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Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)
Revision as of 16:08, 13 June 2009
, 16:08, 13 June 2009→Note 6: markup
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
|
|
<math>\begin{array}{cccl}
<math>\begin{array}{lccl}
\text{Let} &
\text{Let} &
X
X
\\[6pt]
\\[6pt]
&
&
& = & X_1 \times \ldots \times X_k ~\times~ \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.
& = & X_1 \times \ldots \times X_k ~\times~ \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k
\end{array}</math>
\end{array}</math>
|}
|}
For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B}</math> that is defined by the following equation:
For a proposition f : X_1 x ... x X_k -> B,
the (first order) "enlargement" of f is the
proposition Ef : EX -> B that is defined by:
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{array}{l}
\operatorname{E}f(x_1, \ldots, x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k)
\\[6pt]
= \quad f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k)
\\[6pt]
= \quad f( \texttt{(} x_1, \operatorname{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k, \operatorname{d}x_k \texttt{)} )
\end{array}</math>
|}
<pre>
It should be noted that the so-called "differential variables" dx_j
It should be noted that the so-called "differential variables" dx_j
are really just the same type of boolean variables as the other x_j.
are really just the same type of boolean variables as the other x_j.