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.

翻译：由Guzmán和Squier（1990）基于McCarthy的非交换连接词定义的两值与三值条件逻辑（CL），公理化了一种短路逻辑（SCL），该逻辑定义了比MSCL（记忆型SCL）更多的恒等式，而MSCL也有两值与三值变体。这一结论源于以下事实：在CL中，可定义的、规定完全左序合取的连接词是可交换的。我们证明，在CL中，完全左序连接词与否定定义了Bochvar的三值严格逻辑。在两值CL中，完全左序连接词与否定定义了一个比命题逻辑更弱的可交换逻辑，因为吸收律不成立。接着，我们展示了CL的原始等式公理化并非独立，并给出了若干替代的独立公理化。