ONT Re: Differential Logic
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DLOG. Note D50
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Tacit Extension of Conjunction (concl.)
The computational scheme that was shown in Table 36 treated J
as a proposition in U% and formed !e!J as a proposition in EU%.
When J is regarded as a mapping J : U% -> X% then !e!J must be
obtained as a mapping !e!J : EU% -> X%. By default, the tacit
extension of the map J : [u, v] -> [x] is naturally taken to be
a particular map, of the following form:
!e!J : [u, v, du, dv] -> [x] c [x, dx]
This is the map that looks like J when painted in the frame of the
extended source universe and that takes the same thematic variable
in the extended target universe as the one that J already employs.
But the choice of a particular thematic variable, for example "x" for ¢(J),
is a shade more arbitrary than the initial choice of variable names {u, v}.
This means that the map I am calling the "trope extension", specifically:
!h!J : [u, v, du, dv] -> [dx] c [x, dx]
since it looks just the same as !e!J in the way that its
fibers paint the source domain, belongs just as fully to
the family of tacit extensions, generically considered.
These considerations have the practical consequence that all of our
computations and illustrations of !e!J perform the double duty of
capturing an image of !h!J as well. In other words, we are saved
the work of carrying out calculations and drawing figures for the
trope extension !h!J, because the exercise would be identical to
the work already done for !e!J. Since the computations given for
!e!J are expressed solely in terms of the variables {u, v, du, dv},
these variables work equally well for finding !h!J. Furthermore,
since each of the above Figures shows only how the level sets of
!e!J partition the extended source universe EU% = [u, v, du, dv],
all of them serve equally well as portraits of !h!J.
Jon Awbrey
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