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Short-circuit evaluation denotes the semantics of propositional connectivesin which the second argument is evaluated only if the first argument does notsuffice to determine the value of the expression. In programming, short-circuitevaluation is widely used, with left-sequential conjunction and disjunction asprimitive connectives. We consider left-sequential, non-commutativepropositional logic, also known as MSCL (memorising short-circuit logic), andstart from a previously published, equational axiomatisation. First, we extendthis logic with a left-sequential version of the biconditional connective,which allows for an elegant axiomatisation of MSCL. Next, we consider aleft-sequential version of the NAND operator (the Sheffer stroke) and againgive a complete, equational axiomatisation of the corresponding variant ofMSCL. Finally, we consider these logical systems in a three-valued setting witha constant for `undefined', and again provide completeness results.

Non-commutative propositional logic with short-circuited biconditional and NAND