EXTENSIONS OF KLEENE'S THREE-VALUED LOGIC

Here, we prove that proper extensions of Kleene's three-valued logic K3 form a four-element diamond lattice, the classical logic and the consistent but inferentially inconsistent one, defined by any matrix without distinguished values, being incomparable proper extensions of K3. As a by-product of auxiliary tools elaborated for solving the main problem, we also show that K3 is minimally three-valued and has a unique classical extension, while any three-valued matrices defining K3 are isomorphic, whereas any three-valued expansion of K3 is defined by an expansion of any three-valued matrix defining K3.