 In both strong and weak Kleene logics then, \((\phi \vee \neg \phi)\) never has value \(0\), but it doesn’t always have value \(1\) either: if \(\phi\) is assigned value #, so is its negation, but then \((\phi \vee \neg \phi)\) will also have value # on both the strong and weak interpretations of \(\vee\). Thus \((\phi \vee \neg \phi)\) fails to be logically valid in these systems (unless both \(1\) and # are taken to be designated values, as in Priest’s LP, which validates LEM. Priest’s notion of validity is defined later in this section). Other many-valued logics, such as the Łukasiewicz continuum-valued logic, also fail to validate LEM, for similar reasons. In continuum-valued logic, the set of truth values is the set of real numbers between \(0\) and \(1\), where \(1\) stands for completely true, \(0\) for completely false, \(0,5\) for half true, etc. Disjunction and negation are analyzed as \(F_{\vee}(x, y)=max(x,y) \) and \(F_{\neg} (x)= 1-x\) respectively, where \(max\) means “the maximum of”. But then since \(max(x, 1-x)\) need not be equal to \(1\), \((\phi \vee \neg \phi)\) can fail to be completely true in this system.