This is an initial systematic study of the properties
of negation from the point of view of abstract deductive systems. A unifying
framework of multiple-conclusion consequence relations is adopted so as to
allow us to explore symmetry in exposing and matching a great number of positive
contextual sub-classical rules involving this logical constant —among others,
well-known forms of proof by cases, consequentia mirabilis and reductio
ad absurdum. Finer definitions of paraconsistency and the dual paracompleteness
can thus be formulated, allowing for pseudo-scotus and ex contradictione
to be differentiated and for a comprehensive version of the Principle of
Non-Triviality to be presented. A final proposal is made to the effect that
—pure positive rules involving negation being often fallible— a characterization
of what most negations in the literature have in common should rather involve,
in fact, a reduced set of negative rules.
Keywords: negation, abstract deductive systems,
multiple-conclusion logic, paraconsistency
1991 MSC: 03B22, 03B53, 03B60.