Yadayim, Chapter 2, Mishnah 3. This chapter has been occupied with the water used for washing the hands: how much of it is required, whether some task had already been performed with it, and whether the water itself is tahor. Our mishnah takes up the obvious follow-up question: what does a person do when he cannot answer any of those questions with certainty?
All three receive one and the same verdict: "sfeikan tahor". The uncertainty is settled leniently, the water retains the status of tahor, and a person may go ahead and wash his hands with it.
Why the doubt is settled leniently
The mishnah grounds the ruling in an established principle governing doubtful cases that involve hands: "mipnei she'amru, sofek hayadayim" - wherever the purity status of hands is in question, the outcome is "tahor". The principle addresses three distinct questions: "l'tamei", did the hands pick up tumah in the first place; "ul'tamei", did they transmit tumah onward to something else; and "ul'taher", were they in fact rendered tahor. Each of the three is decided in favor of taharah.
Rabbi Yose goes along with the first two but breaks with the third, ruling "l'taher tamei". Where the question is whether hands that had been tamei were successfully purified, he will not let the uncertainty generate taharah, and the hands keep the tamei status they already carried.
What drives Rabbi Yose is consistency: a doubt is not strong enough to dislodge anything from the condition it already occupied. Hands that stood tahor stay tahor when a possible contact with tumah is questionable, and by exactly the same reasoning hands that stood tamei stay tamei when the washing is questionable. So long as no actual change has been established, the prior status remains in force.