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Directory talk:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

Revision as of 20:42, 30 November 2015

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==Operational Representation==
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If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis.  But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> at once overwhelms its discrete and finite powers to grasp them.  But here, in the fully serene idylls of [[zeroth order logic]], we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.
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So let us do just that.
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I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of ''group theory'', and demonstrate its relevance to differential logic in a strikingly apt and useful way.  The data for that account is contained in Table&nbsp;A3.
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<br>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
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|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}\!</math>
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|- style="background:#f0f0ff"
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| width="10%" | &nbsp;
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| width="18%" | <math>f\!</math>
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| width="18%" |
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<p><math>\mathrm{T}_{11} f\!</math></p>
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<p><math>\mathrm{E}f|_{\mathrm{d}p~\mathrm{d}q}\!</math></p>
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| width="18%" |
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<p><math>\mathrm{T}_{10} f\!</math></p>
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<p><math>\mathrm{E}f|_{\mathrm{d}p(\mathrm{d}q)}\!</math></p>
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| width="18%" |
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<p><math>\mathrm{T}_{01} f\!</math></p>
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<p><math>\mathrm{E}f|_{(\mathrm{d}p)\mathrm{d}q}\!</math></p>
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| width="18%" |
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<p><math>\mathrm{T}_{00} f\!</math></p>
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<p><math>\mathrm{E}f|_{(\mathrm{d}p)(\mathrm{d}q)}\!</math></p>
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|-
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| <math>f_0\!</math>
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| <math>(~)\!</math>
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| <math>(~)\!</math>
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| <math>(~)\!</math>
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| <math>(~)\!</math>
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| <math>(~)\!</math>
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|-
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|
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<math>\begin{matrix}
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f_1
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\\[4pt]
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f_2
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\\[4pt]
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f_4
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\\[4pt]
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f_8
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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(p)(q)
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\\[4pt]
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(p)~q~
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\\[4pt]
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~p~(q)
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\\[4pt]
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~p~~q~
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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~p~~q~
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\\[4pt]
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~p~(q)
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\\[4pt]
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(p)~q~
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\\[4pt]
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(p)(q)
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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~p~(q)
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\\[4pt]
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~p~~q~
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\\[4pt]
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(p)(q)
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\\[4pt]
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(p)~q~
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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(p)~q~
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\\[4pt]
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(p)(q)
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\\[4pt]
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~p~~q~
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\\[4pt]
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~p~(q)
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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(p)(q)
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\\[4pt]
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(p)~q~
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\\[4pt]
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~p~(q)
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\\[4pt]
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~p~~q~
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\end{matrix}\!</math>
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|-
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|
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<math>\begin{matrix}
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f_3
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\\[4pt]
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f_{12}
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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(p)
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\\[4pt]
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~p~
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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~p~
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\\[4pt]
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(p)
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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~p~
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\\[4pt]
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(p)
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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(p)
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\\[4pt]
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~p~
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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(p)
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\\[4pt]
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~p~
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\end{matrix}\!</math>
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|-
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|
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<math>\begin{matrix}
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f_6
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\\[4pt]
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f_9
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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~(p,~q)~
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\\[4pt]
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((p,~q))
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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~(p,~q)~
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\\[4pt]
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((p,~q))
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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((p,~q))
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\\[4pt]
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~(p,~q)~
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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((p,~q))
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\\[4pt]
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~(p,~q)~
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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~(p,~q)~
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\\[4pt]
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((p,~q))
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\end{matrix}\!</math>
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|-
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|
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<math>\begin{matrix}
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f_5
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\\[4pt]
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f_{10}
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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(q)
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\\[4pt]
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~q~
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\end{matrix}\!</math>
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<math>\begin{matrix}
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~q~
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\\[4pt]
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(q)
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\end{matrix}\!</math>
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<math>\begin{matrix}
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(q)
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\\[4pt]
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~q~
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\end{matrix}\!</math>
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<math>\begin{matrix}
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~q~
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\\[4pt]
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(q)
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\end{matrix}\!</math>
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<math>\begin{matrix}
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(q)
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\\[4pt]
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~q~
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\end{matrix}\!</math>
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|-
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|
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<math>\begin{matrix}
