The following primer on differential logic uses the cactus graph syntax to represent propositions (boolean functions ) and the operators on propositions that arise in developing the subject of differential propositional calculus.
The cactus graph syntax for propositional calculus is based on minimal negation operators.
- Minimal Negation Operator (InterSciWiki Version)
- Minimal Negation Operator (MyWikiBiz Version)
- Minimal Negation Operator (PlanetMath Version)
- Minimal Negation Operator (Wikiversity Version)
Some work I started on Aristotle, Peirce, Inquiry, Analogy, etc.
Another approach to discrete dynamics is by way of differential logic, where we take the pragmatic or topological attitude of ignoring all the differences that don’t make a difference.
If is a proposition, a boolean variable, that characterizes — holds true of and only of the points in — a region of the relevant state space then is a proposition that characterizes the equivalence class of paths that cross the boundary of in either direction in the relevant interval of time.
The crux of the matter is getting the proper definition of a tangent functor. There are a few expositions at these places:
- Differential Logic : Introduction
- Differential Propositional Calculus
- Differential Logic and Dynamic Systems
The transformation on logical relations that C.S. Peirce described as “hypostatic abstraction” is key to understanding the realm of abstract objects, in particular, mathematical objects. It is an example of a reflective operation, one that links a relation of a given arity to a relation of the next higher arity, and it appears to be involved in many, if not all, acts of reflective practice, formally, if not always consciously.
Here is a blog post I wrote on the subject.
The concept of representation in general is one of the things that Charles Sanders Peirce developed his theory of triadic sign relations to cover, borrowing the name “semiotics” (in various spellings) from Locke and reconstituting a number of earlier traditions within the framework of mathematical relation theory.
Our usual notions of representation in mathematics are a special case of what he had in mind, since they refer to functions and functions are dyadic relations. They fall very roughly into the category of analogical or iconic representations.
I’m not really sure these comments are getting through, so I’ll break off here and try to post a few links later.