Theme One • A Program Of Inquiry : 13


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.

  [see figure, attached]

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

  [see figure, attached]

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.


• Theme One Program • Documentation

• Theme One Program • Exposition

• Theme One Program • Expository Note 14

• Theme One Program • User Guide


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