Interpolation and Three-valued Logics

We consider propositional logic. Three-valued logics are old: the first one is Lukasiewicz three valued logic from 1920 [8]. Gödel in [5] from 1932 studied a hierarchy of finite-valued logics, containing Gödel three-valued logic. Our main interest pays to Kleene three-valued logic [6]. Other threevalued logics will not be considered here. Let us agree that the three truth values are 0, 1 2 , 1 in the natural ordering. All three logics have connectives ∧,∨ interpreted as minimum and maximum; Kleene and Lukasiewicz have Lukasiewicz negation ¬ L, whose truth function is 1−x involutive negation. Lukasiewicz has his implication → L (truth function min(1, 1 − x + y)); it is the residuum of strong Lukasiewicz conjunction max(0, x + y − 1). Kleene’s implication →K may be omitted since it is definable as ¬Lx ∨ y. Gödel’s implication →G has the truth function equal 1 if x ≤ y and equal y otherwise; Gödel’s negation is ¬G0 = 1, ¬Gx = 0 otherwise. We shall not introduce truth constants (⊥,×,>) for our truth values. Call the investigated logics K3, L3, G3.