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

Revision as of 12:41, 6 August 2017

343 bytes added ,  12:41, 6 August 2017
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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%" | <math>\mapsto</math>
 
| width="5%" | <math>\mapsto</math>
 
| <math>\mathrm{false},</math>
 
| <math>\mathrm{false},</math>
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| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
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| <math>\mapsto</math>
 
| <math>\mapsto</math>
 
| <math>\mathrm{true}.</math>
 
| <math>\mathrm{true}.</math>
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| &nbsp;
 
| &nbsp;
 
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| &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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:* Pure Symbols
 
:* Pure Symbols
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:: http://stderr.org/pipermail/inquiry/2005-March/thread.html#2465
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:: [http://web.archive.org/web/20141124153003/http://stderr.org/pipermail/inquiry/2005-March/thread.html#2465 2005 &bull; March]
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:: http://stderr.org/pipermail/inquiry/2005-April/thread.html#2517
+
:: [http://web.archive.org/web/20120601160642/http://stderr.org/pipermail/inquiry/2005-April/thread.html#2517 2005 &bull; April]
    
:* Pure Symbols : Discussion
 
:* Pure Symbols : Discussion
−
:: http://stderr.org/pipermail/inquiry/2005-March/thread.html#2466
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:: [http://web.archive.org/web/20141124153003/http://stderr.org/pipermail/inquiry/2005-March/thread.html#2466 2005 &bull; March]
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:: http://stderr.org/pipermail/inquiry/2005-April/thread.html#2514
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:: [http://web.archive.org/web/20120601160642/http://stderr.org/pipermail/inquiry/2005-April/thread.html#2514 2005 &bull; April]
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:: http://stderr.org/pipermail/inquiry/2005-May/thread.html#2654
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:: [http://web.archive.org/web/20120421003708/http://stderr.org/pipermail/inquiry/2005-May/thread.html#2654 2005 &bull; May]
    
And some will find an ethical principle in this freedom of interpretation.  The act of interpretation bears within it an inalienable degree of freedom.  In consequence of this truth, as far as the activity of interpretation goes, freedom and responsibility are the very same thing.  We cannot blame objects for what we say or what we think.  We cannot blame symbols for what we do.  We cannot escape our response ability.  We cannot escape our freedom.
 
And some will find an ethical principle in this freedom of interpretation.  The act of interpretation bears within it an inalienable degree of freedom.  In consequence of this truth, as far as the activity of interpretation goes, freedom and responsibility are the very same thing.  We cannot blame objects for what we say or what we think.  We cannot blame symbols for what we do.  We cannot escape our response ability.  We cannot escape our freedom.
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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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A ''sort'' of signs is more formally known as an ''equivalence class'' (EC).  There are in general many sorts of sorts of signs that we might wish to consider in this inquiry, but let's begin with the sort of signs all of whose members denote the same object as their referent, a sort of signs to be henceforth referred to as a ''referential equivalence class'' (REC).
 
A ''sort'' of signs is more formally known as an ''equivalence class'' (EC).  There are in general many sorts of sorts of signs that we might wish to consider in this inquiry, but let's begin with the sort of signs all of whose members denote the same object as their referent, a sort of signs to be henceforth referred to as a ''referential equivalence class'' (REC).
   −
:* [http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104 FOLG]
+
:* [http://web.archive.org/web/20150224133200/http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104 Inquiry List &bull; Futures Of Logical Graphs]
−
:* [http://stderr.org/pipermail/inquiry/2005-October/003113.html FOLG 5]
+
:* [http://web.archive.org/web/20120206123011/http://stderr.org/pipermail/inquiry/2005-October/003113.html Inquiry List &bull; Futures Of Logical Graphs &bull; Note 5]
    
Toward the outset of this excursion, I mentioned the distinction between a ''pointwise-restricted iconic map'' or a ''pointedly rigid iconic map'' (PRIM) and a ''system-wide iconic map'' (SWIM).  The time has come to make use of that mention.
 
Toward the outset of this excursion, I mentioned the distinction between a ''pointwise-restricted iconic map'' or a ''pointedly rigid iconic map'' (PRIM) and a ''system-wide iconic map'' (SWIM).  The time has come to make use of that mention.
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| [[Image:Logical_Graph_Figure_33.jpg|500px]]
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| [[File:Logical_Graph_Figure_33.jpg|500px]]
 
|}
 
|}
   Line 1,796: Line 1,796:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (26)
+
| [[File:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (26)
 
|}
 
|}
   Line 1,810: Line 1,810:     
{| align="center" cellpadding="8" style="text-align:center"
 
{| align="center" cellpadding="8" style="text-align:center"
−
| [[Image:Venn Diagram (P (Q)) (P (R)).jpg|500px]] || (27)
+
| [[File:Venn Diagram (P (Q)) (P (R)).jpg|500px]] || (27)
 
|-
 
|-
 
| <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>
Line 1,822: Line 1,822:     
{| align="center" cellpadding="8" style="text-align:center"
 
{| align="center" cellpadding="8" style="text-align:center"
−
| [[Image:Venn Diagram (P (Q R)).jpg|500px]] || (28)
+
| [[File:Venn Diagram (P (Q R)).jpg|500px]] || (28)
 
|-
 
|-
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
Line 1,856: Line 1,856:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg|500px]] || (29)
+
| [[File:Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg|500px]] || (29)
 
|}
 
|}
   Line 1,872: Line 1,872:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]]
+
| [[File:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]]
 
| (30)
 
| (30)
 
|}
 
|}
Line 1,888: Line 1,888:  
{| 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)) 2-1-0.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-0.jpg|500px]]
 
|-
 
|-
−
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-1.jpg|500px]]
+
| [[File:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-1.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)) 2-1-2.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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