ONT Re: Differential Logic
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DLOG. Note D40
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The Secant Operator: $E$
| Mr. Peirce, after pointing out that our beliefs are really
| rules for action, said that, to develop a thought's meaning,
| we need only determine what conduct it is fitted to produce:
| that conduct is for us its sole significance.
|
| William James, 'Pragmatism', [Jam, 46]
Figures 33-i and 33-ii depict two stages in the form of analysis that will be
applied to transformations throughout the remainder of this study. From now
on our interest is staked on an operator denoted "$E$", which receives the
principal investment of analytic attention, and on the constituent parts
of $E$, which derive their shares of significance as developed by the
analysis. In the sequel, I refer to $E$ as the "secant operator",
taking it for granted that a context has been chosen that defines
its type. The secant operator has the component description
$E$ = <!e!, E>, and its active ingredient E is known as the
"enlargement operator". (Here, I have named E after the
literal ancestor of the shift operator in the calculus
of finite differences, defined so that Ef(x) = f(x+1)
for any suitable function f, though, of course the
logical analogue that we take up here must have
a rather different definition.)
U% $E$ $E$U% $E$U% $E$U%
o------------------>o============o============o
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F | | $E$F = | $d$^0.F + | $r$^0.F
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v v v v
o------------------>o============o============o
X% $E$ $E$X% $E$X% $E$X%
Figure 33-i. Analytic Diagram (1)
U% $E$ $E$U% $E$U% $E$U% $E$U%
o------------------>o============o============o============o
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F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F
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v v v v v
o------------------>o============o============o============o
X% $E$ $E$X% $E$X% $E$X% $E$X%
Figure 33-ii. Analytic Diagram (2)
In its action on universes $E$ yields the same result as E, a fact that can
be expressed in equational form by writing $E$U% = EU% for any universe U%.
Notice that the extended universes across the top and bottom of the diagram
are indicated to be strictly identical, rather than requiring a corresponding
decomposition for them. In a certain sense, the functional parts of $E$F are
partitioned into separate contexts that have to be re-integrated again, but the
best image to use is that of making transparent copies of each universe and then
overlapping their functional contents once more at the conclusion of the analysis,
as suggested by the graphic conventions that are used at the top of Figure 30.
Acting on a transformation F from universe U% to universe X%, the operator $E$
determines a transformation $E$F from $E$U% to $E$X%. The map $E$F forms the
main body of evidence to be investigated in performing a differential analysis
of F. Because we shall frequently be focusing on small pieces of this map for
considerable lengths of time, and consequently lose sight of the "big picture",
it is critically important to emphasize that the map $E$F is a transformation
that determines a relation from one extended universe into another. This means
that we should not be satisfied with our understanding of a transformation F
until we can lay out the full "parts diagram" of $E$F along the lines of the
generic frame in Figure 30.
Jon Awbrey
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