We consider a family of two-valued 'fully evaluated left-sequential logics' (FELs), of which Free FEL (defined by Staudt in 2012) is weakest and immune to atomic side effects. Next is Memorising FEL, in which evaluations of subexpressions are memorised. The following stronger logic is Conditional FEL (inspired by Guzmán and Squier's Conditional logic, 1990). The strongest FEL is static FEL, a sequential version of propositional logic. We use evaluation trees as a simple, intuitive semantics and provide complete axiomatisations for closed terms. For each FEL except Static FEL, we also define its three-valued version, with a constant U for 'undefinedness' and again provide complete, independent axiomatisations, each one containing two additional axioms for U on top of the axiomatisations of the two-valued case. In this setting, the strongest FEL is equivalent to Bochvar's logic. Finally, we discuss how the family of FELs is related to the previously defined family of 'short-circuit logics'.
Ponse et al. (2025) studied this question.