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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
Revision as of 21:04, 13 August 2009
, 21:04, 13 August 2009→Variations on a theme of transitivity: TeX markup
Let us now consider the various rules of inference for transitivity in the light of their performance as information-developing actions.
Let us now consider the various rules of inference for transitivity in the light of their performance as information-developing actions.
{| align="center" cellpadding="4" width="90%"
| <big>•</big>
| colspan="3" | '''Transitive Law :''' Implicational Inference
|-
| width="1%" |
| width="1%" |
| colspan="2" |
<math>\begin{array}{l}
~ p \le q
\\
~ q \le r
\\
\overline{~~~~~~~~~~~~~~~}
\\
~ p \le r
\end{array}</math>
|}
: ''p'' ≤ ''q''
:* By itself, the information ''p'' ≤ ''q'' would reduce our uncertainty from log 8 bits to log 6 bits.
* By itself, the information ''p'' ≤ ''q'' would reduce our uncertainty from log 8 bits to log 6 bits.
:* By itself, the information ''q'' ≤ ''r'' would reduce our uncertainty from log 8 bits to log 6 bits.
:* By itself, the information ''p'' ≤ ''r'' would reduce our uncertainty from log 8 bits to log 6 bits.
* By itself, the information ''p'' ≤ ''r'' would reduce our uncertainty from log 8 bits to log 6 bits.
In this situation, the application of the IROI for transitivity to the information ''p'' ≤ ''q'' and the information ''q'' ≤ ''r'' to get the information ''p'' ≤ ''r'' does not increase the measure of information beyond what any one of the three propositions has independently of the other two. In a sense, then, this IROI operates only to move the information around without changing its measure in the slightest bit.
In this situation, the application of the IROI for transitivity to the information ''p'' ≤ ''q'' and the information ''q'' ≤ ''r'' to get the information ''p'' ≤ ''r'' does not increase the measure of information beyond what any one of the three propositions has independently of the other two. In a sense, then, this IROI operates only to move the information around without changing its measure in the slightest bit.
{| align="center" cellpadding="4" width="90%"
| <big>•</big>
| colspan="3" | '''Transitive Law :''' Equational Inference
|-
| width="1%" |
| width="1%" |
| colspan="2" |
<math>\begin{array}{l}
~ p \le q
\\
~ q \le r
\\
=\!=\!=\!=\!=\!=\!=\!=
\\
~ p \le q \le r
\end{array}</math>
|}
The contents and the measures of information that are associated with the propositions ''p'' ≤ ''q'' and ''q'' ≤ ''r'' are the same as before.
The contents and the measures of information that are associated with the propositions ''p'' ≤ ''q'' and ''q'' ≤ ''r'' are the same as before.