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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]]'''
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This is not only a logically equivalent DNF but exactly the same DNF expression that we obtained before, so we have established the given equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.\!</math> Incidentally, one may wish to note that this DNF expression quickly folds into the following form:
This is not only a logically equivalent DNF but exactly the same DNF expression that we obtained before, so we have established the given equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.\!</math> Incidentally, one may wish to note that this DNF expression quickly folds into the following form:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
In other words, <math>{}^{\backprime\backprime} p ~\mathrm{is~equivalent~to}~ p ~\mathrm{and}~ q ~\mathrm{and}~ r {}^{\prime\prime}.\!</math>
In other words, <math>{}^{\backprime\backprime} p ~\mathrm{is~equivalent~to}~ p ~\mathrm{and}~ q ~\mathrm{and}~ r {}^{\prime\prime}.\!</math>
One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''. It says that a bare lobe expression like <math>\texttt{( \_, \_, \ldots )},\!</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(~)}\!</math> that <math>\mathrm{Ex}\!</math> interprets as denoting the logical value <math>\mathrm{false}.\!</math> To depict the rule in graphical form, we have the continuing sequence of equations:
One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''. It says that a bare lobe expression like <math>\texttt{(} \_ \texttt{,} \_ \texttt{,} \ldots \texttt{)},\!</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(} ~ \texttt{)}\!</math> that <math>\mathrm{Ex}~\!</math> interprets as denoting the logical value <math>\mathrm{false}.~\!</math> To depict the rule in graphical form, we have the continuing sequence of equations:
{| align="center" border="0" cellpadding="10"
{| align="center" border="0" cellpadding="10"
e_0 & = &
e_0 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{(~)}
\texttt{( )}
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_1 & = &
e_1 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{~}
\texttt{ }
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_2 & = &
e_2 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}
\texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))}
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_3 & = &
e_3 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{(} p \texttt{~(} q~r \texttt{))}
\texttt{(} p \texttt{ (} q r \texttt{))}
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_4 & = &
e_4 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{(} p~q~r \texttt{~,~(} p \texttt{))}
\texttt{(} p q r \texttt{ , (} p \texttt{))}
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_5 & = &
e_5 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{((~(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))~,~(} p \texttt{~(} q~r \texttt{))~))}
\texttt{(( (} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{)) , (} p \texttt{ (} q r \texttt{)) ))}
{}^{\prime\prime}
{}^{\prime\prime}
\end{array}\!</math>
\end{array}\!</math>
{| 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.
|}
|}
{| align="center" cellpadding="8" width="90%"
{| 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 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==
~ 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>
|}
|}
{| align="center" cellpadding="8" width="90%"
{| 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>
|}
|}
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
|
From <math>{\textit{Expression~1}}\!</math> and <math>{\textit{Expression~2}}\!</math> infer <math>{\textit{Expression~3}}.\!</math>
From <math>{\textit{Expression 1}}\!</math> and <math>{\textit{Expression 2}}\!</math> infer <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>
|-
|-
|
|
| [[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>
|-
|-
|
|
| [[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>
|-
|-
|
|
| [[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 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 — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — 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 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 — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — 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. (1679–1686 ?), “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).
* [[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.]] (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as CP 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–), ''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. (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), “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. (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), “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. (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), “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 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), “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.
* [[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===
