MyWikiBiz, Author Your Legacy — Friday November 01, 2024
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25 bytes added
, 17:34, 13 August 2009
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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> |
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| 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. |