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ONT Re: Functional Logic




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FL.  Note 6

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2.1.6.  Application of Higher Order Propositions to Quantification Theory

Our excursion into the vastening landscape of higher order propositions
has finally come round to the stage where we can bring its returns to
bear on opening up new perspectives for quantificational logic.

There is a question arising next that is still experimental in my mind.
Whether it makes much difference from a purely formal point of view is
not a question I can answer yet, but it does seem to aid the intuition
to invent a slightly different interpretation for the two-valued space
that we use as the target of our basic indicator functions.  Therefore,
let us establish a type of "existence-valued" functions f : B^k -> %E%,
where %E% = {-e-, +e+} = {"empty", "exist"} is a couple of values that
we interpret as indicating whether of not anything exists in the cells
of the underlying universe of discourse, venn diagram, or other domain.
As usual, let us not be too strict about the coding of these functions,
reverting to binary codes whenever the interpretation is clear enough.

With this interpretation in mind we note the following correspondences
between classical quantifications and higher order indicator functions:

Table 16.  Syllogistic Premisses as Higher Order Indicator Functions
o---o---------------------o-----------------o----------------------------o
|   |                     |                 |                            |
| A | Universal Affrmtve  | All   x  is  y  | Indicator of < x (y)> = 0  |
|   |                     |                 |                            |
| E | Universal Negative  | All   x  is (y) | Indicator of < x  y > = 0  |
|   |                     |                 |                            |
| I | Particular Affrmtve | Some  x  is  y  | Indicator of < x  y > = 1  |
|   |                     |                 |                            |
| O | Particular Negative | Some  x  is (y) | Indicator of < x (y)> = 1  |
|   |                     |                 |                            |
o---o---------------------o-----------------o----------------------------o

Tables 17 and 18 develop these ideas in greater detail.

Table 17.  Relation of Quantifiers to Higher Order Propositions
o------------o------------o-----------o-----------o-----------o----------o
| Mnemonic   | Category   | Classical | Alternate | Symmetric | Operator |
|            |            |   Form    |   Form    |   Form    |          |
o============o============o===========o===========o===========o==========o
|     E      | Universal  |  All   x  |           |   No   x  |  (L_11)  |
| Exclusive  |  Negative  |   is  (y) |           |   is   y  |          |
o------------o------------o-----------o-----------o-----------o----------o
|     A      | Universal  |  All   x  |           |   No   x  |  (L_10)  |
| Absolute   |  Affrmtve  |   is   y  |           |   is  (y) |          |
o------------o------------o-----------o-----------o-----------o----------o
|            |            |  All   y  |   No   y  |   No  (x) |  (L_01)  |
|            |            |   is   x  |   is  (x) |   is   y  |          |
o------------o------------o-----------o-----------o-----------o----------o
|            |            |  All  (y) |   No  (y) |   No  (x) |  (L_00)  |
|            |            |   is   x  |   is  (x) |   is  (y) |          |
o------------o------------o-----------o-----------o-----------o----------o
|            |            | Some  (x) |           | Some  (x) |   L_00   |
|            |            |   is  (y) |           |   is  (y) |          |
o------------o------------o-----------o-----------o-----------o----------o
|            |            | Some  (x) |           | Some  (x) |   L_01   |
|            |            |   is   y  |           |   is   y  |          |
o------------o------------o-----------o-----------o-----------o----------o
|     O      | Particular | Some   x  |           | Some   x  |   L_10   |
| Obtrusive  |  Negative  |   is  (y) |           |   is  (y) |          |
o------------o------------o-----------o-----------o-----------o----------o
|     I      | Particular | Some   x  |           | Some   x  |   L_11   |
| Indefinite |  Affrmtve  |   is   y  |           |   is   y  |          |
o------------o------------o-----------o-----------o-----------o----------o

Table 18.  Simple Qualifiers of Propositions (n = 2)
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
|  | x | 1100 |    f     |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 |
|  | y | 1010 |          |no  x|no  x|no ~x|no ~x|sm ~x|sm ~x|sm  x|sm  x|
| f \  |      |          |is  y|is ~y|is  y|is ~y|is ~y|is  y|is ~y|is  y|
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
|      |      |          |                                               |
| f_0  | 0000 |    ()    |  1     1     1     1                          |
|      |      |          |                                               |
| f_1  | 0001 |  (x)(y)  |  1     1     1           1                    |
|      |      |          |                                               |
| f_2  | 0010 |  (x) y   |  1     1           1           1              |
|      |      |          |                                               |
| f_3  | 0011 |  (x)     |  1     1                 1     1              |
|      |      |          |                                               |
| f_4  | 0100 |   x (y)  |  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     1     1        |
|      |      |          |                                               |
| f_8  | 1000 |   x  y   |        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     1           1  |
|      |      |          |                                               |
| f_12 | 1100 |   x      |              1     1                 1     1  |
|      |      |          |                                               |
| f_13 | 1101 | ((x) y)  |              1           1           1     1  |
|      |      |          |                                               |
| f_14 | 1110 | ((x)(y)) |                    1           1     1     1  |
|      |      |          |                                               |
| f_15 | 1111 |   (())   |                          1     1     1     1  |
|      |      |          |                                               |
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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