Fully Evaluated Left-Sequential Logics
This work provides a systematic framework and complete axiomatizations for left-sequential logics, which is of interest to logicians studying side effects and evaluation order in propositional logic.
The paper introduces and axiomatizes a family of fully evaluated left-sequential logics (FELs), ranging from weakest (Free FEL) to strongest (Static FEL), and provides complete axiomatizations for both two-valued and three-valued versions. The strongest three-valued FEL is shown equivalent to Bochvar's strict logic.
We consider a family of two-valued "fully evaluated left-sequential logics" (FELs), of which Free FEL (defined by Staudt in 2012) is most distinguishing (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 (left-sequential propositional expressions). 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 strict logic.