Knots and blanks: The pragmatic foundation of logical principles

Logical principles, in particular the law of noncontradiction and the law of exclusion of middle term, play different roles at different levels of discourse: valid formulae in an axiomatic calculus, methodological requirements (of consistency and completeness) for formalized systems. When postulated as formal laws, —pvp and —(p·—p), they are totally interdefinable and equivalent as well (DeMorgan's transformations are proof of this). If postulated as methodological requirements, the principles are not equivalent, although they could still be said in some sense to be interdefinable (the existence of consistent yet incomplete systems shows that the requirements are not equivalent; still, completeness of a system can be defined in terms of consistency of another system which keeps a definite relationship with the first one).There exists a third level of discourse: scientific praxis. At this level, the principles come even farther apart: they neither have the same logical value nor is one definable in terms of the other. However, they keep a family resemblance which justifies our dealing with them jointly. Let us call the principles at this level pragmatic imperatives. They deal with paradoxes, which are of two types: knots (conflicts) and blanks (gaps in the scientific pattern). The left-hand pragmatic imperative says: Be intolerant with knots, try to remove (dissolve) them. The right-hand pragmatic imperative says: Try to remove (fill) all blanks. The knot-removing and the blank-dissolving imperatives are prior to and more important than the laws of noncontradiction and excluded middle and the requirements of consistency and completeness. Logical principles are not prime categories: pragmatic imperatives are primordial.