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Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.\!</math>
Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.\!</math>
==Variations on a theme of transitivity==
The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived. But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.
In discussing this Example, it is useful to observe that the implication relation indicated by the propositional form <math>x \Rightarrow y\!</math> is equivalent to an order relation <math>x \le y\!</math> on the boolean values <math>0, 1 \in \mathbb{B},\!</math> where <math>0\!</math> is taken to be less than <math>1.\!</math>
{| align="center" cellpadding="8" width="90%"
| width="1%" | <big>•</big>
| colspan="3" | '''Example 2. Transitivity'''
|-
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| width="1%" |
| colspan="2" | ''Information Reducing Inference''
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| width="1%" |
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<math>\begin{array}{l}
~ p \le q
\\
~ q \le r
\\
\overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)}
\\
~ p \le r
\end{array}</math>
|-
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| colspan="2" | ''Information Preserving Inference''
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<math>\begin{array}{l}
~ p \le q
\\
~ q \le r
\\
=\!=\!=\!=\!=\!=\!=\!=
\\
~ p \le q \le r
\end{array}</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 ~\mathrm{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>