hadlakas menorah and chatzi shiur

The Minchas Chinuch frequently raises the question of whether any credit is earned by performing a mitzvas aseh if one has less than the proper shiur. For example, would one get any credit for eating half a k'zayis of matzah, or is it not considered an act of eating unless a proper shiur is consumed? The basis for the safeik seems to be whether the concept of shiur is a quantitative measure (kamus), in which case less than the full amount might still count as some degree of kiyum, or whether shiur is a qualitative measure (eichus), in which case less than the full amount does not count as anything.

One proof cited by the Minchas Chinuch is the gemara (Yoma 39) which says that when times were bad each kohen would get only a small portion the size of a pol of the lechem hapanim or the shtei halechem. Apparently eating this small portion, albeit less than a k'zayis, is still considered a kiyum mitzvah of eating kodshim (see Tosfos Yeshanim).

One can argue on this proof. The Beis haLevi (I:2:7) writes that the mitzvah of eating kodshim is not a chovas hagavra incumbent on the individual to eat kodshim, but is rather a chovas hacheftza for kodshim to be consumed. It is the result, not the act or process, which is important. Therefore, since eating less than a k'zayis contibutes toward achieving the desired outcome, it is considered a mitzvah. The same might not be true of achilas matzah where the chiyuv rests on the individual to engage in an act of eating.

The Beis Yosef famously asks why the holiday of Chanukah is 8 days when there was sufficient oil for at least one day. His first answer is that the kohanim did not anticipate a miracle and divided the oil into 8 equal parts so that they could light something on each night. Even though only a small portion was lit on each night, that small portion miraculously burned the entire night on each of the eight nights.

Perhaps one can bring a proof from this Beis Yosef to resolve the Minchas Chinuch's question. Each lamp of the menorah was usually lit with a shiur of 1/2 a lug of oil. If performing a mitzvah with less than the proper shiur is meaningless, what were the kohanim hoping to accomplish by lighting a small insufficient shiur on each night? It must be that even though less than the shiur was utilized, there was still a kiyum mitzvah in lighting.

This proof is also debatable, but we will save that for another time bl"n.