Jon Awbrey · @Inquiry
164 followers · 961 posts · Server mathstodon.xyzI signed on to the group and list. I don't know if you'd be interested in a side- or sub-project focusing on the propositional layer as I have done some work on #Peirce's #AlphaGraphs and his #LogicOfRelatives.
#LogicOfRelatives #alphagraphs #Peirce
Jon Awbrey · @Inquiry
164 followers · 945 posts · Server mathstodon.xyz@bblfish @josd @semwebpro @hochstenbach
One thing I found out early on is how critical it is to get #AlphaGraphs (#BooleanFunctions, #PropositionalCalculus, #ZerothOrderLogic) down tight. If you do that it changes how you view #FOL (#PredicateCalculus, #QuantificationalLogic). That tends to rub people who view FOL as #GOL (#GodsOwnLogic) the wrong way so you have watch out for that if you go down this road.
Here's a primer on \(\alpha\) #LogicalGraphs as I see them —
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no
#LogicalGraphs #godsownlogic #gol #quantificationallogic #predicatecalculus #fol #zerothorderlogic #PropositionalCalculus #BooleanFunctions #alphagraphs
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
96 followers · 414 posts · Server mathstodon.xyz#LogicalGraphs • 12
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
#Duality • Logical and Topological
Once we make the connection between one of #Peirce's #AlphaGraphs and its character string expression it's not too big a leap to see how the character string codes up the structure of the topological #DualGraph in the space of #RootedTrees.
#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #logic #rootedtrees #dualgraph #alphagraphs #Peirce #duality #LogicalGraphs
Jon Awbrey · @Inquiry
96 followers · 413 posts · Server mathstodon.xyz
For that my (classical) favorites are #PeircesLaw and #Leibniz's #PraeclarumTheorema.
Here's how they look in a variant of #Peirce's #AlphaGraphs.
• https://inquiryintoinquiry.com/2008/10/06/peirces-law/
• https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/
#alphagraphs #Peirce #praeclarumtheorema #leibniz #peirceslaw
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
Jon Awbrey · @Inquiry
94 followers · 377 posts · Server mathstodon.xyz#LogicalGraphs • 7
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
#Duality • Logical and Topological
In using logical graphs there are two kinds of duality — logical and topological — we need to keep in mind.
#Peirce's #AlphaGraphs are conceived as embedded in a continuous manifold, as if a sheet of paper. They differ from the graphs of today's #GraphTheory but they can be textified as strings of parentheses and logical terms which can be parsed to #PointerStructures in computer memory.
#logic #pointerstructures #graphtheory #alphagraphs #Peirce #duality #LogicalGraphs
Jon Awbrey · @Inquiry
90 followers · 359 posts · Server mathstodon.xyz#LogicalGraphs • 3
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no
We begin on a low but expansive plateau of #FormalSystems #Peirce mapped out in his system of #AlphaGraphs \((\alpha),\) a platform so abstract in its mathematical forms as to support at least two interpretations for use in the conduct of logical reasoning. Along the way, we incorporate the later contributions of George #SpencerBrown, who revived and augmented Peirce's system in his book #LawsOfForm.
#ProofTheory #modeltheory #graphtheory #logic #lawsofform #spencerbrown #alphagraphs #Peirce #formalsystems #LogicalGraphs