12,080
edits
Changes
Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
Revision as of 21:30, 13 August 2009
, 21:30, 13 August 2009→Variations on a theme of transitivity: TeX markup
{| align="center" cellpadding="4" width="90%"
{| align="center" cellpadding="4" width="90%"
| <big>•</big>
| <big>•</big>
| colspan="3" | '''Transitive Law :''' Implicational Inference
| colspan="3" | '''Transitive Law''' (Implicational Inference)
|-
|-
| width="1%" |
| width="1%" |
~ p \le r
~ p \le r
\end{array}</math>
\end{array}</math>
|-
| <big>•</big>
| colspan="3" | By itself, the information <math>p \le q</math> would reduce our uncertainty from <math>\log 8\!</math> bits to <math>\log 6\!</math> bits.
|-
| <big>•</big>
| colspan="3" | By itself, the information <math>q \le r</math> would reduce our uncertainty from <math>\log 8\!</math> bits to <math>\log 6\!</math> bits.
|-
| <big>•</big>
| colspan="3" | By itself, the information <math>p \le r</math> would reduce our uncertainty from <math>\log 8\!</math> bits to <math>\log 6\!</math> bits.
|}
|}
In this situation the application of the implicational rule of inference for transitivity to the information <math>p \le q</math> and the information <math>q \le r</math> to get the information <math>p \le r</math> 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, the implicational rule 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%"
{| align="center" cellpadding="4" width="90%"
| <big>•</big>
| <big>•</big>
| colspan="3" | '''Transitive Law :''' Equational Inference
| colspan="3" | '''Transitive Law''' (Equational Inference)
|-
|-
| width="1%" |
| width="1%" |
|}
|}
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 <math>p \le q</math> and <math>q \le r</math> are the same as before.
On its own, the information ''p'' ≤ ''q'' ≤ ''r'' would reduce our uncertainty from log(8) = 3 bits to log(4) = 2 bits, a reduction of 1 bit.
On its own, the information <math>p \le q \le r</math> would reduce our uncertainty from log(8) = 3 bits to log(4) = 2 bits, a reduction of 1 bit.
These are just some of the initial observations that can be made about the dimensions of information and uncertainty in the conduct of logical inference, and there are many issues to be taken up as we get to the thick of it. In particular, we are taking propositions far too literally at the outset, reading their spots at face value, as it were, without yet considering their species character as fallible signs.
These are just some of the initial observations that can be made about the dimensions of information and uncertainty in the conduct of logical inference, and there are many issues to be taken up as we get to the thick of it. In particular, we are taking propositions far too literally at the outset, reading their spots at face value, as it were, without yet considering their species character as fallible signs.