | | 0 |   | 0 |
| 0 0 1 0 0 0 0 | | 0 | = | 0 |
| 0 0 0 0 0 0 1 | | 1 |   | 1 |
| 0 0 1 0 0 0 1 | | 1 |   | 1 |
| 0 0 0 0 0 0 0 | | 1 |   | 0 |

's'w = "servant of a woman" =

| 0 0 0 0 0 0 0 | | 1 |   | 0 |
| 0 0 0 0 0 0 1 | | 0 |   | 0 |
| 0 0 0 0 0 0 0 | | 1 |   | 0 |
| 0 0 1 0 0 0 0 | | 1 | = | 1 |
| 0 0 0 0 0 0 1 | | 0 |   | 0 |
| 0 0 1 0 0 0 1 | | 0 |   | 1 |
| 0 0 0 0 0 0 0 | | 0 |   | 0 |

'ls' = "lover of a servant of ---" =

| 0 1 0 0 0 0 0 | | 0 0 0 0 0 0 0 |   | 0 0 0 0 0 0 1 |
| 1 0 0 0 0 0 0 | | 0 0 0 0 0 0 1 |   | 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 1 | | 0 0 0 0 0 0 0 |   | 0 0 0 0 0 0 0 |
| 0 0 0 0 1 0 0 | | 0 0 1 0 0 0 0 | = | 0 0 0 0 0 0 1 |
| 0 0 0 1 0 0 0 | | 0 0 0 0 0 0 1 |   | 0 0 1 0 0 0 0 |
| 0 0 0 0 0 0 0 | | 0 0 1 0 0 0 1 |   | 0 0 0 0 0 0 0 |
| 0 0 1 0 0 0 0 | | 0 0 0 0 0 0 0 |   | 0 0 0 0 0 0 0 |

'sl' = "servant of a lover of ---" =

| 0 0 0 0 0 0 0 | | 0 1 0 0 0 0 0 |   | 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 1 | | 1 0 0 0 0 0 0 |   | 0 0 1 0 0 0 0 |
| 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 1 |   | 0 0 0 0 0 0 0 |
| 0 0 1 0 0 0 0 | | 0 0 0 0 1 0 0 | = | 0 0 0 0 0 0 1 |
| 0 0 0 0 0 0 1 | | 0 0 0 1 0 0 0 |   | 0 0 1 0 0 0 0 |
| 0 0 1 0 0 0 1 | | 0 0 0 0 0 0 0 |   | 0 0 1 0 0 0 1 |
| 0 0 0 0 0 0 0 | | 0 0 1 0 0 0 0 |   | 0 0 0 0 0 0 0 |

Jon Awbrey

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