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