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update with majority function example & animations
By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, let us examine the proofs of a few essential theorems in the primary algebra.
By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, let us examine the proofs of a few essential theorems in the primary algebra.
====C<sub>1</sub>. Double negation theorem====
====C<sub>1</sub>. Double negation====
The first theorem goes under the names of ''Consequence 1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''.
The first theorem goes under the names of ''Consequence 1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''.
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_26.jpg|500px]] || (26)
| [[Image:Logical_Graph_Figure_26.jpg|500px]] || (26)
|}
The steps of this proof are replayed in the following animation.
{| align="center" cellpadding="8"
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Double Negation 2.0 Animation.gif]]
|}
| (27)
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_27.jpg|500px]] || (27)
| [[Image:Logical_Graph_Figure_27.jpg|500px]] || (28)
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_28.jpg|500px]] || (28)
| [[Image:Logical_Graph_Figure_28.jpg|500px]] || (29)
|}
The steps of this proof are replayed in the following animation.
{| align="center" cellpadding="8"
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Generation Theorem 2.0 Animation.gif]]
|}
| (30)
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_29.jpg|500px]] || (29)
| [[Image:Logical_Graph_Figure_29.jpg|500px]] || (31)
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_30.jpg|500px]] || (30)
| [[Image:Logical_Graph_Figure_30.jpg|500px]] || (32)
|}
|}
====Peirce's law====
====Peirce's law====
: ''Main article'' : [[Peirce's law]]
: ''Main article'' : [http://mywikibiz.com/Peirce's_law Peirce's law]
Peirce's law is commonly written in the following form:
Peirce's law is commonly written in the following form:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_31.jpg|500px]] || (31)
| [[Image:Logical_Graph_Figure_31.jpg|500px]] || (33)
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_32.jpg|500px]] || (32)
| [[Image:Logical_Graph_Figure_32.jpg|500px]] || (34)
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_33.jpg|500px]] || (33)
| [[Image:Logical_Graph_Figure_33.jpg|500px]] || (35)
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_34.jpg|500px]] || (34)
| [[Image:Logical_Graph_Figure_34.jpg|500px]] || (36)
|}
====Two-thirds majority function====
Consider the following equation in boolean algebra, posted as a [http://mathoverflow.net/questions/9292/newbie-boolean-algebra-question problem for proof] at [http://mathoverflow.net/ MathOverFlow].
{| align="center" cellpadding="20"
|
<math>\begin{matrix}
a b \bar{c} + a \bar{b} c + \bar{a} b c + a b c
\\[6pt]
\iff
\\[6pt]
a b + a c + b c
\end{matrix}</math>
| (37)
|}
The required equation can be proven in the medium of logical graphs as shown in the following Figure.
{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Majority Function Example Proof 1 Title.png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 1.jpg|500px]]
|-
| [[Image:Equational Inference Reflect ab, ac, bc.png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 2.jpg|500px]]
|-
| [[Image:Equational Inference Distribute (abc).png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 3.jpg|500px]]
|-
| [[Image:Equational Inference Collect ab, ac, bc.png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 4.jpg|500px]]
|-
| [[Image:Equational Inference Quit (a), (b), (c).png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 5.jpg|500px]]
|-
| [[Image:Equational Inference Cancel (( )).png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 6.jpg|500px]]
|-
| [[Image:Equational Inference Weed ab, ac, bc.png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 7.jpg|500px]]
|-
| [[Image:Equational Inference Delete a, b, c.png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 8.jpg|500px]]
|-
| [[Image:Equational Inference Cancel (( )).png|500px]]
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 9.jpg|500px]]
|-
| [[Image:Equational Inference QED.png|500px]]
|}
| (38)
|}
Here's an animated recap of the graphical transformations that occur in the above proof:
{| align="center" cellpadding="8"
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]]
|}
| (39)
|}
|}
==Bibliography==
==Bibliography==
* [[Gottfried Leibniz|Leibniz, G.W.]] (1679–1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40–46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz: Logical Papers'', Oxford University Press, London, UK.
* [[Gottfried Leibniz|Leibniz, G.W.]] (1679–1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40–46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK.
* [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]].
* [[Charles 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 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 Indianopolis, 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 Indianopolis, 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.
* [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK.
==Resources==
==Resources==
*[http://planetmath.org/ PlanetMath]
* [http://planetmath.org/ PlanetMath]
**[http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : Introduction]
** [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : Introduction]
**[http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : Formal Development]
** [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : Formal Development]
* Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms]
* Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms]
**[http://www.helsinki.fi/science/commens/terms/graphexis.html Existential Graph]
** [http://www.helsinki.fi/science/commens/terms/graphexis.html Existential Graph]
**[http://www.helsinki.fi/science/commens/terms/graphlogi.html Logical Graph]
** [http://www.helsinki.fi/science/commens/terms/graphlogi.html Logical Graph]
*[http://dr-dau.net/index.shtml Dau, Frithjof]
* [http://dr-dau.net/index.shtml Dau, Frithjof]
**[http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links]
** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links]
**[http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] — Computer Animated Proof of Leibniz's Praeclarum Theorema
** [http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] — Computer Animated Proof of Leibniz's Praeclarum Theorema
*[http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.]
* [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.]
**[http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form]
** [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form]
*[http://mathworld.wolfram.com/ MathWorld : A Wolfram Web Resource]
* [http://mathworld.wolfram.com/ MathWorld : A Wolfram Web Resource]
**[http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/Spencer-BrownForm.html Spencer-Brown Form]
** [http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/Spencer-BrownForm.html Spencer-Brown Form]
* Shoup, Richard (ed.), [http://www.lawsofform.org/ Laws of Form Web Site]
* Shoup, Richard (ed.), [http://www.lawsofform.org/ Laws of Form Web Site]
* [http://en.wikipedia.org/wiki/Logical_graph Logical Graph], [http://en.wikipedia.org/ Wikipedia]
* [http://en.wikipedia.org/wiki/Logical_graph Logical Graph], [http://en.wikipedia.org/ Wikipedia]
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