The most fundamental concept in cybernetics is that of “difference”, either that two things are recognisably different or that one thing has changed with time.
The times are rife with distraction, so let’s pause and retrace how we got to this place.
Our last reading in Cybernetics brought us in sight of a convergence or complementarity between the triadic relations in Peirce’s semiotics and those in Ashby’s regulator games. There’s a lot more to explore in that direction and I plan to get back to it soon.
The two threads intertwined here, Cybernetics and Differential Logic, both spun off a thread on Pragmatic Truth, asking what theories of truth are compatible with Peircean disciplines of pragmatic thinking. That’s a topic with a tangled history but the latest local tangle is documented in the following posts and excerpts.
Pragmatic Theory Of Truth • 13
Pragmatic inquiry into a candidate concept of truth would begin by applying the pragmatic maxim to clarify the concept as far as possible and a pragmatic definition of truth, should any result, would find its place within Peirce’s theory of logic as formal semiotics, in other words, stated in terms of a formal theory of triadic sign relations.
Pragmatic Theory Of Truth • 14
There are many conceptions of truth — linguistic, model-theoretic, proof-theoretic — for the moment I’m focused on cybernetics, systems, and experimental sciences and this is where the pragmatic conception of truth fits what we naturally do in those sciences remarkably well.
The main thing in those activities is the relationship among symbol systems, the world, and our actions, whether in thought, among ourselves, or between ourselves and the world. So the notion of truth we want here is predicated on three dimensions: the patch of the world we are dealing with in a given application, the systems of signs we are using to describe that domain, and the transformations of signs we find of good service in bearing information about that piece of the world.
Pragmatic Theory Of Truth • 18
We do not live in axiom systems. We do not live encased in languages, formal or natural. There is no reason to think we will ever have exact and exhaustive theories of what’s out there, and the truth, as we know, is “out there”. Peirce understood there are more truths in mathematics than are dreamt of in logic and Gödel’s realism should have put the last nail in the coffin of logicism, but some ways of thinking just never get a clue.
That brings us to the question —
- What are formalisms and all their embodiments in brains and computers good for?
For that I’ll turn to cybernetics …
Survey of Cybernetics
The Survey linked above recaps the reading of Ashby’s Cybernetics up to the present date.
Meanwhile, the inquiry into Pragmatic Truth branched off at another point when a question from Stephen Paul King demanded an answer in terms of Differential Logic. That point of departure is documented in the following post.
Differential Logic • Comment 4
This updates the state of the threads linking pragmatic truth, cybernetics, and differential logic. Disentangling them to any large extent has always been difficult if not impossible, at least for me.
Resources
- 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
cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science
supplies the groundwork for any number of further extensions, beginning with the first order differential extension 
The construction of 
can be described in the following stages.
is extended by a first order differential alphabet 
resulting in a first order extended alphabet 
defined as follows.
is extended by a first order differential basis 
resulting in a first order extended basis 
defined as follows.
is extended by a first order differential space or tangent space 
at each point of 
resulting in a first order extended space or tangent bundle space 
defined as follows.
![A^\bullet = [ a_1, \ldots, a_n ]](https://s0.wp.com/latex.php?latex=A%5E%5Cbullet+%3D+%5B+a_1%2C+%5Cldots%2C+a_n+%5D&bg=ffffff&fg=000000&s=0&c=20201002)
is extended by a first order differential universe or tangent universe ![\mathrm{d}A^\bullet = [ \mathrm{d}a_1, \ldots, \mathrm{d}a_n ]](https://s0.wp.com/latex.php?latex=%5Cmathrm%7Bd%7DA%5E%5Cbullet+%3D+%5B+%5Cmathrm%7Bd%7Da_1%2C+%5Cldots%2C+%5Cmathrm%7Bd%7Da_n+%5D&bg=ffffff&fg=000000&s=0&c=20201002)
at each point of 
resulting in a first order extended universe or tangent bundle universe ![\mathrm{E}A^\bullet ~=~ [ \mathrm{E}\mathcal{A} ] ~=~ [ \mathcal{A} ~\cup~ \mathrm{d}\mathcal{A} ] ~=~ [ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n ].](https://s0.wp.com/latex.php?latex=%5Cmathrm%7BE%7DA%5E%5Cbullet+%7E%3D%7E+%5B+%5Cmathrm%7BE%7D%5Cmathcal%7BA%7D+%5D+%7E%3D%7E+%5B+%5Cmathcal%7BA%7D+%7E%5Ccup%7E+%5Cmathrm%7Bd%7D%5Cmathcal%7BA%7D+%5D+%7E%3D%7E+%5B+a_1%2C+%5Cldots%2C+a_n%2C+%5Cmathrm%7Bd%7Da_1%2C+%5Cldots%2C+%5Cmathrm%7Bd%7Da_n+%5D.&bg=ffffff&fg=000000&s=0&c=20201002)
![[ \mathbb{B}^n \times \mathbb{D}^n ] ~=~ (\mathbb{B}^n \times \mathbb{D}^n\ +\!\!\to \mathbb{B}) ~=~ (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).](https://s0.wp.com/latex.php?latex=%5B+%5Cmathbb%7BB%7D%5En+%5Ctimes+%5Cmathbb%7BD%7D%5En+%5D+%7E%3D%7E+%28%5Cmathbb%7BB%7D%5En+%5Ctimes+%5Cmathbb%7BD%7D%5En%5C+%2B%5C%21%5C%21%5Cto+%5Cmathbb%7BB%7D%29+%7E%3D%7E+%28%5Cmathbb%7BB%7D%5En+%5Ctimes+%5Cmathbb%7BD%7D%5En%2C+%5Cmathbb%7BB%7D%5En+%5Ctimes+%5Cmathbb%7BD%7D%5En+%5Cto+%5Cmathbb%7BB%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002)
we arrive at the foothills of differential logic.
singular propositions are those singling out the minimal distinct regions of the universe, represented by single cells of the corresponding venn diagram.
there are 
singular propositions. Their venn diagrams are shown in Figure 10.
and identical with the positive proposition of rank 3.
respectively.
function, which may be expressed by the form 
or by a simple 
Inquiry Into Inquiry 