Re: Interpretive Duality in Logical Graphs • 6
The last of our six ways of looking at interpretive duality is arrived at by taking the previous Table of Logical Graphs and Venn Diagrams and sorting it in Orbit Order.
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Re: Interpretive Duality in Logical Graphs • 2
Dualities are symmetries of order two and symmetries bear on complexity by reducing its measure in proportion to their order. The inverse relationship between symmetry and the usual dissymmetries from dispersion and diversity to entropy and uncertainty is governed in cybernetics by the Law of Requisite Variety, the medium of whose exchange C.S. Peirce invested in the formula Information = Comprehension × Extension.
The duality between entitative and existential interpretations of logical graphs is one example of a mathematical symmetry but it’s not unusual to find symmetries within symmetries and it’s always rewarding to find them where they exist. To that end let’s take up our Table of Venn Diagrams and Logical Graphs on Two Variables and sort the rows to bring together diagrams and graphs having similar shapes. What defines their similarity is the action of a mathematical group whose operations transform the elements of each class among one another but intermingle no dissimilar elements. In the jargon of transformation groups those classes are called orbits. We find the sixteen rows partition into seven orbits, as shown below.
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Re: Interpretive Duality in Logical Graphs • 2
A more graphic picture of interpretive duality is given by the next Table, showing how logical graphs map to venn diagrams under entitative and existential interpretations. Column 1 shows the logical graphs for the sixteen boolean functions on two variables. Column 2 shows the venn diagrams associated with the entitative interpretation and Column 3 shows the venn diagrams associated with the existential interpretation.
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Re: Interpretive Duality in Logical Graphs • 1
Another way of looking at interpretive duality in logical graphs is given by the following Table, showing how logical graphs denote the sixteen boolean functions on two variables under entitative and existential interpretations, respectively.
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Re: Interpretive Duality in Logical Graphs • (1) • (2) • (3)
Last time we took up Peirce’s law, 
For convenience, the formal axioms and a few theorems of frequent use are linked below.
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Re: Interpretive Duality in Logical Graphs • (1) • (2)
For a sense of how the choice of interpretation bears on cases beyond the bare minimum complexity let us start with the familiar example of Peirce’s law, commonly expressed in the following form.
The following two formal equations show how Peirce’s law may be expressed in terms of logical graphs, operating under the entitative and existential interpretations, respectively.
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Re: Interpretive Duality in Logical Graphs • 1
A logical concept represented by a boolean variable has its extension, the cases it covers in a designated universe of discourse, and its comprehension (or intension), the properties it implies in a designated hierarchy of predicates.
The formulas and graphs tabulated in the previous post are well‑adapted to articulate the syntactic and intensional aspects of propositional logic. But their very tailoring to those tasks tends to slight the extensional and therefore empirical applications of logic. Venn diagrams, despite their unwieldiness as the number of logical dimensions increases, are indispensable in providing the visual intuition with a solid grounding in the extensions of logical concepts. All that makes it worthwhile to reset our table of boolean functions on two variables to include the corresponding venn diagrams.
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The duality between Entitative and Existential interpretations of logical graphs is a good example of a mathematical symmetry, in this case a symmetry of order two. Symmetries of this and higher orders give us conceptual handles on excess complexity in the manifold of sensuous impressions, making it well worth the effort to seek them out and grasp them where we find them.
Both Peirce and Spencer Brown understood the significance of the mathematical unity underlying the dual interpretation of logical graphs. Peirce began with the Entitative option and later switched to the Existential choice while Spencer Brown exercised the Entitative option in his Laws of Form.
In that vein, here’s a Rosetta Stone to give us a grounding in the relationship between boolean functions and our two readings of logical graphs.
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Re: Operator Variables in Logical Graphs • 11
The rules given in the previous post for evaluating cactus graphs were given in purely formal terms, that is, by referring to the mathematical forms of cacti without mentioning their potential for logical meaning. As it turns out, two ways of mapping cactus graphs to logical meanings are commonly found in practice. These two mappings of mathematical structure to logical meaning are formally dual to each other and known as the Entitative and Existential interpretations respectively. The following Table compares the entitative and existential interpretations of the primary cactus structures, from which the rest of their semantics can be derived.
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Re: Futures Of Logical Graphs • Themes and Variations
This post and the next wrap up the Themes and Variations section of my speculation on Futures of Logical Graphs. I made an effort to “show my work”, reviewing the steps I took to arrive at the present perspective on logical graphs, whistling past the least productive of the blind alleys, cul‑de‑sacs, detours, and forking paths I explored along the way. It can be useful to tell the story that way, partly because others may find things I missed down those roads, but it does call for a recap of the main ideas I would like readers to take away.
Partly through my reflection on Peirce’s use of operator variables I was led to what I called a “reflective extension of logical graphs”, amounting to a graphical formal language called the “cactus language” or “cactus syntax” after its principal graph-theoretic data structure.
The abstract syntax of cactus graphs can be interpreted for logical use in a couple of ways, both of which arise from generalizing the negation operator 
Two rules suffice for evaluating cactus graphs.
-node operator, corresponding to an expression of the form 
is as follows.
-lobe operator, corresponding to an expression of the form 
is as follows.
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Inquiry Into Inquiry 