"as is". For example, the law of excluded middle, , and the law of non-contradiction, are not tautologies in Ł3. However, using the operator defined above, it is possible to state tautologies that are their analogues: (law of excluded fourth) (extended contradiction principle). HT logic The logic of here and there (HT, also referred as Smetanov logic SmT or as Gödel G3 logic), introduced by Heyting in 1930 as a model for studying intuitionistic logic, is a three-valued intermediate logic where the third truth value NF (not false) has the semantics of a proposition that can be intuitionistically proven to not be false, but does not have an intuitionistic proof of correctness. It may be defined either by appending one of the two equivalent axioms or equivalently to the axioms of intuitionistic logic, or by explicit truth tables for its operations. In particular, conjunction and disjunction are the same as for Kleene's and Łukasiewicz's logic, while the negation is different. HT logic is the unique coatom in the lattice of intermediate logics. In this sense it may be viewed as the "second strongest" intermediate logic after classical logic. Bochvar logic Ternary Post logic not(a) = (a + 1) mod 3, or not(a) = (a + 1) mod (n), where (n) is the value of a logic Modular algebras Some 3VL modular algebras have been introduced more recently, motivated by circuit problems rather than philosophical issues: Cohn algebra Pradhan algebra Dubrova and Muzio algebra Applications SQL The database structural query language SQL implements ternary logic as a means of handling comparisons with NULL field content. NULL was originally intended to be used as a sentinel value in SQL to represent missing data in a database, i.e. the assumption that an actual value exists, but that the value is not currently recorded in the database. SQL uses a common fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables. In SQL, the intermediate value is intended to be interpreted as UNKNOWN. Explicit comparisons with NULL, including that of another NULL yields UNKNOWN. However this choice of semantics is abandoned for some set operations, e.g. UNION or INTERSECT, where NULLs are treated as equal with each other. Critics assert that this inconsistency deprives SQL of intuitive semantics in its treatment of NULLs. The SQL standard defines an optional feature called F571, which adds some unary operators, among which is IS