Assume we have a structure \(\langle\bD, \bR_1, \ldots, \bR_n\rangle\), and suppose we have an LPCR language associated with it. A valuation \(v\) in this structure is a mapping from individual variables to members of \(\bD\) and from auxiliary relation symbols to partial relations on \(\bD\). We would like to associate with each valuation \(v\) a mapping \(T_v\) from formulas of LPCR to truth values but since things like the liar sentence are formulable, \(T_v\) must be a partial function, and so we must be careful even about familiar things like propositional connectives. Various three valued logics have been developed; perhaps the most common is Kleene’s strong three-valued logic, motivated by recursion theory and familiar from much work on the Theory of Truth. The following table says how connectives and quantifiers behave. Cases that are not explicitly covered are understood to be those for which a truth valuation is left undefined. (For instance, if the truth value of \(X\) is undefined, the same is the case for \(\lnot X\).)