On the Meaning of Complementary Systems

1. Complementary systems. We say that two proof systems are complementary iff what can be derived in one system is precisely that which cannot be derived in the other, and vice versa [11]. For instance, classical propositional logic and Łukasiewicz's rejection calculus [5] are complementary. If we focus on decid-able logics, given one system there exist of course many algorithms to compute what is not provable in it, e.g., semantic tableaux or refutation trees. Even so, a formulation in terms of axioms and rules of inference is usually not a trivial task, as these may be of a most non-standard sort. For instance, in complementary systems for classical logic standard substitution rules cannot hold, since non-tautologies may become tautologous upon substitution. Moreover, such systems must be paraconsistent, i.e., such as to allow non-trivial derivation of mutually inconsistent sentences. The study of complementary systems of this sort—we argue—presents several points of interest. For one thing, they yield an exhaustive characterization of a logical theory in purely syntactic terms. This weakens, if one will, the view that some amount of semantic reasoning is required in order to fully dominate a logic. Secondly, the combination of complementary notions of derivability discloses interesting perspectives for metamathematics, for instance in connection with stronger forms of decidability of the sort hitherto investigated under the ru-bric of Ł-decidability. Thirdly, these systems have important bearing on certain fundamental issues in computer science, such as non-monotonic reasoning, the semantics of programming languages, and the logical characterization of complexity classes.