Underdeterminacy in Greek participles in terms of their relationship to the rest of the sentence is well known. What has not been studied is the function of certain particles which constrain such relationships and exclude others. The particle kaiper functions in this way, strengthening one assumption while cancelling another (in RT terms). When it is omitted, the relationship, if concessive, must then be the most relevant one and the need for further constraint unnecessary. This paper examines the use of the particle to strengthen a concessive reading of a participle, and to examine some ‘concessive’ participles which lack such a constraint. It focuses particularly on two passages from the letter to the Hebrews.