ONT Re: Differential Logic
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note D48
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Tacit Extension of Conjunction
| I teach straying from me, yet who can stray from me?
| I follow you whoever you are from the present hour;
| My words itch at your ears till you understand them.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 83]
Earlier I defined the tacit extension operators !e! : X% -> Y%
as maps embedding each proposition of a given universe X% in a
more generously given universe Y% containing X%. Of immediate
interest are the tacit extensions !e! : U% -> EU%, that locate
each proposition of U% in the enlarged context of EU%. In its
application to the propositional conjunction J = u v in [u, v],
the tacit extension operator !e! produces the proposition !e!J
in EU% = [u, v, du, dv]. The extended proposition !e!J may be
computed according to the scheme in Table 36, in effect, doing
nothing more than conjoining a tautology of [du, dv] to J in U%.
Table 36. Computation of !e!J
o---------------------------------------------------------------------o
| |
| !e!J = J<u, v> |
| |
| = u v |
| |
| = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv |
| |
o---------------------------------------------------------------------o
| |
| !e!J = u v (du)(dv) + |
| u v (du) dv + |
| u v du (dv) + |
| u v du dv |
| |
o---------------------------------------------------------------------o
The lower portion of the Table contains the dispositional features of !e!J
arranged in such a way that the variety of ordinary features spreads across
the rows and the variety of differential features runs through the columns.
This organization serves to facilitate pattern matching in the remainder
of our computations. Again, the tacit extension is usually so trivial
a concern that we do not always bother to make an explicit note of it,
taking it for granted that any function F that is being employed in
a differential context is equivalent to !e!F, for a suitable !e!.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o