ONT Re: Differential Logic
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DLOG. Note D62
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Analytic Series: A Coordinate Method
| And if he is told that something 'is' the way it is, then he thinks:
| Well, it could probably just as easily be some other way. So the
| sense of possibility might be defined outright as the capacity to
| think how everything could "just as easily" be, and to attach no
| more importance to what is than to what is not.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 12]
Table 50 exhibits a truth table method for computing the analytic series
(or the differential expansion) of a proposition in terms of coordinates.
Table 50. Computation of an Analytic Series in Terms of Coordinates
o-----------o-------------o-------------oo-------------o---------o-------------o
| u v | du dv | u' v' || !e!J EJ | DJ | dJ d^2.J |
o-----------o-------------o-------------oo-------------o---------o-------------o
| | | || | | |
| 0 0 | 0 0 | 0 0 || 0 0 | 0 | 0 0 |
| | | || | | |
| | 0 1 | 0 1 || 0 | 0 | 0 0 |
| | | || | | |
| | 1 0 | 1 0 || 0 | 0 | 0 0 |
| | | || | | |
| | 1 1 | 1 1 || 1 | 1 | 0 1 |
| | | || | | |
o-----------o-------------o-------------oo-------------o---------o-------------o
| | | || | | |
| 0 1 | 0 0 | 0 1 || 0 0 | 0 | 0 0 |
| | | || | | |
| | 0 1 | 0 0 || 0 | 0 | 0 0 |
| | | || | | |
| | 1 0 | 1 1 || 1 | 1 | 1 0 |
| | | || | | |
| | 1 1 | 1 0 || 0 | 0 | 1 1 |
| | | || | | |
o-----------o-------------o-------------oo-------------o---------o-------------o
| | | || | | |
| 1 0 | 0 0 | 1 0 || 0 0 | 0 | 0 0 |
| | | || | | |
| | 0 1 | 1 1 || 1 | 1 | 1 0 |
| | | || | | |
| | 1 0 | 0 0 || 0 | 0 | 0 0 |
| | | || | | |
| | 1 1 | 0 1 || 0 | 0 | 1 1 |
| | | || | | |
o-----------o-------------o-------------oo-------------o---------o-------------o
| | | || | | |
| 1 1 | 0 0 | 1 1 || 1 1 | 0 | 0 0 |
| | | || | | |
| | 0 1 | 1 0 || 0 | 1 | 1 0 |
| | | || | | |
| | 1 0 | 0 1 || 0 | 1 | 1 0 |
| | | || | | |
| | 1 1 | 0 0 || 0 | 1 | 0 1 |
| | | || | | |
o-----------o-------------o-------------oo-------------o---------o-------------o
The first six columns of the Table, taken as a whole, represent the variables of
a construct that I describe as the "contingent universe" [u, v, du, dv, u', v'],
or the bundle of "contingency spaces" [du, dv, u', v'] over the universe [u, v].
Their placement to the left of the double bar indicates that all of them amount
to independent variables, but there is a co-dependency among them, as described
by the following equations:
o-------------------------------------------------o
| |
| u' = u + du = (u, du) |
| |
| v' = v + du = (v, dv) |
| |
o-------------------------------------------------o
These relations correspond to the formal substitutions that are made in
defining EJ and DJ. For now, the whole rigamarole of contingency spaces
can be regarded as a technical device for achieving the effect of these
substitutions, adapted to a setting where functional compositions and
other symbolic manipulations are difficult to contemplate and execute.
The five columns to the right of the double bar in Table 50 contain the
values of the dependent variables {!e!J, EJ, DJ, dJ, d^2.J}. These are
normally interpreted as values of functions WJ : EU -> B or as values of
propositions in the extended universe [u, v, du, dv], but the dependencies
prevailing in the contingent universe make it possible to regard these same
final values as arising via functions on alternative lists of arguments, say,
<u, v, u', v'>.
The column for !e!J is computed as J<u, v> = uv. This, along with the
columns for u and v, illustrates the Table's "structure-sharing" scheme,
listing only the initial entries of each constant block.
The column for EJ is computed by means of the following chain of
identities, where the contingent variables u' and v' are defined
as u' = u + du and v' = v + dv.
o--------------------------------------------------------------o
| |
| EJ<u, v, du, dv> = J<u + dv, v + dv> = J<u', v'> |
| |
o--------------------------------------------------------------o
This makes it easy to determine EJ by inspection, computing the
conjunction J<u', v'> = u' v' from the columns headed u' and v'.
Since all of these forms express the same proposition EJ in EU%,
the dependence on du and dv is still present but merely left
implicit in the final variant J<u', v'>.
NB. On occasion, it is tempting to use the further notation J'<u, v> = J<u', v'>,
especially to suggest a transformation that acts on whole propositions, for example,
taking the proposition J into the proposition J' = EJ. The prime ['] then signifies
an action that is mediated by a field of choices, namely, the values that are picked
out for the contingent variables in sweeping through the initial universe. But this
heaps an unwieldy lot of construed intentions on a rather slight character, and puts
too high a premium on the constant correctness of its interpretation. In practice,
therefore, it is best to avoid this usage.
Given the values of !e!J and EJ, the columns for the remaining functions
can be filled in quickly. The difference map is computed according to the
relation DJ = !e!J + EJ. The first order differential dJ is found by looking
in each block of constant <u, v> and choosing the linear function of <du, dv>
that best approximates DJ in that block. Finally, the remainder is computed
as rJ = DJ + dJ, in this case yielding the second order differential d^2.J.
Jon Awbrey
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