Theme One • A Program Of Inquiry : 13

Re: Laws Of Form Discussions • (1)(2)(3)
Re: Peirce List Discussions • (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 that 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.


This entry was posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Computation, Computational Complexity, Cybernetics, Data Structures, Differential Logic, Form, Formal Languages, Graph Theory, Inquiry, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Learning, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Pragmatics, Programming, Propositional Calculus, Propositional Equation Reasoning Systems, Reasoning, Semantics, Semiotics, Spencer Brown, Syntax, Theorem Proving and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

One Response to Theme One • A Program Of Inquiry : 13

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

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