## Alpha Now, Omega Later : 6

### Minimal Negation Operators and Painted Cacti

Let $\mathbb{B} = \{ 0, 1 \}.$

The objects of penultimate interest are the boolean functions $f : \mathbb{B}^n \to \mathbb{B}$ for $n \in \mathbb{N}.$

A minimal negation operator $\nu_k$ for $k \in \mathbb{N}$ is a boolean function $\nu_k : \mathbb{B}^k \to \mathbb{B}$ that is defined as follows:

• $\nu_0 = 0.$
• $\nu_k (x_1, \ldots, x_k) = 1$ if and only if exactly one of the arguments $x_j$ equals $0.$

The first few of these operators are already enough to generate all boolean functions $f : \mathbb{B}^n \to \mathbb{B}$ via functional composition but the rest of the family is worth keeping around for many practical purposes.

In most contexts $\nu (x_1, \ldots, x_k)$ may be written for $\nu_k (x_1, \ldots, x_k)$ since the number of arguments determines the rank of the operator. In some contexts even the letter $\nu$ may be omitted, writing just the argument list $(x_1, \ldots, x_k),$ in which case it helps to use a distinctive typeface for the list delimiters, as $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}.$

A logical conjunction of $k$ arguments can be expressed in terms of minimal negation operators as $\nu_{k+1} (x_1, x_2, \ldots, x_{k-1}, x_k, 0)$ and this is conveniently abbreviated as a concatenation of arguments $x_1 x_2 \ldots x_{k-1} x_k.$

To be continued …

## Alpha Now, Omega Later : 5

Re: Isomorphism Is Where It’s At • “Are there more good cases of isomorphism to study?”

Just off the top of my head, as Data says, a couple of examples come to mind.

Sign Relations. In computational settings, a sign relation $L$ is a triadic relation of the form $L \subseteq O \times S \times I,$ where $O$ is a set of formal objects under consideration and $S$ and $I$ are two formal languages used to denote those objects. It is common practice to cut one’s teeth on the special case $S = I$ before moving on to more solid diets.

Cactus Graphs. In particular, a variant of cactus graphs known (by me, anyway) as painted and rooted cacti (PARCs) affords us with a very efficient graphical syntax for propositional calculus.

## Alpha Now, Omega Later : 4

It is critically important to distinguish between the objective landscape, the boolean functions as mathematical objects, and the syntactic landscape, the particular formal language we are using as a propositional calculus to denote and compute with those objects. If we do hill-climbing, we must keep our feet on the objective territory, however much we rely on syntactic maps to narrate the travelogue. (Many will be thinking of manifolds here.) The object domain has a fixed structure but the conceptual clarity and computational efficiency of propositional calculi can very likely be improved indefinitely.

## Alpha Now, Omega Later : 3

Bits of Synchronicity …

What kind of information process is scientific inquiry?

What kinds of information process are involved in the various types of inference — abductive, deductive, inductive — that go to make up scientific inquiry?

What kinds of information process are computation and proof?

Many types of deductive inference, including many kinds of computation and proof, don’t really change our state of information so much as increase the clarity of that information. Do we have any way to quantify clarity in the way we define measures of information?

## Alpha Now, Omega Later : 2

Peircers,

It’s been a while since I threaded this thread — and then there were all the delightful distractions of the holiday convergence — so let me refresh my memory as to what drew me back to these environs.

I’m still in the middle of trying to catch up on some long put-off work, but recent discussions of logical graphs and physics and the like on the list have bestirred me from my grindstone long enough to pass on a few links to the things I’ve been doing along those lines. This is all “Alpha” as far as Peirce’s graphology goes, but one of the things we’ve learned in recent decades from computational complexity theory is just how key a role problems like propositional calculus play in solving many other problems of practical interest, so I won’t make any further apology for focusing attention on this “zeroth order” level. I don’t have much to say about physics per se but if we generalize our concept of dynamics and speak of systems theory as a study of media and populations that move through their state spaces over spans of time, then I think it is useful to take up that perspective on the time evolution of logical media informed by logical signs.

Good — logic and time, the time evolution of inquiry driven systems. I’ll do my best to stay focused on that interplay of subjects.

Let me start with a different set of articles this time. You will notice a lot of redundancy among these articles, as I’ve written many invitations to the subject over the years. Some of these were originally written for folks who were already familiar with Spencer-Brown’s Laws of Form, so I jumped right in with the formal equations that would have been recognizable to them.

Regards,

Jon

## Objects, Models, Theories : 4

I need to stay with this problem a while …

### What are objects, models, theories, and how do they relate to one another?

In contemplating this problem I always find it helpful to ruminate on the diagram shown above — I might even call it a mandala for its wealth of symbolic features and its aid in organizing the pro-&-con-fusion of mental impressions.

Here is the corresponding text from Aristotle and the context that leads on to Peirce:

## Objects, Models, Theories : 3

Re: Tom Gollier

Tom,

Here my task is to build bridges between several different classical and contemporary uses of the word “model”, so I don’t have the luxury of complete control over the words in play but have to start from the customary senses in the various communities of interpretation. Of course I’m slyly working from a sign-relational backdrop, but I have to be sleight-handed about that and not hit people over the head with it.

You can probably guess that I’m using “object” to cover sign-relational objects, and “theories” are clearly syntacked together from complexes of sign-relational signs, so all we have left to pin down is where the various kinds of “model” sit at the table set with the labels of Object, Sign, Interpretant.

In its theoretical sense, a model of a theory is anything the theory is true of, anything that satisfies the theory. In that sense, a model is very like an object. It is whatever the theory is talking about. In the order of nature, indeed, models come before theories. But there is another order, the order of art, and one may construct artificial models out of almost any stuff, even the stuff of signs. So you see the kind of wiggle room we have to work with.

Things are easier outside of logic, in applied mathematics and the special sciences, where models are just things like analogues, icons, simulations, and similar representations of objects. But that makes them objects serving as signs of other objects, and so you may find some semiotic subtlety lurking there.

Well, enough for now …

Jon