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{{DISPLAYTITLE:Propositional Equation Reasoning Systems}}

{{DISPLAYTITLE:Propositional Equation Reasoning Systems}}

'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''

'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''

{| align="center" cellpadding="8" width="90%"

{| align="center" cellpadding="8" width="90%"

| <math>e_0 = {}^{\backprime\backprime} \texttt{(~)} {}^{\prime\prime}\!</math> expresses the logical constant <math>\mathrm{false}.\!</math>

| <math>e_0 = {}^{\backprime\backprime} \texttt{( )} {}^{\prime\prime}\!</math> expresses the logical constant <math>\mathrm{false}.\!</math>

|-

|-

| <math>e_1 = {}^{\backprime\backprime} \texttt{~} {}^{\prime\prime}\!</math> expresses the logical constant <math>\mathrm{true}.\!</math>

| <math>e_1 = {}^{\backprime\backprime} \texttt{ } {}^{\prime\prime}\!</math> expresses the logical constant <math>\mathrm{true}.\!</math>

|-

|-

| <math>e_2 = {}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q,\!</math> <math>\mathrm{and~not}~ p ~\mathrm{without}~ r {}^{\prime\prime}.\!</math>

| <math>e_2 = {}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q,\!</math> <math>\mathrm{and~not}~ p ~\mathrm{without}~ r {}^{\prime\prime}.\!</math>

|-

|-

| <math>e_3 = {}^{\backprime\backprime} \texttt{(} p \texttt{~(} q~r \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q ~\mathrm{and}~ r {}^{\prime\prime}.\!</math>

| <math>e_3 = {}^{\backprime\backprime} \texttt{(} p \texttt{ (} q~r \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q ~\mathrm{and}~ r {}^{\prime\prime}.\!</math>

|-

|-

| <math>e_4 = {}^{\backprime\backprime} \texttt{(} p~q~r \texttt{~,~(} p \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} p ~\mathrm{and}~ q ~\mathrm{and}~ r,\!</math> <math>~\mathrm{or~else~not}~ p{}^{\prime\prime}.\!</math>

| <math>e_4 = {}^{\backprime\backprime} \texttt{(} p~q~r \texttt{ , (} p \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} p ~\mathrm{and}~ q ~\mathrm{and}~ r,\!</math> <math>~\mathrm{or~else~not}~ p{}^{\prime\prime}.\!</math>

|-

|-

| <math>e_5 = {}^{\backprime\backprime} \texttt{((~(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))~,~(} p \texttt{~(} q~r \texttt{))~))} {}^{\prime\prime}\!</math> says that <math>e_2\!</math> and <math>e_3\!</math> say the same thing.

| <math>e_5 = {}^{\backprime\backprime} \texttt{(( (} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{)) , (} p \texttt{ (} q~r \texttt{)) ))} {}^{\prime\prime}\!</math> says that <math>e_2\!</math> and <math>e_3\!</math> say the same thing.

|}

|}

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{| align="center" cellpadding="8" width="90%"

| <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&nbsp;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&nbsp;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&nbsp;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&nbsp;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&nbsp;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&nbsp;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&nbsp;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&nbsp;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==

~ p

~ p

\\

\\

\overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)}

\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}

\\

\\

~ q

~ q

~ p

~ p

\\

\\

=\!=\!=\!=\!=\!=\!=\!=

\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

\\

\\

~ p ~ q

~ p ~ q

|

|

<math>\begin{array}{l}

<math>\begin{array}{l}

~ \textit{Expression~1}

~ \textit{Expression 1}

\\

\\

~ \textit{Expression~2}

~ \textit{Expression 2}

\\

\\

\overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)}

\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}

\\

\\

~ \textit{Expression~3}

~ \textit{Expression 3}

\end{array}</math>

\end{array}</math>

|}

|}

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{| align="center" cellpadding="8" width="90%"

|

|

<math>\textit{Premiss~1}, \textit{Premiss~2} ~\vdash~ \textit{Conclusion}.\!</math>

<math>\textit{Premiss 1}, \textit{Premiss 2} ~\vdash~ \textit{Conclusion}.\!</math>

|}

|}

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{| align="center" cellpadding="8" width="90%"

|

|

From &nbsp; <math>{\textit{Expression~1}}\!</math> &nbsp; and &nbsp; <math>{\textit{Expression~2}}\!</math> &nbsp; infer &nbsp; <math>{\textit{Expression~3}}.\!</math>

From &nbsp; <math>{\textit{Expression 1}}\!</math> &nbsp; and &nbsp; <math>{\textit{Expression 2}}\!</math> &nbsp; infer &nbsp; <math>{\textit{Expression 3}}.\!</math>

|}

|}

~ q \le r

~ q \le r

\\

\\

\overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)}

\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}

\\

\\

~ p \le r

~ p \le r

~ q \le r

~ q \le r

\\

\\

=\!=\!=\!=\!=\!=\!=\!=

\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

\\

\\

~ p \le q \le r

~ p \le q \le r

|

|

<math>\begin{matrix}

<math>\begin{matrix}

\texttt{(} p \texttt{~(} q \texttt{))}

\texttt{(} p \texttt{ (} q \texttt{))}

\\[4pt]

\\[4pt]

\texttt{(} q \texttt{~(} r \texttt{))}

\texttt{(} q \texttt{ (} r \texttt{))}

\\[4pt]

\\[4pt]

