One of the insights coming out of Peirce’s logical work is the fact that negative operations are more powerful than positive operations in the sense that negative operations can generate all possible operations while positive operations by themselves do not suffice. This is epitomized by his discovery of the amphecks as sole sufficient operators for propositional logic.
The propositional logic algorithm I wrote for my Theme One Program turns this principle to good effect in two ways:
- The graph-theoretic syntax is based on a graph-theoretic operator, a type of controlled negation called the minimal negation operator, that generalizes Peirce’s graph-theoretic operator for negation.
- It turns out that recognizing contradictions quickly makes for a high degree of efficiency in finding the “models” or satisfying interpretations of a propositional formula.
Relations of contradiction are also critical in statistical inference, but I’ll need to save that for another day.