Re: Differential Logic • The Logic of Change and Difference
Re: Systems Science Working Group • Paola Di Maio

PDM:

Subject: Differential Logic —
A point of contact with AI Knowledge Representation

Dear Jon,

Thank you for keeping the bell tolling — your framing of differential logic as the logic of variation arrives at a propitious moment.

For the past year I have been working at the intersection of knowledge representation, non‑logical reasoning, and AI systems, partly through the W3C AI Knowledge Representation Community Group (which I chair) and partly through independent research.  One of the persistent problems we encounter is that classical propositional and first order logic, however powerful for static state description, cannot represent the dynamics of reasoning systems — what changes, how fast, under what perturbation.

Your formulation cuts right to it:  ordinary propositional calculus describes positions in logical space; differential propositional calculus describes movement through it.  The analogy to Leibniz–Newton augmenting Descartes marks a categorical shift.

This connects directly to work I have been developing on what I call the five‑corners framework, extending Nagarjuna’s “catuskoti” (the four‑cornered logic:  true, false, both, neither — with Graham Priest’s fifth corner as refusal of the frame) toward a relational and co‑evolutionary account of knowledge.  The catuskoti gives us positions;  your differential extension gives us the calculus of transitions between them.  The five corners are attractors;  differential logic describes the manifold on which the system moves.

I am attaching a recent research note —

• “Beyond Formal Logic:  Non‑Logical Forms of Valid Reasoning and Their Implications for AI Knowledge Representation”.  Online.

It documents three classes of reasoning that produce valid outcomes yet resist formalization in FOL:  embodied ecological reasoning, somatic‑intuitive reasoning, and transrational insight.

I suspect your differential extension of propositional calculus may offer formal traction on at least the first two, precisely because it can represent how a reasoning agent’s truth‑value assignments shift as context changes.

I also noticed your reference to neural network activation states and competition constraints in relation to the boundary operator.

• Differential Logic • Various Sketches

This is terrain I am actively exploring in connection with oscillatory network models and a citizen science project on anomalous luminous phenomena (where the signal is change, not static state).  I may have to write a paper on that.

Jotted down some thoughts —

With collegial regards,

Paola Di Maio
Chair, W3C AI Knowledge Representation Community Group
Research Lead, Center for Systems, Knowledge Representation and Neuroscience, Ronin Institute

Dear Paola,

Many thanks for your kind reply and comments.

I was getting ready to devote a blog post (or two or three) by way of responding to your very substantial comments and I see you addressed the Systems Science Working Group but your post did not make it through to the web interface.  Did you intend to post it there?  It would help if I had a list link in my response if you did so.  Otherwise, if it’s okay with you, I could just quote the whole of your remarks on my blog.  Please let me know what you prefer.

Regards,

Jon

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