Jon Awbrey · @Inquiry
129 followers · 624 posts · Server mathstodon.xyz
#LogicalGraphs • 17
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
#Duality • Logical and Topological
Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 8 to bring out the dual graphs by themselves affords a view of the first #InitialEquation shown in Figure 9.
Figure 9
• https://oeis.org/w/images/8/85/Logical_Graph_Figure_9_Visible_Frame.jpg
#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #Peirce #logic #initialequation #dualgraphs #alphagraphs #duality #LogicalGraphs
Jon Awbrey · @Inquiry
109 followers · 499 posts · Server mathstodon.xyz
#LogicalGraphs • 16
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
#Duality • Logical and Topological
Turning now to the #InitialEquation or #LogicalAxiom whose text expression is \(\texttt{(}~\texttt{)(}~\texttt{)}=\texttt{(}~\texttt{)}\), Figure 8 shows the planar maps and their #DualGraphs superimposed.
Figure 8
• https://oeis.org/w/images/0/09/Logical_Graph_Figure_8_Visible_Frame.jpg
#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #Peirce #logic #dualgraphs #logicalaxiom #initialequation #duality #LogicalGraphs
Jon Awbrey · @Inquiry
100 followers · 428 posts · Server mathstodon.xyz
#LogicalGraphs • 15
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
We have treated in some detail various forms of the #InitialEquation or #LogicalAxiom whose text expression is \(``\texttt{((}~\texttt{))}~=~".\) For comparison, let's record the plane-embedded & #TopologicalDual forms of the axiom whose text expression is \(``\texttt{(}~\texttt{)(}~\texttt{)}=\texttt{(}~\texttt{)}".\)
Figure 7 reproduces the planar form of the equation we first saw in Figure 1.
Figure 7
• https://oeis.org/w/images/b/b4/Logical_Graph_Figure_7_Visible_Frame.jpg
#Peirce #logic #topologicaldual #logicalaxiom #initialequation #LogicalGraphs
Jon Awbrey · @Inquiry
96 followers · 408 posts · Server mathstodon.xyz
#LogicalGraphs • 11
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 4 to bring out the dual graphs by themselves affords a view of the second #InitialEquation shown in Figure 5.
Figure 5
• https://oeis.org/w/images/4/46/Logical_Graph_Figure_5_Visible_Frame.jpg
#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #Peirce #logic #initialequation #dualgraphs #alphagraphs #LogicalGraphs
Jon Awbrey · @Inquiry
96 followers · 397 posts · Server mathstodon.xyz
#LogicalGraphs • 10
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
Figure 4 illustrates the mapping of #AlphaGraphs to #DualGraphs by overlaying the alpha graphs of Figure 2 with their corresponding dual graphs.
Figure 4. Alpha Graphs and Dual Graphs for the Second #InitialEquation
• https://oeis.org/w/images/3/3f/Logical_Graph_Figure_4_Visible_Frame.jpg
It is usual to think of ourselves as observing alpha graphs from the outermost region of the plane and we mark that by mapping that region to a node singled out as the “root” of the dual.
#Peirce #logic #initialequation #dualgraphs #alphagraphs #LogicalGraphs