Ohalos, Chapter 6, Mishnah 4. Our mishnah takes up the parts of a building that lie on the border between two spaces: the wall dividing one house from its neighbor, and the layer separating a lower story from the story constructed above it. Each of these serves two areas at the same time, and the question before us is which of the two areas an object found inside such a divider belongs to. The Mishnah presents its cases in matched pairs: first where the tumah itself is lodged within the divider, and afterward where the tumah occupies one of the two spaces while keilim are lodged within the divider.
Tumah inside a wall between two houses
"Kosel shebein shnei batim vehatumah b'socho": a wall stands between two houses, and there is tumah lodged inside the thickness of that wall. The Mishnah rules: "bayis hakarov latumah, tamei." The house on the side to which the tumah lies nearer is tamei, and everything in it becomes tamei. "Vehakarov lataharah, tahor." The house on the far side, the one from which the tumah is more distant, remains tahor. In other words, tumah embedded in a shared wall is treated as though it were lying inside whichever of the two houses it is closer to.
What if the tumah rests "mechtza lemechtza," precisely at the wall's midpoint, so that the distance to either side is identical? The ruling is "shneihem tmeiyim": each of the two houses is rendered tamei. Since the wall serves the two houses in equal measure, tumah standing at its exact center cannot be attributed to this side rather than that one, and so we view it as if it were situated within each of them.
Tumah in a house and keilim in the wall
The Mishnah now inverts the arrangement. "Tumah b'echad meihen vekeilim bakosel": tumah is present in one of the two houses, while keilim have been placed in a cavity within the dividing wall. The same reasoning governs the outcome. "Mechetzya ul'chlapei tumah, tmeiyim": keilim positioned on the tumah side of the midpoint are tamei, for they are reckoned as sitting inside the house that holds the tumah. "Umechetzya ul'chlapei taharah, tehorin": keilim positioned on the far side of the midpoint, facing the house that contains no tumah, stay tahor, because they are counted as being within that tahor space.
"Mechtza lemechtza, harei hein tmeiyim." Keilim resting exactly at the halfway point, no nearer to one house than to the other, are tamei. Standing precisely in the middle, they are treated as though they lie in both houses at once, and since one of those houses contains tumah, the keilim contract tumah.
The ma'azivah between the house and the aliyah
The Mishnah now turns to a new question: "ma'azivah shebein habayis la'aliyah." The Bartenura explains that the ma'azivah is the tikrah, the covering that separates the lower story from the upper one. As we will see in the following mishnah, this covering is made of crossbeams with a layer of plaster spread over them. Picture the construction: one first builds the walls of the lower house, lays beams across the top of them, and spreads plaster over the beams. That completed layer is the ma'azivah, serving simultaneously as the ceiling of the house below and the floor of the aliyah above, where people live.
"Tumah b'sochah": there is tumah inside the thickness of the ma'azivah itself. "Mechetzya ul'matan, habayis tamei veha'aliyah tehorah." If the tumah lies in the lower half, from the midpoint downward, the house below is tamei and the upper story is tahor. "Mechetzya ul'maalah, ha'aliyah tmeiah vehabayis tahor." If the tumah lies in the upper half, the aliyah is tamei and the house beneath is tahor. Practically speaking, one must measure the thickness of the ma'azivah and determine whether the tumah sits in the lower portion or the upper one.
"Mechtza lemechtza, shneihen tmeiyim." Tumah lying exactly at the center of the layer renders both the lower house and the aliyah tamei, since at that point we regard it as though it were present in both stories together.
Tumah in one story and keilim in the ma'azivah
Once more the Mishnah turns the case around: "tumah b'echad meihen vekeilim bama'azivah." Tumah occupies one of the two stories, whether the lower or the upper, while keilim sit inside the ma'azivah. "Mechetzya ul'chlapei tumah, tmeiyim": any keili in the portion of the layer that faces the tumah becomes tamei. Where the tumah is downstairs, that means the lower portion; where it is upstairs, the upper portion. "Mechetzya ul'chlapei taharah, tehorin": keilim in the portion facing the story that has no tumah stay tahor, for they are reckoned as lying within that tahor space.
"Mechtza lemechtza, harei hein tmeiyim." Keilim set exactly in the middle of the ma'azivah, equally distant from its upper and lower surfaces, are tamei. Since this layer functions equally as ceiling for the house below and as floor for the story above, an object at its precise center is treated as lying in both stories, and one of those stories contains tumah.
The view of Reb Yehuda
"Reb Yehuda omer: kulo hama'azivah la'elyonah." In Reb Yehuda's view, this layer in its full depth, the very same one that roofs the lower house and floors the upper one, is assigned to the aliyah alone. Even tumah situated in the layer's bottom part, right up against the ceiling of the room beneath, counts as being in the aliyah: the upper story becomes tamei while the house underneath stays tahor.
This principle of Reb Yehuda affects both of the ma'azivah rulings. When tumah is lodged within the layer, it is the aliyah alone that becomes tamei. And in the reversed case, where tumah occupies one story and keilim sit in the layer: tumah in the aliyah renders every keili in the ma'azivah tamei, including one resting at its very bottom, since the layer in its entirety is part of the upper story. Should the tumah instead be downstairs, not a single keili in the ma'azivah becomes tamei, even one at its lowest point, for according to Reb Yehuda this layer forms no part of the house below.