A modality for recursion

We propose a modal logic that enables us to handle self-referential formulae, including ones with negative self-references, which on one hand, would introduce a logical contradiction, namely Russell's paradox, in the conventional setting, while on the other hand, are necessary to capture a certain class of programs such as fixed point combinators and objects with so-called binary methods in object oriented programming. Our logic provides a basis for axiomatic semantics of such a wider range of programs and a new framework for natural construction of recursive programs in the proofs-as-programs paradigm.