Differential Logic • 2

Cactus Language for Propositional Logic

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of boolean-valued functions and elementary logical propositions.  One very efficient calculus on both conceptual and computational grounds is based on just two types of logical connectives, both of variable $k$-ary scope.  The syntactic formulas of this calculus map into a family of graph-theoretic structures called “painted and rooted cacti” which lend visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence of propositional expressions, written $\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}$ and meaning exactly one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is false, in short, their minimal negation is true.  An expression of this form maps into a cactus structure called a lobe, in this case, “painted” with the colors $e_1, e_2, \ldots, e_{k-1}, e_k$ as shown below.

The second kind of connective is a concatenated sequence of propositional expressions, written $e_1\ e_2\ \ldots\ e_{k-1}\ e_k$ and meaning all the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ are true, in short, their logical conjunction is true.  An expression of this form maps into a cactus structure called a node, in this case, “painted” with the colors $e_1, e_2, \ldots, e_{k-1}, e_k$ as shown below.

All other propositional connectives can be obtained through combinations of these two forms.  As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it’s convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms.  While working with expressions solely in propositional calculus, it’s easiest to use plain parentheses for logical connectives.  In contexts where ordinary parentheses are needed for other purposes an alternate typeface $\texttt{(} \ldots \texttt{)}$ may be used for the logical operators.

Introduction

Differential logic is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.  To the extent a logical inquiry makes use of a formal system, its differential component treats the principles governing the use of a differential logical calculus, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by differential propositional calculi.  A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.  Such a calculus augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Differential Logic • Overview

The previous series of blog posts on Differential Propositional Calculus brought us to the threshold of the subject without quite stepping over but I wanted to lay out the necessary ingredients in the most concrete, intuitive, and visual way possible before taking up the abstract forms.

One of my readers on Facebook told me “venn diagrams are obsolete” and of course we all know they become unwieldy as our universes of discourse expand beyond four or five dimensions.  Indeed, one of the first lessons I learned when I set about implementing Peirce’s graphs and Spencer Brown’s forms on the computer is that 2-dimensional representations of logic are a death trap on numerous conceptual and computational counts.  Still, venn diagrams do us good service in visualizing the relationships among extensional, functional, and intensional aspects of logic.  A facility with those connections is critical to the computational applications and statistical generalizations of propositional logic commonly used in mathematical and empirical practice.

At any rate, intrepid readers will have provisioned their visual imaginations fully enough at this point to pick their way through the cactus patch ahead.  The outline below links to my last, best introduction to Differential Logic, which I’ll be working to improve as I serialize it to this blog.

Document History

Differential Propositional Calculus • Discussion 2

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.

W. Ross Ashby • An Introduction to Cybernetics

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

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science

Survey of Cybernetics • 1

Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation (αρετης κυβερνητικης), do you perceive what must happen to him and his fellow sailors?

Plato • Alcibiades • 135 A

This is a Survey of blog and wiki posts relating to Cybernetics.

Ashby’s Introduction to Cybernetics

• Chapter 11 • Requisite Variety

Abduction, Deduction, Induction, Analogy, Inquiry : 29

Questions about Abduction in AI and Computer Science raised in the Ontolog Forum prompted me to look up previous discussions tracing the integral relationship among information, inquiry, and the three types of inference.  Here’s a sample of links.

Differential Propositional Calculus • Discussion 1

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.

W. Ross Ashby • An Introduction to Cybernetics

KK:
To me, differences are the result of drawing distinctions.  They don’t exist unless you actively draw them.  So, the act of drawing distinctions is more fundamental than the differences thereby created.

I often return to that line from Ashby.  This time I thought it made an apt segue from the scene of propositional calculus, where universes of discourse are ruled by collections of distinctive features, to the differential extension of propositional calculus, which enables us to describe trajectories within and transformations between our logical universes.

So I agree with Klaus Krippendorff about “which came first”, the distinctions drawn or the states distinguished in space or time.  The primitive character of distinctions is especially salient in this setting since our formalism for propositional calculus is built on the forms of distinction pioneered by C.S. Peirce and augmented by George Spencer Brown.

Resources

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science