Conditional logic as a short-circuit logic
Jan A. Bergstra, Alban Ponse
TL;DR
This work analyzes conditional logic ($CL$) through the lens of short-circuit logic (SCL), distinguishing two- and three-valued variants and showing how full left-sequential connectives interact with negation to yield Bochvar's $S_3$. It introduces and formalises two-valued ($CL_2$) and three-valued ($CL_3$) axiomatisations, proving completeness for the corresponding two- and three-valued SCLs ($ ext{C$ ext{L}$SCL}_2$ and $ ext{C$ ext{L}$SCL}$) via CL-basic and mem-basic forms. The paper also demonstrates that the original CL axiomatisation is not independent, offering several independent axiomatisations and establishing hierarchy results such as $ extup{$ ext{MSCL}$} ext{ extless } extup{$ ext{C$ ext{L}$SCL}_2$}$ and $ extup{$ ext{MSCL}$}^{ extsf{U}} ext{ extless } extup{$ ext{C$ ext{L}$SCL}$}$. Additionally, it discusses SCLs without constants and the consequences for the commutativity and absorption properties of left-sequential connectives, connecting the two-valued and three-valued theories and outlining directions for future work, including fracterm calculus and higher-order semantics. Overall, the results clarify the structure and independence of CL-style logics within the broader family of SCLs and reveal Bochvar-like behavior emerging from full left-sequential evaluation in $CL_3$.
Abstract
Three-valued conditional logic (CL) is defined by Guzmán and Squier (1990), and based on McCarthy's noncommutative connectives, axiomatises a short-circuit logic (SCL) that defines more identities than three-valued MSCL (Memorising SCL, which also has a two-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. We observe that CL also has a two-valued variant of which 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.