ONT Re: Differential Logic -- Series A
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DLOG. Note A7
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If you think that I linger in the realm of logical difference calculus
out of sheer vacillation about getting down to the differential proper,
it is probably out of a prior expectation that you derive from the art
or the long-engrained practice of real analysis. But the fact is that
ordinary calculus only rushes on to the sundry orders of approximation
because the strain of comprehending the full import of E and D at once
whelm over its discrete and finite powers to grasp them. But here, in
the fully serene idylls of ZOL, we find ourselves fit with the compass
of a wit that is all we'd ever wish to explore their effects with care.
So let us do just that.
I will first rationalize the novel grouping of propositional forms
in the last set of Tables, as that will extend a gentle invitation
to the mathematical subject of "group theory", and demonstrate its
relevance to differential logic in a strikingly apt and useful way.
The data for that account is contained in Table 4.
Table 4. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
| | | | | | |
| f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| | | | | | |
| f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| | | | | | |
| f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
The shift operator E can be understood as enacting a substitution operation
on the proposition that is given as its argument. In our immediate example,
we have the following data and definition:
E : (U -> B) -> (EU -> B),
E : f(x, y) -> Ef(x, y, dx, dy),
Ef(x, y, dx, dy) = f(x + dx, y + dy).
Therefore, if we evaluate Ef at particular values of dx and dy,
for example, dx = i and dy = j, where i, j are in B, we obtain:
E_ij : (U -> B) -> (U -> B),
E_ij : f -> E_ij f,
E_ij f = Ef | <dx = i, dy = j> = f(x + i, y + j).
The notation is a little bit awkward, but the data of the Table should
make the sense clear. The important thing to observe is that E_ij has
the effect of transforming each proposition f : U -> B into some other
proposition f' : U -> B. As it happens, the action is one-to-one and
onto for each E_ij, so the gang of four operators {E_ij : i, j in B}
is an example of what is called a "transformation group" on the set
of sixteen propositions. Bowing to a longstanding local and linear
tradition, I will therefore redub the four elements of this group
as T_00, T_01, T_10, T_11, to bear in mind their transformative
character, or nature, as the case may be. Abstractly viewed,
this group of order four has the following operation table:
o----------o----------o----------o----------o----------o
| % | | | |
| * % T_00 | T_01 | T_10 | T_11 |
| % | | | |
o==========o==========o==========o==========o==========o
| % | | | |
| T_00 % T_00 | T_01 | T_10 | T_11 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_01 % T_01 | T_00 | T_11 | T_10 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_10 % T_10 | T_11 | T_00 | T_01 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_11 % T_11 | T_10 | T_01 | T_00 |
| % | | | |
o----------o----------o----------o----------o----------o
It happens that there are just two possible groups of 4 elements.
One is the cyclic group Z_4 (German "Zyklus"), which this is not.
The other is Klein's four-group V_4 (German "Vier"), which it is.
More concretely viewed, the group as a whole pushes the set
of sixteen propositions around in such a way that they fall
into seven natural classes, called "orbits". One says that
the orbits are preserved by the action of the group. There
is an "Orbit Lemma" of immense utility to "those who count"
which, depending on your upbringing, you may associate with
the names of Burnside, Cauchy, Frobenius, or some subset or
superset of these three, vouching that the number of orbits
is equal to the mean number of fixed points, in other words,
the total number of points (in our case, propositions) that
are left unmoved by the separate operations, divided by the
order of the group. In this instance, T_00 operates as the
group identity, fixing all 16 propositions, while the other
three group elements fix 4 propositions each, and so we get:
Number of orbits = (4 + 4 + 4 + 16) / 4 = 7. Amazing!
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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