Theme One • A Program Of Inquiry 13

Re: Peirce List • (1)(2)(3)

Logical Cacti (cont.)

The abstract character of the cactus language relative to its logical interpretations makes it possible to give abstract rules of equivalence for transforming one cactus into another which partition the space of cacti into formal equivalence classes.  These transformation rules and the resulting equivalence classes are “purely formal” in the sense of being indifferent to the logical interpretation, entitative or existential, one happens to choose.

Two definitions are useful here:

  • A reduction is an equivalence transformation that applies in the direction of decreasing graphical complexity.
  • A basic reduction is a reduction that applies to a basic connective, either a node connective or a lobe connective.

The two kinds of basic reductions are described as follows:

  • A node reduction is permitted if and only if every component cactus joined to a node itself reduces to a node.


    Node Reduction

  • A lobe reduction is permitted if and only if exactly one component cactus listed in a lobe reduces to an edge.


    Lobe Reduction

That is roughly the gist of the rules.  More formal definitions can wait for the day when we have to explain all this to a computer.

Resources

This entry was posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Cognition, Computation, Constraint Satisfaction Problems, Data Structures, Differential Logic, Equational Inference, Formal Languages, Graph Theory, Inquiry Driven Systems, Laws of Form, Learning Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Semiotics, Spencer Brown, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

3 Responses to Theme One • A Program Of Inquiry 13

  1. Pingback: Survey of Theme One Program • 2 | Inquiry Into Inquiry

  2. Pingback: Survey of Theme One Program • 2 | Inquiry Into Inquiry

  3. Pingback: Survey of Theme One Program • 3 | Inquiry Into Inquiry

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.