TITLE: ``Disjunctive Logic and Semantics of Disjunctive Databases and Logic Programs''

ABSTRACT:

In common sense reasoning with incomplete knowledge/data bases, we make conclusions by default. However, we observe that when the negation-by-default operator ``not'' is defined as ``not provable'', the database {a or b, not a, not b} should be consistent because a being not provable and b being not provable does not imply (a or b) being not provable. Such an observation is significant for non-monotonic reasoning, but none of the major current non-monotonic formalisms are able to support it because they are all based on classical first-order logic in which (not a) and (not b) implies not(a or b). In this paper, we develop a new first-order logic (``Disjunctive Logic'') that fully complies with this observation. Based on such a logic we define (disjunctive) databases and (generalized disjunctive) logic programs and establish intended semantics for them. A new closed world rule (``Consistent Closed World Assumption'') is introduced that consistently negates all unprovable ground positive clauses in databases and all underivable ground positive clauses in logic programs. Our non-monotonic theory is able to formalize and solve some paradoxical problems, such as the lottery paradox and the bird-flying paradox.

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