Talking Talmud

2 mishnayot! The first one includes the details about a non-believing Jew who participates in shechitah, and whether that shechitah would be kosher. And the mishnah is preceded by Gemara that includes a discussion of a Jewish apostate. The mishnah includes what manner of disposing of the blood of kosher shechitah are acceptable and which are problematic because they smack of idolatry. The second mishnah lists ways of dedicating one's shechitah in ways that make it invalid. Plus other key variations on intent and the way one's actions might be misinterpreted.

A new mishnah! One who slaughters an animal in the name of worship of natural bodies (mountains, rivers, seas, etc.), then the slaughtering is not kosher shechitah. Even if two people are doing the slaughtering together and one of them has correct "kosher" intent, the shechitah is still not valid. But it's also not fully idolatry either - except for a beraita that presents comparable cases as being idolatry (with slight differences that make room to say they are different). Also, if one's animal is lying in front of idolatry, then the moment one cuts that animal, it's prohibited - because it smacks of idolatry, even if it was not intended as such (live, the animal isn't a concern of idolatry, or not until a physical act is done to it). But does that really work? To make another person's animal problematic? The distinction is drawn between a "zevach" (or a shelamim/peace-offering) and a "chatat" - sin-offering.

If one slaughters an animal with intent to use the blood for idolatrous purposes, is the rest of the animal forbidden for any benefit or not? Yes, it's a machloket. With hypothetical parallels to pigul. Unless you think nothing in the Temple can be used to infer halakhic details for that which does not pertain to the Temple service. Also, a husband who writes a bill of divorce that was written when he is healthy as a safeguard against the future, and then he dies -- but if he had no knowledge of that risk to his life, then his intent in writing the divorce is absent and the divorce is not a divorce. Conversely, one who inherits a contract of slaves cannot free them by protesting the ownership.

An animal that is close to death demonstrates its vitality by some form of movement. Of course, the degree and nature of movement is subject to dispute. Also, a new mishnah - if/when one slaughters an animal for a non-Jew, that shechitah should be kosher, depending on the idolatry factor.

A new mishnah! Establishing the category of an animal that is close to death (misukenet) -- in contrast to a treyfa, where it becomes clear that the animal had something wrong internally and would likely have died within the year. This case is an animal that is weak, not damaged internally. Thus, signs of vitality during shechitah become essential, to be certain that the animal didn't die of its own accord just prior to the shechitah. Also, how the Gemara ensures the distinction between a treyfa and an animal that is close to death already, with the prohibited fat (cheilev) as a key point.