Conditional logic as a short-circuit logic

Both two-valued and three-valued conditional logic (CL), defined by Guzm\'an and Squier (1990) and based on McCarthy's non-commutative connectives, axiomatise a short-circuit logic (SCL) that defines more identities than MSCL (Memorising SCL), which also has a two- and a three-valued variant. This follows from the fact that the definable connective that prescribes full left-sequential conjunction is commutative in CL. We show that in CL, the full left-sequential connectives and negation define Bochvar's three-valued strict logic. In two-valued CL, 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.