Seeing as how quasi-neural models and the recurring issues of logical-symbolic vs. quantitative-connectionist paradigms have come round again, as they do every dozen or twenty years or so, I thought I might refer again to work I started initially in that context, investigating logical-qualitative-symbolic analogues of systems proposed by McClelland, Rumelhart, and the Parallel Distributed Processing Group, and especially Stephen Grossberg’s competition-cooperation models.
Cf: Differential Logic, Dynamic Systems, Tangent Functors • 1 (copied below)
People interested in category theory as applied to systems may wish to check out the following article, reporting work I carried out while engaged in a systems engineering program at Oakland University.
The problem addressed is a longstanding one, that of building bridges to negotiate the gap between qualitative and quantitative descriptions of complex phenomena, like those we meet in analyzing and engineering systems, especially intelligent systems endowed with a capacity for processing information and acquiring knowledge of objective reality.
One of the ways this problem arises has to do with describing change in logical, qualitative, or symbolic terms, long before we grasp the reality beneath the appearances firmly enough to cast it in measured, quantitative, real number form.
Development on the quantitative shore got no further than a Sisyphean beachhead until the discovery/invention of differential calculus by Leibniz and Newton, after which things advanced by leaps and bounds.
And there’s our clue what we need to do on the qualitative shore, namely, to discover/invent the missing logical analogue of differential calculus.
With that preamble …
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.
The reading continues here: Differential Logic and Dynamic Systems
- Differential Propositional Calculus • Part 1 • Part 2
- Differential Logic • Part 1 • Part 2 • Part 3
- Differential Logic and Dynamic Systems
• Part 1 • Part 2 • Part 3 • Part 4 • Part 5