  Validity for this brand of temporal logic can now be defined. A frame \(\langle W, R\rangle\) is a pair consisting of a non-empty set \(W\) (of worlds) and a binary relation \(R\) on \(W\). A model \(\langle F, v\rangle\) consists of a frame \(F\), and a valuation \(v\) that assigns truth values to each atomic sentence at each world in \(W\). Given a model, the values of all complex sentences can be determined using \(({\sim}), (\rightarrow)\), and \((K)\). An argument is \(\bK\)-valid just in case any model whose valuation assigns the premises \(T\) at a world also assigns the conclusion \(T\) at the same world. As the reader may have guessed from our use of ‘\(\bK\)’, it has been shown that the simplest modal logic \(\bK\) is both sound and complete for \(\bK\)-validity.