 This semantics generalizes the possible worlds semantics for normal modal logics. Therefore, it is important to note that the situations are not maximally consistent theories, rather they are theories possessing the property that for any pair of formulas they contain a formula that implies both of them. Equivalently, the situations may be taken to be dual ideals on the Lindenbaum algebra of \(\text{CL}_\triangleright\). These situations are typically consistent in the sense that they do not contain all the terms in all but one case. (The notion of negation consistency, of course, cannot be defined for \(\text{CL}_\triangleright\) or for \(\text{CL}_=\).)