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| ==Note 15== | | ==Note 15== |
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− | <br>
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| {| cellpadding="2" cellspacing="2" width="100%" | | {| cellpadding="2" cellspacing="2" width="100%" |
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| | — ''Winter's Tale'', 3.2.43–44 | | | — ''Winter's Tale'', 3.2.43–44 |
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− | <br>
| + | We've talked about differentials long enough that I think it's way past time we met with some. |
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− | We've talked about differentials long enough that I think it's past time we met with some. | |
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| 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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− | <pre>
| + | 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
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− | "localizing" part of the task by subtracting out the forms
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− | 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. | |
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| This raises the question: What is a linear proposition? | | This raises the question: What is a linear proposition? |
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− | 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. | |
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− | 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. | |
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− | 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). | |
− | </pre> | |
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| ==Note 16== | | ==Note 16== |