Definitional Reasoning in Proof-Theoretic Semantics and the Square of Opposition

Within a framework of reasoning with respect to clausal definitions of atoms, four forms of judgement are distinguished: Direct and indirect assertion, and direct and indirect denial. Whereas direct assertion and direct denial are established by directly applying a definitional clause (“definitional closure”), indirect assertion and indirect denial result from showing that all possible premisses of the opposite judgement can be refuted (“definitional reflection”). The deductive relationships between these four forms of judgement correspond to those represented in the square of opposition, if direct assertion and direct denial are placed in the A and E corners, and indirect assertion and indirect denial in the I and O corners of the square.