three cases. If numeric values, e.g. balanced ternary values, are assigned to false, unknown and true such that false is less than unknown and unknown is less than true, then A AND B AND C... = MIN(A, B, C ...) and A OR B OR C ... = MAX(A, B, C...). Material implication for Kleene logic can be defined as: , and its truth table is which differs from that for Łukasiewicz logic (described below). Kleene logic has no tautologies (valid formulas) because whenever all of the atomic components of a well-formed formula are assigned the value Unknown, the formula itself must also have the value Unknown. (And the only designated truth value for Kleene logic is True.) However, the lack of valid formulas does not mean that it lacks valid arguments and/or inference rules. An argument is semantically valid in Kleene logic if, whenever (for any interpretation/model) all of its premises are True, the conclusion must also be True. (Note that the Logic of Paradox (LP) has the same truth tables as Kleene logic, but it has two designated truth values instead of one; these are: True and Both (the analogue of Unknown), so that LP does have tautologies but it has fewer valid inference rules). Łukasiewicz logic The Łukasiewicz Ł3 has the same tables for AND, OR, and NOT as the Kleene logic given above, but differs in its definition of implication in that "unknown implies unknown" is true. This section follows the presentation from Malinowski's chapter of the Handbook of the History of Logic, vol 8. Material implication for Łukasiewicz logic truth table is In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as: It is also possible to derive a few other useful unary operators (first derived by Tarski in 1921): They have the following truth tables: M is read as "it is not false that..." or in the (unsuccessful) Tarski–Łukasiewicz attempt to axiomatize modal logic using a three-valued logic, "it is possible that..." L is read "it is true that..." or "it is necessary that..." Finally I is read "it is unknown that..." or "it is contingent that..." In Łukasiewicz's Ł3 the designated value is True, meaning that only a proposition having this value everywhere is considered a tautology. For example, and are tautologies in Ł3 and also in classical logic. Not all tautologies of classical logic lift to Ł3