An axiomatization of Herzberger's 2-dimensional presuppositional semantics

The purpose of this paper* is to axiomatize two 4-valued propositional logics suggested by Herzberger in [l], section VI. They are of philosophical interest because their interpretation makes use of two ideas inspired by Jean Buridan: (1) a proposition may correspond to the world and yet be untrue because it is semantically deviant, and (2) logically valid arguments preserve correspondence with reality, not truth. If the two non-classical truth-values of these systems are identified, the resulting tables for the classical connectives are the weak and strong systems of Kleene. Unlike Kleene's system, the 4-valued ones offer a choice of designated values that renders semantic entailment perfectly classical. Compare Herzberger [2] and Martin [5].