Christian Lawson-Perfect @christianp

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**Jon Awbrey** @Inquiry · Dec 18, 2022 *

Dec 18, 2022 *

#LogicalGraphs • 16
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Turning to the #InitialEquation or #LogicalAxiom whose text expression is $‘ ‘ ()() = () ",$ Figure 8 shows the planar maps and their corresponding #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

Figure 8 is a composite picture for the Formal Equation in Figure 1. It illustrates the mapping of alpha graphs to dual graphs by overlaying the alpha graphs in Figure 1 with their corresponding dual graphs.

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

Layer 1 shows the alpha graphs for the First Initial Equation, previously shown in Figure 1 and again in Figure 7. 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 consists of two circles drawn next to each other a space apart.

The form on the right consists of a single circle.

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 is a graph-theoretic rooted tree, in this case consisting of two edges sharing a common root. The root is placed outside and below the two circles. The first edge extends to the interior of the circle on the left, where it meets its terminal node. The second edge extends to the interior of the circle on the right, where it meets its terminal node.

The form on the right consists of a single rooted edge, or a rooted path of length one. It extends from its root outside the single circle to its terminal node inside the circle.

Continued thread

**Jon Awbrey** @Inquiry · Dec 13, 2022 *

Dec 13, 2022 *

Jon Awbrey @Inquiry

#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 $‘ ‘ (()) = " .$ For comparison, let's record the plane-embedded and #TopologicalDual forms of the axiom whose text expression is $‘ ‘ ()() = () " .$

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

Figure 7 reproduces Figure 1. It shows what we call a “Formal Equation”, that is, an equation between two abstract forms. Proceeding from left to right in the Figure, there is (1) a form on the left hand side, (2) an equal sign, and (3) a form on the right hand side.

The form on the left hand side consists of two circles drawn next to each other a space apart.

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

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

What's going on here is an attempt to convey an abstract idea in a concrete picture. The picture uses geometric shapes to convey the idea but the idea is really more abstract than geometry.

#Logic #Peirce

**Jon Awbrey** @Inquiry · Dec 3, 2022 *

Dec 3, 2022 *

Jon Awbrey @Inquiry

#AllLiarNoParadox
• https://inquiryintoinquiry.com/2015/08/01/all-liar-no-paradox/

A statement $S_{0}$ asserts that a statement $S_{1}$ is a statement that $S_{1}$ is false.

The statement $S_{0}$ violates an #Axiom of #Logic, so it doesn’t really matter whether the #OstensibleStatement $S_{1},$ the so-called #Liar, really is a statement or has a #TruthValue.

#LogicalAxiom #LawOfLogic #LogicalGraph
#LiarParadox #Epimenides #EpimenidesParadox
#Peirce #Semiotics #Semeiotics #SignRelations

Inquiry Into InquiryAll Liar, No ParadoxBy Jon Awbrey

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