Set-Oriented Logical Connectives: Syntax and Semantics

Of the common commutative binary logical connectives, only and and ormay be used as operators that take arbitrary numbers of arguments with order and multiplicity being irrelevant, that is, as connectives that take sets of arguments. This is especially evident in the Common Logic Interchange Format, in which it is easy for operators to be given arbitrary numbers of arguments. The reason is that and and or are associative and idempotent, as well as commutative. We extend the ability of taking sets of arguments to the other common commutative connectives by defining generalized versions of nand, nor, xor, and iff, as well as the additional, parameterized connectives andor and thresh. We prove that andor is expressively complete—all the other connectives may be considered abbreviations of it.