Summary: We consider the problem of defining a logic that captures exactly the polynomial time computable properties of finite structures, by extending first-order logic (FO) and fixed-point logic (FP) by means of Lindström quantifiers. In particular, we define infinite uniform sequences of quantifiers and show that these correspond to a natural notion of logical reducibility. We show that if there is any logic that captures the complexity class PTIME, in the sense of a recursive enumeration of PTIME properties, then there is one that is an extension of FO by a uniform sequence of quantifiers. This is established through a general result linking the existence of complete problems for a complexity class to the existence of recursive index sets for the class, for a wide variety of complexity classes.