Inquiry Into Inquiry

Re: Peirce’s Law • (1) • (2) • (3) • (4) • (5) • (6) • (7)
Re: Logical Graphs • Interpretive Duality • (1) • (2) • (3)

Last time we took up Peirce’s law, and saw how it might be expressed in two different ways, under the entitative and existential interpretations, respectively.  The next thing to do is see how our choice of interpretation bears on the patterns of proof we might find.  To that purpose the following table shows a pair of proofs, one of each kind, in parallel array.

For convenience, the formal axioms and a few theorems of frequent use are linked below.

• Axiom I₁ • Condense/Protract
• Axiom I₂ • Cancel/Elicit
• Axiom J₁ • Delete/Insert
• Axiom J₂ • Collect/Distribute
• C₁ • Double Negation Theorem • Reflect/Reflect
• C₂ • Generation Theorem • Regenerate/Degenerate
• C₃ • Dominant Form Theorem • Quit/Quip

Resource

• Logical Graphs • Formal Development

cc: FB | Logical Graphs • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Re: Peirce’s Law • (1) • (2) • (3) • (4) • (5) • (6) • (7)
Re: Logical Graphs • Interpretive Duality • (1) • (2)

To see how our choice of interpretation bears on cases beyond the bare minimum 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.

cc: FB | Logical Graphs • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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 previous posts 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.

cc: FB | Logical Graphs • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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.

cc: FB | Logical Graphs • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science