Conditional logic as a short-circuit logic

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.

20 pages, 4 tables. Differences with v2: 1) The two-valued variant of CL does not come from Guzmán and Squier's paper, but is an observation of ours. 2) A typo in Definition 4.1 has been corrected (U is also exported). 3) A typo in Theorem 5.1.(iii) has been corrected. 4) The axiomatisation of MSCL_0 (p.18) has been described more clearly