(a) RASHI (DH Ela) says that since Beis Hillel holds that the Isur of
Hotza'ah, carrying into Reshus ha'Rabim, applies to Yom Tov in principle,
and it is only permitted because of "Mitoch," there remains an Isur
d'Rabanan of carrying non-essential items (such as stones) on Yom Tov.
Rashi's words imply that even carrying stones into Reshus ha'Rabim is
prohibited only mid'Rabanan.
TOSFOS (DH Hachi Garsinan) points out that the common text of the Gemara in
their days (in contrast to the text which appears in our Gemaras) read that
since Hotza'ah is Asur on Yom Tov and the only Heter is because of "Mitoch,"
one is *Chayav* [mid'Oraisa] for carrying stones into Reshus ha'Rabim (and
not that it is merely an Isur d'Rabanan). Tosfos maintains that Rashi
changed this Girsa intentionally, because Rashi held that once "Mitoch"
applies, one will not be Chayav for carrying out stones to Reshus ha'Rabim
on Yom Tov; it will only be Asur mid'Rabanan, even though the act is done
for no purpose. According to Rashi, "Mitoch" permits doing the Melachah
under *all* circumstances.
However, how does Rashi understand the Gemara later (21a, and in Pesachim
46b) which states that if one bakes on Yom Tov for the weekday, one is
Chayav Malkus? Why should he get Malkus if the principle of "Mitoch" permits
baking on Yom Tov under all circumstances?
The RAN (who asserts that the RIF also holds like Rashi, in contrast to the
ROSH, who understands the Rif to be learning like TOSFOS) answers that it is
true that Rashi does not require that the Melachah be done for a necessary
purpose on Yom Tov in order for it to be permitted. However, if one is doing
it because he specifically needs it *for tomorrow*, that is worse than doing
it simply with no need for it today, and in such a case one is Chayav.
(b) TOSFOS and most other Rishonim argue with Rashi and maintain that the
principle of "Mitoch" only permits Melachos that are done for a need on Yom
Tov, either because one enjoys doing the act and benefits from it on Yom Tov
(such as taking a stroll outside while carrying one's child), or because one
fulfills a Mitzvah of Yom Tov by doing that act (such as carrying a Lulav or
a Sefer Torah).
(c) RABEINU CHANANEL on the Mishnah says that the Mishnah gives the specific
examples of carrying a child, Lulav, and Sefer Torah, because all three are
objects with which a Mitzvah is done (a Lulav on Sukos, and Sefer Torah to
read, and a child to perform Milah upon him). Tosfos learns that this
implies that "Mitoch" only permits something that is a Mitzvah to do on that
day (such as Milah), but not something that is not a specific Mitzvah which
must be performed on that day (such as carrying a child as one strolls
(It is also possible that Rabeinu Chananel agrees with Tosfos that all types
of pleasurable acts on Yom Tov are in within the bounds of Mitoch. Rabeinu
Chananel is adding, though, that *even* for the performance of Mitzvos, from
which one does not derive physical pleasure, Mitoch can be applied -- as
long as the Mitzvah is one that must be done on Yom Tov.)
(d) The RAMBAM (Hilchos Yom Tov 1:4) writes that there are only two Melachos
to which "Mitoch" applies: Hotza'ah and Hav'arah (carrying into Reshus
ha'Rabim and kindling a flame). Only those two Melachos may be performed not
for the sake of food preparation. All other Melachos are permitted only for
the sake of food preparation.
(It could be that the Rambam does not mean that these two Melachos are
permitted because of the principle of "Mitoch." Rather, the Rambam might be
ruling like the opinion in our Gemara that suggests that "Ein Hotza'ah b'Yom
Tov" -- there is no Isur whatsoever of Hotza'ah (and Hav'arah, see Pesachim
5b) on Yom Tov. The Rambam is merely borrowing the term "Mitoch" from the
Gemara, but the Heter is not because of the principle of "Mitoch," but
because there is no Isur at all on Yom Tov. (We find that the Rambam
occasionally borrows terms from the Gemara and uses them in ways different
from the Gemara, see, for instance, Hil. Isurei Bi'ah 1:3.)
This way of understanding the Rambam has its advantages, because if the
Rambam is ruling like the opinion in our Gemara which holds of "Mitoch,"
then why does the Rambam limit it to Hotza'ah and Hav'arah? The Gemara says
that according to the opinion which permits Melachos because of "Mitoch,"
the principle of "Mitoch" applies to the Melachos of Bishul and Shechitah
(as well as Hotza'ah and Hav'arah), permitting them even when not done for
the needs of Yom Tov. If the Rambam is not ruling like the opinion which
holds of "Mitoch," then it makes sense why he does not include Bishul and
Shechitah in his list of Melachos that are Mutar on Yom Tov even when not
done for the sake of food preparation.)