Alpha Now, Omega Later • 6

Posted on December 6, 2013 by Jon Awbrey

Re: Alpha Now, Omega Later • Theorems From Physics? • Isomorphism Is Where It’s At

In the late 1970s a number of problems in combinatorics and graph theory that I really wanted to know the answers to had driven me to the desperate measures of trying to write a theorem-proving program to help with the work.  Being familiar with the conceptual efficiencies of Peirce’s logical graphs and inspired by George Spencer Brown’s more recent resurrection of Peirce’s ideas, I naturally turned to those resources for the initial implements of my computational prospecting. The succession of computers and programming languages that I quested with over the years taught me a lot about the things that work and the things that do not. Dissolving for now to the present scene, I will use the next few posts to outline, as succinctly as I can, the basic constructs that developed through that line of inquiry.

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} 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.

See the following article for more information.

To be continued …

This entry was posted in C.S. Peirce, Differential Logic, Dynamical Systems, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Propositional Calculus, Semiotics, Sign Relations, Spencer Brown and tagged , , , , , , , , , , , . Bookmark the permalink.

5 Responses to Alpha Now, Omega Later • 6

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