Complex sentence: 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.
Split into simpler sentences: The "standard" local consistency conditions all require that all consistent partial evaluations can be extended to another variable in such a way the resulting assignment is consistent. A partial evaluation is consistent if it satisfies all constraints whose scope is a subset of the assigned variables.