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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
Revision as of 16:25, 8 November 2016
, 16:25, 8 November 2016→Proof as semiosis: work around for MathJax's inconsistent treatment of "~" in different environments
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| <math>\texttt{(} p \texttt{~(} q \texttt{))(} p \texttt{~(} r \texttt{))} = \texttt{(} p \texttt{~(} q~r \texttt{))}.\!</math>
| <math>\texttt{(} p \texttt{ (} q \texttt{))(} p \texttt{ (} r \texttt{))} = \texttt{(} p \texttt{ (} q~r \texttt{))}.\!</math>
|}
|}
'''Proof 1''' proceeded by the ''straightforward approach'', starting with <math>e_2\!</math> as <math>s_1\!</math> and ending with <math>e_3\!</math> as <math>s_n\!.</math>
'''Proof 1''' proceeded by the ''straightforward approach'', starting with <math>e_2\!</math> as <math>s_1\!</math> and ending with <math>e_3\!</math> as <math>s_n\!.</math>
: That is, Proof 1 commenced from the sign <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> and ended up at the sign <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q~r \texttt{))} {}^{\prime\prime}\!</math> by legal moves.
: That is, Proof 1 commenced from the sign <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> and ended up at the sign <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q~r \texttt{))} {}^{\prime\prime}\!</math> by legal moves.
'''Proof 2''' lit on by ''burning the candle at both ends'', changing <math>e_2\!</math> into a normal form that reduced to <math>e_4,\!</math> and changing <math>e_3\!</math> into a normal form that also reduced to <math>e_4,\!</math> in this way tethering <math>e_2\!</math> and <math>e_3\!</math> to a common stake.
'''Proof 2''' lit on by ''burning the candle at both ends'', changing <math>e_2\!</math> into a normal form that reduced to <math>e_4,\!</math> and changing <math>e_3\!</math> into a normal form that also reduced to <math>e_4,\!</math> in this way tethering <math>e_2\!</math> and <math>e_3\!</math> to a common stake.
: Filling in the details, one route went from <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> to <math>{}^{\backprime\backprime} \texttt{(} p~q~r \texttt{~,~(} p \texttt{))} {}^{\prime\prime},\!</math> and another went from <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q~r \texttt{))} {}^{\prime\prime}\!</math> to <math>{}^{\backprime\backprime} \texttt{(} p~q~r \texttt{~,~(} p \texttt{))} {}^{\prime\prime},\!</math> thus equating the two points of departure.
: Filling in the details, one route went from <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> to <math>{}^{\backprime\backprime} \texttt{(} p~q~r \texttt{ , (} p \texttt{))} {}^{\prime\prime},\!</math> and another went from <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q~r \texttt{))} {}^{\prime\prime}\!</math> to <math>{}^{\backprime\backprime} \texttt{(} p~q~r \texttt{ , (} p \texttt{))} {}^{\prime\prime},\!</math> thus equating the two points of departure.
'''Proof 3''' took the path of reflection, expressing the meta-equation between <math>e_2\!</math> and <math>e_3\!</math> in the form of the naturalized equation <math>e_5,\!</math> then taking <math>e_5\!</math> as <math>s_1\!</math> and exchanging it by dint of value preserving steps for <math>e_1\!</math> as <math>s_n.\!</math>
'''Proof 3''' took the path of reflection, expressing the meta-equation between <math>e_2\!</math> and <math>e_3\!</math> in the form of the naturalized equation <math>e_5,\!</math> then taking <math>e_5\!</math> as <math>s_1\!</math> and exchanging it by dint of value preserving steps for <math>e_1\!</math> as <math>s_n.\!</math>
: This way of proceeding went from <math>e_5 = {}^{\backprime\backprime} \texttt{((~(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))~,~(} p \texttt{~(} q~r \texttt{))~))} {}^{\prime\prime}\!</math> to the blank expression that <math>\mathrm{Ex}\!</math> recognizes as the value <math>{\mathrm{true}}.\!</math>
: This way of proceeding went from <math>e_5 = {}^{\backprime\backprime} \texttt{(( (} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{)) , (} p \texttt{ (} q~r \texttt{)) ))} {}^{\prime\prime}\!</math> to the blank expression that <math>\mathrm{Ex}\!</math> recognizes as the value <math>{\mathrm{true}}.\!</math>
==Computation and inference as semiosis==
==Computation and inference as semiosis==