  Integrated information theory (IIT) starts from consciousness itself and
identifies a set of properties (axioms) that are true of every conceivable
experience. The axioms are translated into a set of postulates about the
substrate of consciousness (called a complex), which are then used to formulate
a mathematical framework for assessing both the quality and quantity of
experience. The explanatory identity proposed by IIT is that an experience is
identical to the cause-effect structure unfolded from a maximally irreducible
substrate (a $\Phi$-structure). In this work we introduce a definition for the
integrated information of a system ($\varphi_s$) that is based on the
existence, intrinsicality, information, and integration postulates of IIT. We
explore how notions of determinism, degeneracy, and fault lines in the
connectivity impact system integrated information. We then demonstrate how the
proposed measure identifies complexes as systems whose $\varphi_s$ is greater
than the $\varphi_s$ of any overlapping candidate systems.
