  We consider an extension of first-order logic with a recursion operator that
corresponds to allowing formulas to refer to themselves. We investigate the
obtained language under two different systems of semantics, thereby obtaining
two closely related but different logics. We provide a natural deduction system
that is complete for validities for both of these logics, and we also
investigate a range of related basic decision problems. For example, the
validity problems of the two-variable fragments of the logics are shown
coNexpTime-complete, which is in stark contrast with the high undecidability of
two-variable logic extended with least fixed points. We also argue for the
naturalness and benefits of the investigated approach to recursion and
self-reference by, for example, relating the new logics to Lindstrom's Second
Theorem.
