Chapter Three, Mishnah Three. We are returning to a topic we dealt with earlier, and essentially this is a review of what we have already learned, except that one additional detail has been added here regarding the view of Rabbi Eliezer.
An intention of eating in place of burning, and an intention of burning in place of eating:
To eat what is designated for burning - One slaughters the animal with the intention to eat a part that is not designated for eating but rather for burning on the Altar, such as the eimurim and the kidneys.
To burn what is designated for eating - One slaughters with the intention to burn on the Altar, not in its proper time or not in its proper place, a part that is not designated for burning at all, such as the meat of a shelamim offering that is meant for human consumption.
The ruling in both of these cases is "kasher" - valid. These are not intentions that invalidate the offering, because they do not apply to what the parts were designated for: what is designated for human consumption he intended for the Altar, and what is designated for the Altar he intended for eating.
The basis for the distinction between the two types of eating is rooted in the fact that the verse in Leviticus (Chapter 7, Verse 18) employs, as it were, a double expression of eating and consuming. The Tanna Kamma holds, therefore, that these are two different types of eating, and they are not interchangeable.
The view of Rabbi Eliezer:
Rabbi Eliezer disagrees and maintains that these two expressions come together, and they are interchangeable. Therefore, one who intends to eat what is designated for burning, or to burn what is designated for eating, and his intention is to do so in the wrong place or at the wrong time - indeed invalidates the offering in doing so. The halachah does not follow his view.
Intending for less than a k'zayis:
The Mishnah continues with a rule we have already discussed: One who has intentions regarding the eating of the offering or its burning, but his intent is to do so with only a small portion, an amount less than a k'zayis - it is valid, and this intention does not invalidate it. The reason, as has been explained, is that an act of eating requires a minimum measure of a k'zayis; with less than a k'zayis it is not considered an act of eating, and therefore there is no invalidating intention of eating here.
Two halves of a k'zayis:
Nevertheless, the Mishnah continues and establishes: One who has an intention regarding two halves of a k'zayis - to eat half a k'zayis at the wrong time and to burn another half a k'zayis at the wrong time - "einam mitztarpin" - they do not combine. The two halves do not combine to form a whole, because we differentiate between eating and burning: eating with the mouth, and the Altar's consuming through burning upon it, do not combine with each other.
And why did the Mishnah need to repeat a law that was already stated explicitly several mishnayos earlier? The answer is that even Rabbi Eliezer - who holds that the two types of eating are interchangeable, to the point that one who intends to eat what is designated for burning or to burn what is designated for eating invalidates the offering - agrees that two halves cannot be combined. Half a k'zayis of eating and half a k'zayis of burning do not combine to a whole k'zayis to invalidate the offering with an intention of out of its proper place or out of its proper time.
In summary: In this Mishnah, we learned that an intention to eat something designated for burning, and an intention to burn something designated for eating, do not invalidate the offering according to the Tanna Kamma, who derives from the double expression in the verse that these are two types of eating that are not interchangeable; Rabbi Eliezer disagrees and holds that they are interchangeable and do invalidate, and the halachah does not follow his opinion. We also learned that an intention for less than a kezayis does not invalidate, and that two half-kezayis amounts - one regarding eating and one regarding burning - do not combine, and this is true even according to Rabbi Eliezer.