We define a proper inductive semantics for LTL which includes those finite partial traces for which a definitive answer can be given: the “definitive prefixes” of the property. Sets of definitive prefixes are called definitive sets. Defining a semantics for LTL in terms of definitive sets gives us something equivalent to LTL3, a formulation popular in runtime verification. Our theory of definitive sets is sufficient to give a neat characterisation of (co-)safety and (co-)liveness properties, allowing us to straightforwardly prove Alpern and Schneider’s result that every property is the intersection of a safety and liveness property by simpler means.