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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
Revision as of 15:10, 14 August 2009
, 15:10, 14 August 2009rename cellular automata q_i to f_i
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<math>\begin{matrix}
<math>\begin{matrix}
f_{207}
\\[4pt]
\\[4pt]
f_{187}
\\[4pt]
\\[4pt]
f_{175}
\\[4pt]
\\[4pt]
f_{139}
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
f_{11001111}
\\[4pt]
\\[4pt]
f_{10111011}
\\[4pt]
\\[4pt]
f_{10101111}
\\[4pt]
\\[4pt]
f_{10001011}
\end{matrix}</math>
\end{matrix}</math>
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<pre>
<pre>
[| q_207 |] = [| p =< q |]
[| f_207 |] = [| p =< q |]
o---------o---------o---------o
o---------o---------o---------o
| p | q | r |
| p | q | r |
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<pre>
<pre>
[| q_187 |] = [| q =< r |]
[| f_187 |] = [| q =< r |]
o---------o---------o---------o
o---------o---------o---------o
| p | q | r |
| p | q | r |
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<pre>
<pre>
[| q_175 |] = [| p =< r |]
[| f_175 |] = [| p =< r |]
o---------o---------o---------o
o---------o---------o---------o
| p | q | r |
| p | q | r |
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<pre>
<pre>
[| q_139 |] = [| p =< q =< r |]
[| f_139 |] = [| p =< q =< r |]
o---------o---------o---------o
o---------o---------o---------o
| p | q | r |
| p | q | r |
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.
For ease of reference during the rest of this discussion, let us refer to the propositional form <math>f : \mathbb{B}^3 \to \mathbb{B}</math> such that <math>f(p, q, r) = q_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> as the ''syllogism mapping'', written as <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to the fiber <math>\operatorname{syll}^{-1}(1) \subseteq \mathbb{B}^3</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math> Table 25-a shows <math>\operatorname{Syll}</math> as a relational dataset.
For ease of reference during the rest of this discussion, let us refer to the propositional form <math>f : \mathbb{B}^3 \to \mathbb{B}</math> such that <math>f(p, q, r) = f_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> as the ''syllogism mapping'', written as <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to the fiber <math>\operatorname{syll}^{-1}(1) \subseteq \mathbb{B}^3</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math> Table 25-a shows <math>\operatorname{Syll}</math> as a relational dataset.
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" cellpadding="10" style="text-align:center; width:90%"
We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference — folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.
We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference — folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.
The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules <math>q_{139}, q_{175}, q_{187}, q_{207}.\!</math>
The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules <math>f_{139}, f_{175}, f_{187}, f_{207}.\!</math>
The function <math>q_{139} : \mathbb{B}^3 \to \mathbb{B}</math> and its fiber <math>[| q_{139} |] \subseteq \mathbb{B}^3</math> appeared to be key to many structures in this setting, and so I singled them out under the new names of <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B}</math> and <math>\operatorname{Syll} \subseteq \mathbb{B}^3,</math> respectively.
The function <math>f_{139} : \mathbb{B}^3 \to \mathbb{B}</math> and its fiber <math>[| f_{139} |] \subseteq \mathbb{B}^3</math> appeared to be key to many structures in this setting, and so I singled them out under the new names of <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B}</math> and <math>\operatorname{Syll} \subseteq \mathbb{B}^3,</math> respectively.
Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations". The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.
Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations". The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.
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We now compute the tacit extensions of the 2-adic projections of <math>\operatorname{Syll},</math> alias <math>q_{139},\!</math> and this makes manifest its relationship to the other functions and fibers, namely, <math>q_{175}, q_{187}, q_{207}.\!</math>
We now compute the tacit extensions of the 2-adic projections of <math>\operatorname{Syll},</math> alias <math>f_{139},\!</math> and this makes manifest its relationship to the other functions and fibers, namely, <math>f_{175}, f_{187}, f_{207}.\!</math>
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" cellpadding="10" style="text-align:center; width:90%"
| [| (p (q)) |] | | [| (p (r)) |] | | [| (q (r)) |] |
| [| (p (q)) |] | | [| (p (r)) |] | | [| (q (r)) |] |
o---------------o o---------------o o---------------o
o---------------o o---------------o o---------------o
| [| q_207 |] | | [| q_175 |] | | [| q_187 |] |
| [| f_207 |] | | [| f_175 |] | | [| f_187 |] |
o---------------o o---------------o o---------------o
o---------------o o---------------o o---------------o
</pre>
</pre>
| (p (q)) | | (q (r)) |
| (p (q)) | | (q (r)) |
o-------------------o o-------------------o
o-------------------o o-------------------o
| q_207 | | q_187 |
| f_207 | | f_187 |
o---------o---------o o---------o---------o
o---------o---------o o---------o---------o
\ /
\ /
| (p (q)) (q (r)) |
| (p (q)) (q (r)) |
o-------------------o
o-------------------o
| q_139 |
| f_139 |
o---------o---------o
o---------o---------o
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|
| (p (r)) |
| (p (r)) |
o-------------------o
o-------------------o
| q_175 |
| f_175 |
o-------------------o
o-------------------o
|TE(Syll_12) c B:B:B| |TE(Syll_23) c B:B:B|
|TE(Syll_12) c B:B:B| |TE(Syll_23) c B:B:B|
o-------------------o o-------------------o
o-------------------o o-------------------o
| [| q_207 |] | | [| q_187 |] |
| [| f_207 |] | | [| f_187 |] |
o----o---------o----o o----o---------o----o
o----o---------o----o o----o---------o----o
^ \ / ^
^ \ / ^
| | Syll c B:B:B | |
| | Syll c B:B:B | |
| o-------------------o |
| o-------------------o |
| | [| q_139 |] | |
| | [| f_139 |] | |
| o---------o---------o |
| o---------o---------o |
| | |
| | |
|TE(Syll_12) c B:B:B| |TE(Syll_23) c B:B:B|
|TE(Syll_12) c B:B:B| |TE(Syll_23) c B:B:B|
o-------------------o o-------------------o
o-------------------o o-------------------o
| [| q_207 |] | | [| q_187 |] |
| [| f_207 |] | | [| f_187 |] |
o----o---------o----o o----o---------o----o
o----o---------o----o o----o---------o----o
^ \ / ^
^ \ / ^
| | Syll c P‡ Q‡ R‡ | |
| | Syll c P‡ Q‡ R‡ | |
| o-------------------o |
| o-------------------o |
| | [| q_139 |] | |
| | [| f_139 |] | |
| o---------o---------o |
| o---------o---------o |
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