Rambam - 1 Chapter a Day

The Rambam lists the twelve situations here and goes on to elaborate upon them in the subsequent halachot in this and the following chapters.

Water that was drawn using a container and then used by humans to fill a pool. This term is explained at length in Hilchot Mikvaot, ch. 4.

A log is 344 cc according to Shiurei Torah and 600 cc according to Chazon Ish.

Three luggim.

In his Commentary to the Mishnah (Taharot 4:8; see also the following halachah and Chapter 15, Halachah 8), the Rambam emphasizes that this applies only in a private domain. In a public domain, there would be no need for such a leniency, for its status would be pure regardless.

There is an allusion to this in the Torah. Leviticus 11:29, the source that teaches that the carcass of a crawling animal imparts impurity, speaks of “a crawling animal that crawls on the earth.” Since this carcass is not on the earth, but floating, there is room for leniency [the Rambam’s Commentary to the Mishnah (op. cit.)].

Although the Mishnah (Taharot 4:7-8) speaks of “impurity floating,” from Nazir 64a, it is apparent that the principle applies only with regard to the carcass of a crawling animal.

Since they are being held by a person, they are considered as placed down and not floating (Kessef Mishneh).

Chapter 15:7-8. As stated there, this principle applies with regard to people and also with regard to keilim or foods.
This halachah points to an issue that requires resolution. The rationale for two of the twelve situations involving a doubt — a doubt whether an object touched a carcass of a crawling animal that was floating and a doubt that such a carcass that was discarded — is seemingly the same: Since the impurity does not have a definite place, we rule leniently. Why, then, are they considered as two different situations?
In resolution, it can be explained that the ruling here involves humans and thus there is room for stringency, because a human has the knowledge to inquire about his status. In contrast, the doubt involving a discarded carcass involves food where there is greater room for leniency, because the food cannot question its status.

Since the ashes of the red heifer had been sprinkled upon it, it is thicker than other water and can be considered as a separate entity.

The commentaries question: What is the distinction between the paper and the container or semen mentioned in Halachah 4?

Since the kneading trough is impure, the water contracts impurity because of it and is considered as a primary derivative. Hence, if the water would come in contact with the loaf, it would render it impure.

Which would render it impure.

I.e., if there was a question whether a loaf of bread that was terumah touched the carcass.

For the leniency mentioned by our Sages involved a carcass floating on water, not on wine.

Who were perushim and sought to maintain a state of purity at all times (Kessef Mishneh).

For liquids impart impurity to other substances only due to Rabbinic decree and when a doubt arises regarding a point of Rabbinic Law, we rule leniently (Kessef Mishneh).

The fact that the impure person inserted his hand into the inner space of the jug does not render it impure, as stated in Hilchot Metamei Mishkav UMoshav 8:4. Nor does he contract impurity from the liquids for the reason stated by the Rambam.

Due to the principle stated in the previous halachah.

I.e., if the liquids are not touched by an impure entity, their status depends on the status of the jug. If it becomes impure, they are also deemed impure. If it remains pure, so do they.

For the liquids in the jug derive questionable impurity from the liquids that enter it.

In the Talmudic era, the floors were often simple earth and water would be sprinkled upon it to keep the dust down.

Because the question concerns articles that contract impurity from liquids.

See Chapter 15, Halachah 8; Chapter 16, Halachot 1-2. Although loaves of bread which cannot inquire about their status are involved, since it is known that a source of impurity was present, there is no room for leniency.

The Ra’avad opines that it is impure, only because of the impression that might be created (marit ayin). An onlooker will see that a question is being raised about the status of the loaf, but he will not know about the liquids that caused the question to be raised. Were the loaf to be deemed pure, he might receive a wrong impression regarding questions of impurity in a private domain in general.

I.e., a situation where the person himself is not impure, but his hands are impure due to Rabbinic decree, as stated in ch. 8. Since the entire question is a matter of Rabbinic Law, when a doubt arises, we rule leniently.

This refers to loaves that are a primary derivative of impurity. Hands do not contract impurity from a secondary derivative or lower (Chapter 8, Halachah 1).

Would it be known that he had touched them, he would have contracted impurity.

I.e., the impurity of the hands was definite and the doubt is whether or not they were purified.

I.e., drawn water disqualifies a mikveh and there are four factors that disqualify water for washing hands (e.g., that work was performed with the water, see Hilchot Mikveot 11:11, Hilchot Berachot 6:7).

40 se’ah for a mikveh, a revi’it for washing.

Which would disqualify the immersion or the washing.

The Kessef Mishneh compares this to Halachah 2. Other commentaries (see Mishneh LeMelech, Hilchot Mikveot 10:6) make a distinction between the two.

If, however, he touches them with both hands before washing his hands, they are impure, for he will have certainly have touched them with his impure hand.