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Revision as of 18:25, 30 November 2015
, 18:25, 30 November 2015→Operational Representation: try next piece of section
| <math>4\!</math>
| <math>4\!</math>
| <math>16\!</math>
| <math>16\!</math>
|}
<br>
The shift operator <math>\mathrm{E}\!</math> can be understood as enacting a substitution operation on the propositional form <math>f(p, q)\!</math> that is given as its argument. In our present focus on propositional forms that involve two variables, we have the following type specifications and definitions:
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{array}{lcl}
\mathrm{E} ~:~ (X \to \mathbb{B})
& \to &
(\mathrm{E}X \to \mathbb{B})
\\[6pt]
\mathrm{E} ~:~ f(p, q)
& \mapsto &
\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
\\[6pt]
\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
& = &
f(p + \mathrm{d}p, q + \mathrm{d}q)
\\[6pt]
& = &
f( \texttt{(} p, \mathrm{d}p \texttt{)}, \texttt{(} q, \mathrm{d}q \texttt{)} )
\end{array}\!</math>
|}
Evaluating <math>\mathrm{E}f\!</math> at particular values of <math>\mathrm{d}p\!</math> and <math>\mathrm{d}q,\!</math> for example, <math>\mathrm{d}p = i\!</math> and <math>\mathrm{d}q = j,\!</math> where <math>i\!</math> and <math>j\!</math> are values in <math>\mathbb{B},\!</math> produces the following result:
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{array}{lclcl}
\mathrm{E}_{ij}
& : &
(X \to \mathbb{B})
& \to &
(X \to \mathbb{B})
\\[6pt]
\mathrm{E}_{ij}
& : &
f
& \mapsto &
\mathrm{E}_{ij}f
\\[6pt]
\mathrm{E}_{ij}f
& = &
\mathrm{E}f|_{\mathrm{d}p = i, \mathrm{d}q = j}
& = &
f(p + i, q + j)
\\[6pt]
& &
& = &
f( \texttt{(} p, i \texttt{)}, \texttt{(} q, j \texttt{)} )
\end{array}\!</math>
|}
The notation is a little awkward, but the data of Table A3 should make the sense clear. The important thing to observe is that <math>\mathrm{E}_{ij}\!</math> has the effect of transforming each proposition <math>f : X \to \mathbb{B}\!</math> into a proposition <math>f^\prime : X \to \mathbb{B}.\!</math> As it happens, the action of each <math>\mathrm{E}_{ij}\!</math> is one-to-one and onto, so the gang of four operators <math>\{ \mathrm{E}_{ij} : i, j \in \mathbb{B} \}\!</math> is an example of what is called a ''transformation group'' on the set of sixteen propositions. Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as <math>\mathrm{T}_{00}, \mathrm{T}_{01}, \mathrm{T}_{10}, \mathrm{T}_{11},\!</math> to bear in mind their transformative character, or nature, as the case may be. Abstractly viewed, this group of order four has the following operation table:
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" |
<math>\cdot\!</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{T}_{00}\!</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{T}_{01}\!</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{T}_{10}\!</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{T}_{11}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{T}_{00}\!</math>
| <math>\mathrm{T}_{00}\!</math>
| <math>\mathrm{T}_{01}\!</math>
| <math>\mathrm{T}_{10}\!</math>
| <math>\mathrm{T}_{11}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{T}_{01}\!</math>
| <math>\mathrm{T}_{01}\!</math>
| <math>\mathrm{T}_{00}\!</math>
| <math>\mathrm{T}_{11}\!</math>
| <math>\mathrm{T}_{10}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{T}_{10}\!</math>
| <math>\mathrm{T}_{10}\!</math>
| <math>\mathrm{T}_{11}\!</math>
| <math>\mathrm{T}_{00}\!</math>
| <math>\mathrm{T}_{01}\!</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{T}_{11}\!</math>
| <math>\mathrm{T}_{11}\!</math>
| <math>\mathrm{T}_{10}\!</math>
| <math>\mathrm{T}_{01}\!</math>
| <math>\mathrm{T}_{00}\!</math>
|}
|}
