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The presumption of priestly descent in Judaism is the attribution to a kohen of equivalent position as if there was proven descent from the priestly family of Aaron. This is evidenced not by genealogical records but by de facto priestly behavior as defined in rabbinical halakhic texts.
In the Land of Israel it was the raising up of hands in the priestly benediction, and sharing heave offering at the threshing floor, whereas in Syria and Babylonia the raising up of hands constituted adequate grounds, but not sharing heave offering at the threshing floor as stated in the Jerusalem Talmud Ketubot 2:7 and other texts.
The "presumed kohen" is a rabbinic title which legitimates kohen status to a kohen who—among multiple criteria—exhibits conduct exemplary of and is recognized by his peers and community as such.
The tannaitic rabbi Jose ben Halafta extolled the soundness of the said "presumption" by calling it a basis for the entire halakhic concept of chazakah. It is based on this presumption that all poskim agree, unanimously, to forbid presumptive kohanim from marrying a divorcee. Of note is that from among the opinions of the Acharonim, the mentioned presumption is given the title "a sound presumption".