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Revision as of 17:34, 13 August 2009
, 17:34, 13 August 2009→Variations on a theme of transitivity: spacing
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<math>\begin{array}{l}
<math>\begin{array}{l}
p \le q
~ p \le q
\\
\\
q \le r
~ q \le r
\\
\\
\overline{~~~~~~~~~~~~~~~~}
\overline{~~~~~~~~~~~~~~~}
\\
\\
p \le r
~ p \le r
\end{array}</math>
\end{array}</math>
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<math>\begin{array}{l}
<math>\begin{array}{l}
p \le q
~ p \le q
\\
\\
q \le r
~ q \le r
\\
\\
=\!=\!=\!=\!=\!=\!=\!=
=\!=\!=\!=\!=\!=\!=\!=
\\
\\
p \le q \le r
~ p \le q \le r
\end{array}</math>
\end{array}</math>
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In stating the information-preserving analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a concatenated syntax. Thus, <math>p \le q \le r</math> means <math>p \le q ~\operatorname{and}~ q \le r.</math> The claim that this 3-adic relation holds among the 3 propositions <math>p, q, r\!</math> is a stronger claim — holds more information — than the claim that the 2-adic relation <math>p \le r</math> holds between the 2 propositions <math>p\!</math> and <math>r.\!</math>
In stating the information-preserving analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a concatenated syntax. Thus, <math>p \le q \le r</math> means <math>p \le q ~\operatorname{and}~ q \le r.</math> The claim that this 3-adic order relation holds among the three propositions <math>p, q, r\!</math> is a stronger claim — conveys more information — than the claim that the 2-adic relation <math>p \le r</math> holds between the two propositions <math>p\!</math> and <math>r.\!</math>
To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.
To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.
