The "standard" local consistency conditions all require that all partial assignments satisfying a constraint can be extended to another variable in such a way the resulting assignment is consistent; such consistency requires satisfaction of all constraints whose scope is a subset of the assigned variables.
The "standard" local consistency conditions all require that all partial assignments satisfying a constraint can be extended to another variable in such a way the resulting assignment is consistent. Such consistency requires satisfaction of all constraints. All constraints's scope is a subset of the assigned variables.