[Inquiry] Re: Introduction to Inquiry Driven Systems

[Jon Awbrey]

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3.2.3.  Measure for Measure

An acquaintance with the functions of the umpire operator can
be gained from Tables 10 and 11, where the 2-dimensional case
is worked out in full.

The auxiliary notations:

   !a!_i f  =  !Y!(f_i, f),

   !b!_i f  =  !Y!(f, f_i),

define two series of measures:

   !a!_i, !b!_i  :  (B^2 -> B) -> B,

incidentally providing compact names for
the column headings of the next two Tables.

Table 10.  Qualifiers of Implication Ordering:  !a!_i f  =  !Y!(f_i => f)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|  | x | 1100 |    f     |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
|  | y | 1010 |          |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
| f \  |      |          |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|      |      |          |                                               |
| f_0  | 0000 |    ()    |                                             1 |
|      |      |          |                                               |
| f_1  | 0001 |  (x)(y)  |                                          1  1 |
|      |      |          |                                               |
| f_2  | 0010 |  (x) y   |                                       1     1 |
|      |      |          |                                               |
| f_3  | 0011 |  (x)     |                                    1  1  1  1 |
|      |      |          |                                               |
| f_4  | 0100 |   x (y)  |                                 1           1 |
|      |      |          |                                               |
| f_5  | 0101 |     (y)  |                              1  1        1  1 |
|      |      |          |                                               |
| f_6  | 0110 |  (x, y)  |                           1     1     1     1 |
|      |      |          |                                               |
| f_7  | 0111 |  (x  y)  |                        1  1  1  1  1  1  1  1 |
|      |      |          |                                               |
| f_8  | 1000 |   x  y   |                     1                       1 |
|      |      |          |                                               |
| f_9  | 1001 | ((x, y)) |                  1  1                    1  1 |
|      |      |          |                                               |
| f_10 | 1010 |      y   |               1     1                 1     1 |
|      |      |          |                                               |
| f_11 | 1011 |  (x (y)) |            1  1  1  1              1  1  1  1 |
|      |      |          |                                               |
| f_12 | 1100 |   x      |         1           1           1           1 |
|      |      |          |                                               |
| f_13 | 1101 | ((x) y)  |      1  1        1  1        1  1        1  1 |
|      |      |          |                                               |
| f_14 | 1110 | ((x)(y)) |   1     1     1     1     1     1     1     1 |
|      |      |          |                                               |
| f_15 | 1111 |   (())   |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
|      |      |          |                                               |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

Table 11.  Qualifiers of Implication Ordering:  !b!_i f  =  !Y!(f => f_i)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|  | x | 1100 |    f     |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
|  | y | 1010 |          |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 |
| f \  |      |          |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|      |      |          |                                               |
| f_0  | 0000 |    ()    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
|      |      |          |                                               |
| f_1  | 0001 |  (x)(y)  |   1     1     1     1     1     1     1     1 |
|      |      |          |                                               |
| f_2  | 0010 |  (x) y   |      1  1        1  1        1  1        1  1 |
|      |      |          |                                               |
| f_3  | 0011 |  (x)     |         1           1           1           1 |
|      |      |          |                                               |
| f_4  | 0100 |   x (y)  |            1  1  1  1              1  1  1  1 |
|      |      |          |                                               |
| f_5  | 0101 |     (y)  |               1     1                 1     1 |
|      |      |          |                                               |
| f_6  | 0110 |  (x, y)  |                  1  1                    1  1 |
|      |      |          |                                               |
| f_7  | 0111 |  (x  y)  |                     1                       1 |
|      |      |          |                                               |
| f_8  | 1000 |   x  y   |                        1  1  1  1  1  1  1  1 |
|      |      |          |                                               |
| f_9  | 1001 | ((x, y)) |                           1     1     1     1 |
|      |      |          |                                               |
| f_10 | 1010 |      y   |                              1  1        1  1 |
|      |      |          |                                               |
| f_11 | 1011 |  (x (y)) |                                 1           1 |
|      |      |          |                                               |
| f_12 | 1100 |   x      |                                    1  1  1  1 |
|      |      |          |                                               |
| f_13 | 1101 | ((x) y)  |                                       1     1 |
|      |      |          |                                               |
| f_14 | 1110 | ((x)(y)) |                                          1  1 |
|      |      |          |                                               |
| f_15 | 1111 |   (())   |                                             1 |
|      |      |          |                                               |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

Applied to a given proposition f, the qualifiers !a!_i and !b!_i tell whether
f rests "above f_i" or "below f_i", respectively, in the implication ordering.
By way of example, let us trace the effects of several such measures, namely,
those that occupy the limiting positions of the Tables.

   !a!_00 f = 1  iff  f_00 => f,  iff  0 => f, hence !a!_00 f = 1 for all f.

   !a!_15 f = 1  iff  f_15 => f,  iff  1 => f, hence !a!_15 f = 1 iff f = 1.

   !b!_00 f = 1  iff  f => f_00,  iff  f => 0, hence !b!_00 f = 1 iff f = 0.

   !b!_15 f = 1  iff  f => f_15,  iff  f => 1, hence !b!_15 f = 1 for all f.

Thus, !a!_0 = !b!_15 is a totally indiscriminate measure,
one that accepts all propositions f : B^2 -> B, whereas
!a!_15 and !b!_0 are measures that value the constant
propositions 1 : B^2 -> B and 0 : B^2 -> %B%,
respectively, above all others.

Finally, in conformity with the use of the fiber notation to
indicate sets of models, it is natural to use notations like:

   [| !a!_i |]  =  (!a!_i)^(-1)(1),

   [| !b!_i |]  =  (!b!_i)^(-1)(1),

   [| !Y!_p |]  =  (!Y!_p)^(-1)(1),

to denote sets of propositions that satisfy the umpires in question.

Jon Awbrey

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