Lmst

@hochstenbach @josd

I 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.

@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 • 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

Figure 9 is derived from the composite picture of alpha graphs and dual graphs in Figure 8 by editing it to show just the dual graphs themselves. Proceeding from left to right in the Figure, there is a form on the left hand side, an equal sign, and a form on the right hand side.

The form on the left hand side is a graph-theoretic rooted tree, in this case consisting of two edges sharing a common root node.

The form on the right hand side consists of a single rooted edge, or a rooted path of length one.

The formal equation says these two configurations are equivalent in some sense — a sense to be explained further on.

#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

@DavisSmith @philosophy

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/

#LogicalGraphs • 11
• 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 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

Figure 5 is derived from the composite picture of alpha graphs and dual graphs in Figure 4 by editing it to show just the dual graphs themselves. Proceeding from left to right in the Figure, there is a form on the left hand side, an equal sign, and a form on the right hand side.

The form on the left hand side is a graph-theoretic rooted tree, in this case a rooted path of length 2. Proceeding upward from bottom to top, there is a distinguished root node, an edge (or a line segment), a second node, then a second edge, and finally a terminal node.

The form on the right hand side consists of a single root node.

The formal equation says these two configurations are equivalent in some sense — a sense to be explained further on.

#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.

#Logic #Peirce

Figure 4 is a composite picture for the Formal Equation in Figure 2. It illustrates the mapping of alpha graphs to dual graphs by overlaying the alpha graphs in Figure 2 with their corresponding dual graphs.

We may think of the picture as composed of two layers.

Layer 1 shows the alpha graphs for the Second Initial Equation, previously shown in Figure 2 and again in Figure 3. Proceeding from left to right in Layer 1, there is a form on the left hand side, an equal sign, and a form on the right hand side.

The form on the left hand side consists of two concentric circles.

The form on the right hand side is shown as white space or a blank area.

Layer 2 shows the dual graphs corresponding to the alpha graphs in Layer 1. Proceeding from left to right in Layer 2, there is a form on the left hand side, an equal sign, and a form on the right hand side.

The form on the left hand side is a graph-theoretic rooted tree, in this case a rooted path of length 2. Proceeding upward from bottom to top, there is a distinguished root node, an edge (or a line segment), a second node, then a second edge, and finally a terminal node.

The form on the right hand side consists of a single root node.

#LogicalGraphs • 9
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

From the #AlphaGraph and its text expression we turn to representing their common form in computer memory, where it can be manipulated with the greatest of ease. We begin by transforming the alpha graph into its #TopologicalDual. Planar regions of the alpha graph are mapped into points (or nodes) of the #DualGraph & boundaries between those planar regions are mapped into lines (or edges) connecting those points.

#Logic #Peirce #AlphaGraphs #Duality

#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 separately 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 converted to text as strings of parentheses and logical terms which can be parsed to #PointerStructures in computer memory.

#Logic

#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.

#Logic #GraphTheory #ModelTheory #ProofTheory

#AlphaGraphs

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