Hartry Field and Paracompleteness

Abstract

The paracomplete solution to the semantic paradoxes championed by Hartry Field might seem to be more promising than its paraconsistent competitor. Embracing some logical contradictions is more initially implausible than rejecting both disjuncts of some instances of the LEM. On closer investigation, however, this intuitive disadvantage disappears. The LEM turns out to be as important to our reasoning practices as the LNC. The importance is simply less obvious because implicit appeals to the LEM tend to have in the background of arguments rather than at flashy and obvious points such as the discovery of a contradiction in an opponent’s views. Moreover, while incompleteness is normally far less of a strike against a theory than inconsistency, paracompleteness is a very different beast than mere incompleteness. In fact, to even make sense of what it means to reject propositions without accepting their negations we have to abandon the probability calculus that is essential to our understanding of subjects ranging from weather prediction to the mysteries of quantum mechanics.