ONT Re: Differential Logic
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DLOG. Note D21
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Back to the Feature
| I guess it must be the flag of my disposition, out of hopeful
| green stuff woven.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 31]
Let us assume that the sense intended for differential features is well enough
established in the intuition, for now, that I may continue with outlining the
structure of the differential extension [E!X!] = [A, dA]. Over the extended
alphabet E!X! = {x_1, dx_1} = {A, dA}, of cardinality 2^n = 2, one generates
the set of points, EX, of cardinality of cardinality 2^2n = 4, that has the
following equivalent descriptions:
EX = <|A, dA|>
= {(A), A} x {(dA), dA}
= {(A)(dA), (A) dA, A (dA), A dA}.
The space EX may be assigned the mnemonic type B x D, which is really
no different than B x B = B^2. An individual element of EX may be
regarded as a "disposition at a point" or a "situated direction",
in effect, a singular mode of change occurring at a single point
in the universe of discourse. In applications, the modality of
this change can be interpreted in various ways, for example,
as an expectation, an intention, or an observation with
respect the behavior of a system.
To complete the construction of the extended universe of discourse
EX% = [x_1, dx_1] = [A, dA], one must add the set of differential
propositions EX^ = {g : EX -> B} ~=~ (B x D -> B) to the set of
dispositions in EX. There are 2^2^2n = 16 propositions in EX^,
as detailed in Table 14.
Table 14. Differential Propositions
o-------o--------o---------o-----------o-------------------o----------o
| | A : 1 1 0 0 | | | |
| | dA : 1 0 1 0 | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| f_0 | g_0 | 0 0 0 0 | () | False | 0 |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| | g_1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ~A & ~dA |
| | | | | | |
| | g_2 | 0 0 1 0 | (A) dA | Not A but dA | ~A & dA |
| | | | | | |
| | g_4 | 0 1 0 0 | A (dA) | A but not dA | A & ~dA |
| | | | | | |
| | g_8 | 1 0 0 0 | A dA | A and dA | A & dA |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| f_1 | g_3 | 0 0 1 1 | (A) | Not A | ~A |
| | | | | | |
| f_2 | g_12 | 1 1 0 0 | A | A | A |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| | g_6 | 0 1 1 0 | (A, dA) | A not equal to dA | A + dA |
| | | | | | |
| | g_9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| | g_5 | 0 1 0 1 | (dA) | Not dA | ~dA |
| | | | | | |
| | g_10 | 1 0 1 0 | dA | dA | dA |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| | g_7 | 0 1 1 1 | (A dA) | Not both A and dA | ~A v ~dA |
| | | | | | |
| | g_11 | 1 0 1 1 | (A (dA)) | Not A without dA | A => dA |
| | | | | | |
| | g_13 | 1 1 0 1 | ((A) dA) | Not dA without A | A <= dA |
| | | | | | |
| | g_14 | 1 1 1 0 | ((A)(dA)) | A or dA | A v dA |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| f_3 | g_15 | 1 1 1 1 | (()) | True | 1 |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
Aside from changing the names of variables and shuffling the order of rows,
this Table follows the format that was used previously for boolean functions
of two variables. The rows are grouped to reflect natural similarity classes
among the propositions. In a future discussion, these classes will be given
additional explanation and motivation as the orbits of a certain transformation
group acting on the set of 16 propositions. Notice that four of the propositions,
in their logical expressions, resemble those given in the table for X^. Thus the
first set of propositions {f_i} is automatically embedded in the present set {g_j},
and the corresponding inclusions are indicated at the far left margin of the table.
Jon Awbrey
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