The pure logic of necessitation

In this paper we discuss the pure logic of necessitation N, a modal logic containing classical propositional calculus, with modus ponens and necessitation as inference rules, but without any axioms for manipulating modalities. We develop a theory of the logic N. We propose a sound and complete Kripke-like semantics for N and build a tableaux system for testing whether a formula is provable from a theory in the logic N. An alternative method to compute modal-free consequences of a finite theory is also given. Our main motivation to consider the logic N comes from the area of nonmonotonic reasoning. The nonmonotonic variant of N seems to be particularly useful in investigations of knowledge sets built when only partial information is available. In particular, this logic N is deeply connected with the default logic. In this paper, we apply our results to problems in nonmonotonic reasoning and we design algorithms for building the nonmonotonic consequence operator associated with N.