model). However, the converse does not hold generally. There are Kripke incomplete normal modal logics, which is unproblematic, because most of the modal systems studied are complete of classes of frames described by simple conditions. A normal modal logic L corresponds to a class of frames C, if C = Mod(L). In other words, C is the largest class of frames such that L is sound wrt C. It follows that L is Kripke complete if and only if it is complete of its corresponding class. Consider the schema T : . T is valid in any reflexive frame : if , then since w R w. On the other hand, a frame which validates T has to be reflexive: fix w ∈ W, and define satisfaction of a propositional variable p as follows: if and only if w R u. Then , thus by T, which means w R w using the definition of . T corresponds to the class of reflexive Kripke frames. It is often much easier to characterize the corresponding class of L than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose L1 ⊆ L2 are normal modal logics that correspond to the same class of frames, but L1 does not prove all theorems of L2. Then L1 is Kripke incomplete. For example, the schema generates an incomplete logic, as it corresponds to the same class of frames as GL (viz. transitive and converse well-founded frames), but does not prove the GL-tautology . Canonical models For any normal modal logic L, a Kripke model (called the canonical model) can be constructed, which validates precisely the theorems of L, by an adaptation of the standard technique of using maximal consistent sets as models. Canonical Kripke models play a role similar to the Lindenbaum–Tarski algebra construction in algebraic semantics. A set of formulas is L-consistent if no contradiction can be derived from them using the axioms of L, and modus ponens. A maximal L-consistent set (an L-MCS for short) is an L-consistent set which has no proper L-consistent superset. The canonical model of L is a Kripke model , where W is the set of all L-MCS, and the relations R and are as follows: if and only if for every formula , if then , if and only if . The