Re: Alpha Now, Omega Later

Post   : Alpha Now, Omega Later : 6
URL    : http://inquiryintoinquiry.com/2013/12/06/alpha-now-omega-later-6/
Posted : December 6, 2013 at 3:00 pm
Author : Jon Awbrey

Re: Alpha Now, Omega Later
http://web.archive.org/web/20131117023028/http://permalink.gmane.org/gmane.science.philosophy.peirce/11108

Peircers,

In the late 70s 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 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 in this 
line of inquiry.  (P.S. The text below is formatted much more nicely at the blog post linked above.)

Minimal Negation Operators and Painted Cacti
--------------------------------------------

Let B = {0, 1}.

The mathematical objects of penultimate interest are the
boolean functions f : B^n → B for n ∈ N = {0, 1, 2, 3, …}.

A ''minimal negation operator'' ν_k for k in N is a boolean function
ν_k : B^k → B that is defined as follows:

• ν_0 = 0.

• ν_k (x_1, …, x_k) = 1 if and only if
   exactly one of the arguments x_j is 0.

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

In most contexts ν(x_1, …, x_k) may be written for ν_k (x_1, …, x_k)
since the number of arguments determines the rank of the operator.
In some contexts even the letter ν may be omitted, writing just
the argument list (x_1, …, x_k), in which case it helps to use
a distinctive typeface for the list delimiters, as ❨x_1, …, x_k❩.

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

See the following article for more information:

• Minimal Negation Operators
= http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator

To be continued …

-- 

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