Conditional Logic as a Short-Circuit Logic

Three-valued conditional logic (CL), defined by Guzm ́an and Squier (1990) and based on McCarthy’s noncommutative connectives, axiomatises a short-circuit logic (SCL), that is, a logic that prescribes shortcircuit evaluation of conjunction and disjunction. CL defines more identities than three-valued MSCL (Memorising SCL, which also has a two-valued variant). This follows from the fact that the definable connective that prescribes full left-sequential conjunction is commutative in CL. We observe that CL also has a two-valued variant of which the full left-sequential connectives and negation define a commutative logic that is weaker than propositional logic because the absorption laws do not hold. Next, we show that the original, equational axiomatisation of CL is not independent and give several alternative, independent axiomatisations. Finally, we show that in CL, the full left-sequential connectives and negation define Bochvar’s three-valued logic. The paper ends with an appendix on the use of Prover9 and Mace4.