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f_7
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\\[4pt]
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f_{11}
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\\[4pt]
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f_{13}
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\\[4pt]
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f_{14}
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\end{matrix}\!</math>
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<math>\begin{matrix}
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(~p~~q~)
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\\[4pt]
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(~p~(q))
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\\[4pt]
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((p)~q~)
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\\[4pt]
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((p)(q))
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\end{matrix}\!</math>
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<math>\begin{matrix}
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((p)(q))
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\\[4pt]
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((p)~q~)
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\\[4pt]
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(~p~(q))
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\\[4pt]
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(~p~~q~)
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\end{matrix}\!</math>
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<math>\begin{matrix}
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((p)~q~)
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\\[4pt]
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((p)(q))
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\\[4pt]
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(~p~~q~)
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\\[4pt]
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(~p~(q))
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\end{matrix}\!</math>
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<math>\begin{matrix}
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(~p~(q))
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\\[4pt]
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(~p~~q~)
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\\[4pt]
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((p)(q))
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\\[4pt]
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((p)~q~)
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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(~p~~q~)
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\\[4pt]
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(~p~(q))
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\\[4pt]
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((p)~q~)
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\\[4pt]
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((p)(q))
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\end{matrix}\!</math>
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|-
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| <math>f_{15}\!</math>
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| <math>((~))\!</math>
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| <math>((~))\!</math>
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| <math>((~))\!</math>
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| <math>((~))\!</math>
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| <math>((~))\!</math>
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|- style="background:#f0f0ff"
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| colspan="2" | <math>\text{Fixed Point Total}\!</math>
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| <math>4\!</math>
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| <math>4\!</math>
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| <math>4\!</math>
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| <math>16\!</math>
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|}
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<br>
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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:
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{| align="center" cellpadding="6" width="90%"
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|
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<math>\begin{array}{lcl}
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\mathrm{E} ~:~ (X \to \mathbb{B})
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& \to &
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(\mathrm{E}X \to \mathbb{B})
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\\[6pt]
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\mathrm{E} ~:~ f(p, q)
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& \mapsto &
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\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
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\\[6pt]
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\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
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& = &
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f(p + \mathrm{d}p, q + \mathrm{d}q)
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\\[6pt]
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& = &
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f( \texttt{(} p, \mathrm{d}p \texttt{)}, \texttt{(} q, \mathrm{d}q \texttt{)} )
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\end{array}\!</math>
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|}
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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:
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{| align="center" cellpadding="6" width="90%"
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|
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<math>\begin{array}{lclcl}
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\mathrm{E}_{ij}
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& : &
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(X \to \mathbb{B})
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& \to &
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(X \to \mathbb{B})
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\\[6pt]
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\mathrm{E}_{ij}
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& : &
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f
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& \mapsto &
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\mathrm{E}_{ij}f
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\\[6pt]
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\mathrm{E}_{ij}f
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& = &
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\mathrm{E}f|_{\mathrm{d}p = i, \mathrm{d}q = j}
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& = &
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f(p + i, q + j)
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\\[6pt]
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&  &
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& = &
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f( \texttt{(} p, i \texttt{)}, \texttt{(} q, j \texttt{)} )
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\end{array}\!</math>
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|}
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The notation is a little awkward, but the data of Table&nbsp;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:
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<br>
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{| 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%"
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|- style="height:50px"
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| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" |
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<math>\cdot\!</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\mathrm{T}_{00}\!</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\mathrm{T}_{01}\!</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\mathrm{T}_{10}\!</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\mathrm{T}_{11}\!</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\mathrm{T}_{00}\!</math>
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| <math>\mathrm{T}_{00}\!</math>
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| <math>\mathrm{T}_{01}\!</math>
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| <math>\mathrm{T}_{10}\!</math>
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| <math>\mathrm{T}_{11}\!</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\mathrm{T}_{01}\!</math>
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| <math>\mathrm{T}_{01}\!</math>
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| <math>\mathrm{T}_{00}\!</math>
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| <math>\mathrm{T}_{11}\!</math>
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| <math>\mathrm{T}_{10}\!</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\mathrm{T}_{10}\!</math>
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| <math>\mathrm{T}_{10}\!</math>
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| <math>\mathrm{T}_{11}\!</math>
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| <math>\mathrm{T}_{00}\!</math>
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| <math>\mathrm{T}_{01}\!</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\mathrm{T}_{11}\!</math>
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| <math>\mathrm{T}_{11}\!</math>
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| <math>\mathrm{T}_{10}\!</math>
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| <math>\mathrm{T}_{01}\!</math>
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| <math>\mathrm{T}_{00}\!</math>
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|}
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<br>
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It happens that there are just two possible groups of 4 elements.  One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not.  The other is the Klein four-group <math>V_4\!</math> (from German ''Vier''), which this is.
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More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''.  One says that the orbits are preserved by the action of the group.  There is an ''Orbit Lemma'' of immense utility to &ldquo;those who count&rdquo; which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group.  In this instance, <math>\mathrm{T}_{00}\!</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get: &nbsp; Number of Orbits &nbsp;=&nbsp; (4 + 4 + 4 + 16) &divide; 4 &nbsp;=&nbsp; 7. &nbsp; Amazing!
    
==Logical Cacti==
 
==Logical Cacti==
Jon Awbrey
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