Changes

==Note 15==

==Note 15==

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| — ''Winter's Tale'', 3.2.43–44

| — ''Winter's Tale'', 3.2.43–44

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We've talked about differentials long enough that I think it's way past time we met with some.

 

We've talked about differentials long enough that I think it's past time we met with some.

When the term is being used with its more exact sense, a ''differential'' is a locally linear approximation to a function, in the context of this logical discussion, then, a locally linear approximation to a proposition.

When the term is being used with its more exact sense, a ''differential'' is a locally linear approximation to a function, in the context of this logical discussion, then, a locally linear approximation to a proposition.

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As a matter of fact, computing the symmetric differences <math>\operatorname{D}f = f + \operatorname{E}f</math> and <math>\operatorname{D}g = g + \operatorname{E}g</math> has already taken care of the ''localizing'' part of the task by subtracting out the forms of <math>f\!</math> and <math>g\!</math> from the forms of <math>\operatorname{E}f</math> and <math>\operatorname{E}g,</math> respectively.  Thus all we have left to do is to decide what linear propositions best approximate the difference maps <math>\operatorname{D}f</math> and <math>\operatorname{D}g,</math> respectively.

As a matter of fact, computing the symmetric differences

Df = f + Ef and Dg = g + Eg has already taken care of the

"localizing" part of the task by subtracting out the forms

of f and g from the forms of Ef and Eg, respectively.  Thus

all we have left to do is to decide what linear propositions

best approximate the difference maps Df and Dg, respectively.

This raises the question:  What is a linear proposition?

This raises the question:  What is a linear proposition?

The answer that makes the most sense in this context is this:

The answer that makes the most sense in this context is this: A proposition is just a boolean-valued function, so a linear proposition is a linear function into the boolean space <math>\mathbb{B}.</math>

A proposition is just a boolean-valued function, so a linear

proposition is a linear function into the boolean space B.

In particular, the linear functions that we want will be

In particular, the linear functions that we want will be linear functions in the differential variables <math>du\!</math> and <math>dv.\!</math>

linear functions in the differential variables du and dv.

As it turns out, there are just four linear propositions

As it turns out, there are just four linear propositions in the associated ''differential universe'' <math>\operatorname{d}U^\circ = [du, dv],</math> and these are the propositions that are commonly denoted: <math>\texttt{0}, \texttt{du}, \texttt{dv}, \texttt{du + dv},</math> in other words, \texttt{()}, \texttt{du}, \texttt{dv}, \texttt{(du,~dv)}.</math>

in the associated "differential universe" dU% = [du, dv],

and these are the propositions that are commonly denoted:

0, du, dv, du + dv, in other words, (), du, dv, (du, dv).

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==Note 16==

==Note 16==