In a model that has false
represented by 0 and true
represented by 1 the discrepancy corresponds to addition modulo 2 and conjunction corresponds to multiplication; these are the operations of a boolean ring
. We are happy to use quantifier notation for addition and it should be uncontroversial, I think, to use it for addition modulo 2.
In fact, the original poster provides broader reasons for admitting quantifier notation for discrepancy: the operation is symmetric, associative, and has a unit (or neutral) element. It is defined for any finite bag of arguments. I have understood Dijkstra's notation for quantified expressions to apply in precisely that situation.
-Bas Braams (new here and therefore replying a bit late)