Commentary On Small Models • 2

The consequence of dealing with “practically infinite extensions” becomes crucial in building neural network systems capable of learning and adapting, since the adaptive competence of any intelligent system is limited to the objects and domains it is able to represent.  If we seek to design systems which operate intelligently with the full deck of propositions dealt by intact universes of discourse then we must supply those systems with succinct representations and efficient transformations in that domain.

Beyond the ability to learn and adapt, which taken at the ebb so often devolves into bare conformity and confirmation bias, the ability to inquire and reason makes even more demands on propositional representation.  The project of constructing inquiry driven systems forces us to contemplate the level of generality embodied in logical propositions.  We can see that because the progress of inquiry is driven by the manifest discrepancies occurring among expectations, intentions, and observations, and each of those components of systematic knowledge takes on the fully generic character of an empirical summary or an axiomatic theory.

A compression scheme by any other name is a symbolic representation — and that is what the differential extension of propositional calculus is intended to supply.  But why is that particular program of mental calisthenics worth carrying out in general?

The provision of a uniform logical framework for describing time‑evolving systems makes the task of understanding complex systems easier than it would otherwise be when we try to tackle each new system de novo, “from scratch” as we say.  Having a uniform medium ready to hand helps both in looking for invariant representations of individual cases and also in finding points of comparison among diverse structures otherwise appearing to be isolated systems.  All that goes to facilitate the search for compact knowledge, to apply what is learned from individual cases to the general realm.

Resources

• Differential Logic and Dynamic Systems
• Differential Logic • Commentary On Small Models

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