In this talk, Porter will frame a number of definitions of algorithmic randomness as instances of what he refers to as the logical approach to randomness.  In order to better understand this logical approach, he will contrast it with one of the standard approaches to defining randomness in classical mathematics, which he calls the valuative approach to randomness.  He will focus specifically on two potential problems faced by the logical approach that threaten to trivialize this approach to defining randomness. He will address these two potential problems, arguing that the logical approach fills an important role, namely, that of supplementing the valuative approach by yielding additional information about classically random objects, which is unearthed when we bring the tools of mathematical logic to bear on the study of randomness.