Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems - Revision history

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Jon Awbrey: /* Proof as semiosis */ work around for MathJax's inconsistent treatment of "~" in different environments

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Proof as semiosis: work around for MathJax's inconsistent treatment of "~" in different environments

Jon Awbrey: /* Proof as semiosis */ MathJax treats "~" in \mathem as blank space but as printing character in \texttt

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Proof as semiosis: MathJax treats "~" in \mathem as blank space but as printing character in \texttt

@@ Line 2,696: / Line 2,696: @@
 * 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.
-* 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==

@@ Line 2,677: / Line 2,677: @@
 </pre>
 |}
 ==References==
@@ Line 2,701: / Line 2,699: @@
 ==See also==
 ===Related essays and projects===

@@ Line 2,692: / Line 2,692: @@
 * 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), &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 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), &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.
@@ Line 2,698: / Line 2,698: @@
 * 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.
-* 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==

@@ Line 2,680: / Line 2,680: @@
 ==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).
+==References==
-* [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]].
+* 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|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 (Bibliography)|Peirce, C.S., Bibliography]].
-* 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.
+* [[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. (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. (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. (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. (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. (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. (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 4, Mathematical Philosophy'', Mouton, The Hague.
-* 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 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.
-* [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK.
+* 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.
 ==See also==

← Older revision		Revision as of 20:10, 8 November 2016
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	{{DISPLAYTITLE:Propositional Equation Reasoning Systems}}		{{DISPLAYTITLE:Propositional Equation Reasoning Systems}}
−	* '''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems InterSciWiki version].
−
	'''Author: [[User:Jon Awbrey\|Jon Awbrey]]'''		'''Author: [[User:Jon Awbrey\|Jon Awbrey]]'''

@@ Line 1,656: / Line 1,656: @@
+|
 <math>\begin{matrix}
-\texttt{(} p \texttt{~(} q \texttt{))}
+\texttt{(} p \texttt{ (} q \texttt{))}
 \\[4pt]
-\texttt{(} q \texttt{~(} r \texttt{))}
+\texttt{(} q \texttt{ (} r \texttt{))}
 \\[4pt]
-\texttt{(} p \texttt{~(} r \texttt{))}
+\texttt{(} p \texttt{ (} r \texttt{))}
 \\[4pt]
-\texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}
+\texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}
 \end{matrix}</math>
+|
@@ Line 1,683: / Line 1,683: @@
 | [[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;
@@ Line 1,689: / Line 1,689: @@
 | [[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;
@@ Line 1,695: / Line 1,695: @@
 | [[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;
@@ Line 1,701: / Line 1,701: @@
 | [[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>
 |}
@@ Line 1,824: / Line 1,824: @@
 ~ 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
@@ Line 1,853: / Line 1,853: @@
 ~ q \le r
 \\
-=\!=\!=\!=\!=\!=\!=\!=
+\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
 \\
 ~ p \le q \le r
@@ Line 1,921: / Line 1,921: @@
 |- style="height:40px; background:#f0f0ff"
 | <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>
 |}
@@ Line 2,431: / Line 2,431: @@
 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"

@@ Line 1,475: / Line 1,475: @@
 ~ p
 \\
-\overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)}
+\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
 \\
 ~ q
@@ Line 1,493: / Line 1,493: @@
 ~ p
 \\
-=\!=\!=\!=\!=\!=\!=\!=
+\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
 \\
 ~ p ~ q
@@ Line 1,504: / Line 1,504: @@
+|
 <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>
 |}
@@ Line 1,518: / Line 1,518: @@
 {| 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>
 |}
@@ Line 1,525: / Line 1,525: @@
 {| 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>
 |}

@@ Line 1,431: / Line 1,431: @@
 {| 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>
 |}
@@ Line 1,442: / Line 1,442: @@
 '''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.
-: 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>
-: 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==

@@ Line 1,415: / Line 1,415: @@
 {| 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.
 |}