\texttt{(} p \texttt{~(} r \texttt{))}

\texttt{(} p \texttt{ (} r \texttt{))}

\\[4pt]

\\[4pt]

\texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}

\texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}

\end{matrix}</math>

\end{matrix}</math>

|

|

| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52)

| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52)

|-

|-

| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}\!</math>

| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{))}\!</math>

|-

|-

| &nbsp;

| &nbsp;

| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)

| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)

|-

|-

| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}\!</math>

| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{ (} r \texttt{))}\!</math>

|-

|-

| &nbsp;

| &nbsp;

| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)

| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)

|-

|-

| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}\!</math>

| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{ (} r \texttt{))}\!</math>

|-

|-

| &nbsp;

| &nbsp;

| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55)

| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55)

|-

|-

| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}\!</math>

| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}\!</math>

|}

|}

~ q \le r

~ q \le r

\\

\\

\overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)}

\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}

\\

\\

~ p \le r

~ p \le r

~ q \le r

~ q \le r

\\

\\

=\!=\!=\!=\!=\!=\!=\!=

\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

\\

\\

~ p \le q \le r

~ p \le q \le r

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| <math>p \le q \le r\!</math>

| <math>p \le q \le r\!</math>

| <math>\texttt{(} p \texttt{~(} q \texttt{))}\!</math>

| <math>\texttt{(} p \texttt{ (} q \texttt{))}\!</math>

| <math>\texttt{(} p \texttt{~(} r \texttt{))}\!</math>

| <math>\texttt{(} p \texttt{ (} r \texttt{))}\!</math>

| <math>\texttt{(} q \texttt{~(} r \texttt{))}\!</math>

| <math>\texttt{(} q \texttt{ (} r \texttt{))}\!</math>

|}

|}

In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.

In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.

At first sight, the relationships seem easy enough to write out.  Figure&nbsp;75 shows how the various logical expressions are related to each other:  The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> are conjoined in a purely syntactic fashion &mdash; much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about &mdash; and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}{}^{\prime\prime}\!</math> that would the canonical result of an equational or reversible rule of inference.  From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> by the most conventional implication.

At first sight, the relationships seem easy enough to write out.  Figure&nbsp;75 shows how the various logical expressions are related to each other:  The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{))} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> are conjoined in a purely syntactic fashion &mdash; much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about &mdash; and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}{}^{\prime\prime}\!</math> that would the canonical result of an equational or reversible rule of inference.  From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> by the most conventional implication.

{| align="center" border="0" cellpadding="10"

{| align="center" border="0" cellpadding="10"

==References==

==References==

* [[Gottfried Leibniz|Leibniz, G.W.]] (1679–1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40–46 in Parkinson, G.H.R. (ed.), ''Leibniz : Logical Papers'', Oxford University Press, London, UK, 1966.  (Cf. Gerhardt, 7, p. 223).

* Leibniz, G.W. (1679&ndash;1686 ?), &ldquo;Addenda to the Specimen of the Universal Calculus&rdquo;, pp. 40&ndash;46 in Parkinson, G.H.R. (ed.), ''Leibniz : Logical Papers'', Oxford University Press, London, UK, 1966.  (Cf. Gerhardt, 7, p. 223).

* [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]].

* [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]].

* [[Charles Peirce|Peirce, C.S.]] (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]] (eds.), vols. 7–8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA.  Cited as CP volume.paragraph.

* [[Charles Sanders Peirce|Peirce, C.S.]] (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA.  Cited as CP&nbsp;volume.paragraph.

* Peirce, C.S. (1981–), ''Writings of Charles S. Peirce: A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianoplis, IN.  Cited as CE volume, page.

* Peirce, C.S. (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN.  Cited as CE&nbsp;volume, page.

* Peirce, C.S. (1885), "On the Algebra of Logic: A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180–202.  Reprinted as CP 3.359–403 and CE 5, 162–190.

* Peirce, C.S. (1885), &ldquo;On the Algebra of Logic : A Contribution to the Philosophy of Notation&rdquo;, ''American Journal of Mathematics'' 7 (1885), 180&ndash;202.  Reprinted as CP&nbsp;3.359&ndash;403 and CE&nbsp;5, 162&ndash;190.

* Peirce, C.S. (c. 1886), "Qualitative Logic", MS 736.  Published as pp. 101–115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague.

* Peirce, C.S. (c. 1886), &ldquo;Qualitative Logic&rdquo;, MS&nbsp;736.  Published as pp. 101&ndash;115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume&nbsp;4, Mathematical Philosophy'', Mouton, The Hague.

* Peirce, C.S. (1886 a), "Qualitative Logic", MS 582.  Published as pp. 323–371 in ''Writings of Charles S. Peirce: A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.

* Peirce, C.S. (1886 a), &ldquo;Qualitative Logic&rdquo;, MS&nbsp;582.  Published as pp. 323&ndash;371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume&nbsp;5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.

* Peirce, C.S. (1886 b), "The Logic of Relatives: Qualitative and Quantitative", MS 584.  Published as pp. 372–378 in ''Writings of Charles S. Peirce: A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.

* Peirce, C.S. (1886 b), &ldquo;The Logic of Relatives : Qualitative and Quantitative&rdquo;, MS&nbsp;584.  Published as pp. 372&ndash;378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume&nbsp;5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.

* [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK.

* Spencer Brown, George (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK.

==See also==

==See also==

===Related essays and projects===

===Related essays and projects===