Quantitative semantics of Signal Temporal Logic (STL) play an important role in both the falsification and control synthesis for dynamical systems by assigning numerical quantities to truth values. Recently, several different quantitative semantics have been proposed, offering better performance in many cases. Yet a general, systematic understanding of the structure and properties of quantitative semantics is missing. In this paper, we develop a general framework to model quantitative semantics. We focus mainly on soundness, which requires that the quantitative semantics of a statement is positive when the statement is true, and negative when the statement is false. This ensures that counterexamples will not be missed during verification. We derive simple, necessary conditions in our framework for soundness. We show how several recently proposed quantitative semantics fit in our framework, and how others do not, typically because they do not strictly satisfy soundness. We implement various quantitative semantics, including existing semantics from literature, in our framework and compare their effectiveness as objective functions for optimization-based falsification on both novel and existing benchmarks.