Re: R.J. Lipton and K.W. Regan • Mounting Or Solving Open Problems

Comment 1

Sometimes the programme must simply be to keep developing our understanding of the ground on which the mountains rest.

Comment 2

It might be observed that the concept of a research programme is closely related to the concept of a research paradigm, about which much has been written.

Comment 3

As long as we’re brainstorming in a laid back sort of way …

Folks who find propositional logic — and the whole space between zeroth order logic and first order logic — more of a fascinating playground for exploration than a dog run for the questying beast might find it fun to look at the graph-theoretic calculi for propositions and boolean functions that derive from C.S. Peirce’s logical graphs.  There is an extension of Peirce’s tree-form graphs to cactus graphs that presents many interesting possibilities for efficient expression and inference.  And the resulting cactus calculus facilitates the development of differential logic, extending propositional calculus analogous to the way differential calculus extends analytic geometry.

Comment 4

The following primer on differential logic uses the cactus graph syntax to represent propositions (boolean functions $latex f : \mathbb{B}^k \to \mathbb{B}) and the operators on propositions that arise in developing the subject of differential propositional calculus.

• Differential Logic • Part 1 • Part 2 • Part 3

The cactus graph syntax for propositional calculus is based on minimal negation operators.

• Minimal Negation Operator (OEIS Wiki)
• Minimal Negation Operator (MyWikiBiz
• Minimal Negation Operator (PlanetMath)
• Minimal Negation Operator (Wikiversity)