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Directory:Jon Awbrey/Papers/Futures Of Logical Graphs (view source)

Revision as of 18:18, 4 August 2017

143 bytes removed ,  18:18, 4 August 2017
sub "File" for "Image" + use "Rooted Node Big" & "Rooted Edge Big"
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| width="5%" | <math>\mathrm{En}_\text{sem} :</math>
 
| width="5%" | <math>\mathrm{En}_\text{sem} :</math>
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| width="5%" align="center" | [[Image:Rooted Node Big.jpg|16px]]
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| width="5%" | <math>\mapsto</math>
 
| width="5%" | <math>\mapsto</math>
 
| <math>\mathrm{false},</math>
 
| <math>\mathrm{false},</math>
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| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
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| align="center" | [[File:Rooted Edge Big.jpg|12px]]
 
| <math>\mapsto</math>
 
| <math>\mapsto</math>
 
| <math>\mathrm{true}.</math>
 
| <math>\mathrm{true}.</math>
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| &nbsp;
 
| &nbsp;
 
| <math>\mathrm{Ex}_\text{sem} :</math>
 
| <math>\mathrm{Ex}_\text{sem} :</math>
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| <math>\mapsto</math>
 
| <math>\mathrm{true},</math>
 
| <math>\mathrm{true},</math>
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| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
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The functional images of the syntactic reduction map <math>\mathrm{E}_\text{syn} : Y \to Y_0</math> are the two simplest signs or the most reduced pair of expressions, regarded as the rooted trees [[Image:Rooted Node.jpg|16px]] and [[Image:Rooted Edge.jpg|12px]], and these may be treated as the canonical representatives of their respective equivalence classes.
+
The functional images of the syntactic reduction map <math>\mathrm{E}_\text{syn} : Y \to Y_0</math> are the two simplest signs or the most reduced pair of expressions, regarded as the rooted trees [[File:Rooted Node Big.jpg|16px]] and [[File:Rooted Edge Big.jpg|12px]], and these may be treated as the canonical representatives of their respective equivalence classes.
    
The more Peirce-sistent among you, on contemplating that last picture, will naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"
 
The more Peirce-sistent among you, on contemplating that last picture, will naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"
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| <math>S_0~\!</math>
 
| <math>S_0~\!</math>
 
| <math>=~\!</math>
 
| <math>=~\!</math>
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| <math>\{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}~\!</math>
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| <math>\{ \ominus, \vert \} = \{</math>[[File:Rooted Node Big.jpg|16px]], [[File:Rooted Edge Big.jpg|12px]]<math>\}~\!</math>
 
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   −
Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;.
+
Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[File:Rooted Node Big.jpg|16px]] or else to a rooted edge [[File:Rooted Edge Big.jpg|12px]]&nbsp;.
    
For example, consider the reduction that proceeds as follows:
 
For example, consider the reduction that proceeds as follows:
    
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| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (26)
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| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math>
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math>
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| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
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| (30)
 
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|-
−
| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast P.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-2.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-2.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
+
| [[File:Equational Inference Bar -- Domination ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-3.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-3.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-4.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-4.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cast Q.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast Q.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-5.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-5.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-6.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-6.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
+
| [[File:Equational Inference Bar -- Domination ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-7.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-7.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cast R.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast R.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-8.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-8.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-9.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-9.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
+
| [[File:Equational Inference Bar -- DNF.jpg|500px]]
 
|}
 
|}
 
| (31)
 
| (31)
Line 1,932: Line 1,932:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| [[Image:Logical Graph (P (Q)) (P (R)) DNF.jpg|500px]]
+
| [[File:Logical Graph (P (Q)) (P (R)) DNF.jpg|500px]]
 
| (32)
 
| (32)
 
|}
 
|}
Line 1,960: Line 1,960:  
{| 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"
 
{| 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:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-0.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-0.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-1.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-1.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast P.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2 ISW.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2 ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
+
| [[File:Equational Inference Bar -- Domination ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-4.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-4.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cast Q.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast Q.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
+
| [[File:Equational Inference Bar -- Domination ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6 ISW.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6 ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-7.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-7.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cast R.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast R.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-8.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-8.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-9.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-9.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
+
| [[File:Equational Inference Bar -- DNF.jpg|500px]]
 
|}
 
|}
 
| (33)
 
| (33)
Line 2,004: Line 2,004:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (34)
+
| [[File:Logical Graph (P Q R , (P)).jpg|500px]] || (34)
 
|}
 
|}
   Line 2,010: Line 2,010:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (35)
+
| [[File:Logical Graph ((P , P Q R)).jpg|500px]] || (35)
 
|}
 
|}
   Line 2,114: Line 2,114:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| [[Image:Logical Graph (( (P (Q)) (P (R)) , (P (Q R)) )).jpg|500px]]
+
| [[File:Logical Graph (( (P (Q)) (P (R)) , (P (Q R)) )).jpg|500px]]
 
| (39)
 
| (39)
 
|}
 
|}
Line 2,124: Line 2,124:  
{| 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"
 
{| 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:Proof (P (Q)) (P (R)) = (P (Q R)) 3-00.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-00.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-01.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-01.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast P.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-02.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-02.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
+
| [[File:Equational Inference Bar -- Domination ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-03.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-03.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-04.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-04.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Emptiness.jpg|500px]]
+
| [[File:Equational Inference Bar -- Emptiness.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-05.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-05.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-06.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-06.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cast Q.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast Q.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-07.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-07.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-08.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-08.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
+
| [[File:Equational Inference Bar -- Domination ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-09.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-09.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-10.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-10.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Spike.jpg|500px]]
+
| [[File:Equational Inference Bar -- Spike.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-11.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-11.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-12.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-12.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cast R.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cast R.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-13.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-13.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-14 ISW.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-14 ISW.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Emptiness.jpg|500px]]
+
| [[File:Equational Inference Bar -- Emptiness.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-15.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-15.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Spike.jpg|500px]]
+
| [[File:Equational Inference Bar -- Spike.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-16.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-16.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
+
| [[File:Equational Inference Bar -- Cancellation.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-17.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 3-17.jpg|500px]]
 
|-
 
|-
−
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
+
| [[File:Equational Inference Bar -- QED.jpg|500px]]
 
|}
 
|}
 
| (40)
 
| (40)
Jon Awbrey
12,080

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