designated truth value, while in both and are (a logical formula is considered a tautology if it evaluates to a designated truth value). In Kleene's logic can be interpreted as being "underdetermined", being neither true nor false, while in Priest's logic can be interpreted as being "overdetermined", being both true and false. does not have any tautologies, while has the same tautologies as classical two-valued logic. Bochvar's internal three-valued logic Another logic is Dmitry Bochvar's "internal" three-valued logic , also called Kleene's weak three-valued logic. Except for negation and biconditional, its truth tables are all different from the above. The intermediate truth value in Bochvar's "internal" logic can be described as "contagious" because it propagates in a formula regardless of the value of any other variable. Belnap logic () Belnap's logic combines and . The overdetermined truth value is here denoted as B and the underdetermined truth value as N. Gödel logics Gk and G∞ In 1932 Gödel defined a family of many-valued logics, with finitely many truth values , for example has the truth values and has . In a similar manner he defined a logic with infinitely many truth values, , in which the truth values are all the real numbers in the interval . The designated truth value in these logics is 1. The conjunction and the disjunction are defined respectively as the minimum and maximum of the operands: Negation and implication are defined as follows: Gödel logics are completely axiomatisable, that is to say it is possible to define a logical calculus in which all tautologies are provable. The implication above is the unique Heyting implication defined by the fact that the suprema and minima operations form a complete lattice with an infinite distributive law, which defines a unique complete Heyting algebra structure on the lattice. Łukasiewicz logics and Implication and negation were defined by Jan Łukasiewicz through the following functions: At first Łukasiewicz used these definitions in 1920 for his three-valued logic , with truth values . In 1922 he developed a logic with infinitely many values , in which the truth values spanned the real numbers in the interval . In both cases the designated truth value was 1. By adopting truth values defined in the same way as for Gödel logics , it is possible to create a finitely-valued family of logics , the abovementioned and the logic , in which the truth