Gemara

27a (Mishnah): The four species (altogether) taken with the Lulav are Me'akev each other.

It says about the four species including the Lulav - "u'Lkachtem". The Kichah (taking) must be Tamah (complete);

65b (Beraisa): "U'Sfartem Lachem" - it is a Mitzvah for every individual to count (the Omer).

Sukah 31a (Beraisa): If one did not find an Esrog, he does not bring a quince or pomegranate or anything else.

If they are dry, they are Pasul;

R. Yehudah says, even if they are dry (they are Kosher).

R. Yehudah: A case occurred in which city dwellers bequeathed their (totally dry) Lulavim to their grandchildren (because there were no fresh Lulavim)!

Rabanan: We do not learn from pressed circumstances.

Question: Why must the Beraisa teach that if one did not find an Esrog, he does not bring a quince or pomegranate? This is obvious!

Answer: One might have thought that one brings it, lest the Mitzvah of Esrog be forgotten. The Beraisa teaches that we do not bring, lest disaster result. People might continue to do so (even when Esrogim are available).

Rishonim

Rambam (Hilchos Lulav 7:5): The four Minim are one Mitzvah. They are Me'akev each other. If one of them is not found, we do not bring a similar species.

Rosh (Sukah 3:14): We learned that if one did not find an Esrog, he does not bring a quince or pomegranate or anything else. Some derive from here that if one did not find a Kosher Esrog, he does not bring other species, but he may bless on a Pasul Esrog. The Ri rejects this. The Beraisa teaches about a quince or pomegranate to teach that even if it is Hadar (beautiful) and whole (one may not take it). We can say that it is permitted, just like a Pasul Lulav according to Rabanan who permit in pressed circumstances. The Gemara connotes that in pressed circumstances, one is Yotzei even with Pasul Minim. R. Yeshayah said similarly, and also the Ra'avad.

Rosh (ibid., citing the Ra'avad): I retracted. Whether or not it is pressed circumstances, one is not Yotzei with dry Minim. Whenever it says 'Pasul', this is even b'Di'eved, even in pressed circumstances. How can one be Yotzei and bless l'Chatchilah in pressed circumstances, but when it is not pressed circumstances, even b'Di'eved one is not Yotzei?! Dry is like dead. It is as if it does not have the required Shi'ur, and it is not here. We do not bless, but we take it in the hands in pressed circumstances, lest the Mitzvah of Lulav and Esrog be forgotten. This is why Chachamim say that we cannot learn from pressed circumstances. I.e. it was done only lest the Mitzvah be forgotten. This is like we say about dry Minim. We take them for a mere commemoration. Likewise, when one has a Lulav but not an Esrog, even though they are Me'akev each other, and one does not bless on a Lulav without an Esrog, it is a Mitzvah to take the Lulav by itself for a mere commemoration, lest the Mitzvah be forgotten.

Note: Presumably, 'Lulav by itself' refers to the Lulav bundle, with Hadasim and Aravos.

Rosh (ibid.): The Beraisa said that we do not bring, lest disaster result. Even when Esrogim are available, people might abandon them and bring these other species. We infer that when this is not a concern, like here (when only Pasul Minim can be found), one must take them, lest the Mitzvah be forgotten. Also, the Gemara said 'he should bring, lest the Mitzvah be forgotten.' If he does not take a Lulav, why does he bring a quince, lest the Mitzvah be forgotten? Is Esrog more important than Lulav?! Rather, he takes a Lulav even if he has no Esrog. One might have thought that he holds that a quince or pomegranate with it, to commemorate the Esrog. The Beraisa teaches that this is not so. It is better to take the Lulav without another species in place of the Esrog, lest disaster result.

Suggestion: Disaster can result also through talking the Lulav alone. Perhaps both will be available, and one will take only the Lulav!

Rejection (ibid.): People will not err about this, for the Torah explicitly says to take four Minim with (i.e. including) the Lulav. However, the Torah did not specify that Esrog is one of them, and people could think that a quince or pomegranate is a "Pri Etz Hadar." We do not decree not to take dry, lest people do so even when moist are available. People would not bring dry if moist if available.

Rebuttal (of Ra'avad - Rosh): 'We do not learn from pressed circumstances' connotes unlike this. R. Yehudah proved that dry is Kosher and we bless on it from city dwellers who blessed on dry Lulavim. Chachamim said that we do not learn from pressed circumstances. This shows that they agree that we bless in pressed circumstances! If not, they should have said that even in pressed circumstances we do not bless! The Torah empowered Chachamim to rule about Pesulim. They said that when it is not pressed circumstances, even b'Di'eved one is not Yotzei, so that Yisrael will be zealous about Mitzvos. In pressed circumstances, Chachamim said that they are Kosher. Since there is no alternative, we bless on them.

Rosh (Sukah 3:3): It was clear (Sukah 29b) that Hadar applies every day, but "Lachem" (it must be yours) is only on the first day. Similarly, "ul'Kachtem", from which we expound Lekichah Tamah, is only on the first day. R. Chanina used a Chaser (deficient) Esrog after the first day (Sukah 36b). This is astounding. Why do we distinguish? All are written in one verse - "ul'Kachtem Lachem ba'Yom ha'Rishon Pri Etz Hadar." Perhaps "Yom Rishon" applies only to what is written before it; Chachamim did not enact those matters on other days. Hadar is written after "Yom Rishon", so they enacted it for all days. Still, this is difficult, for the four Minim are Me'akev each other due to Lekichah Tamah. "Ul'Kachtem" also teaches that everyone must take by himself (Sukah 43a). This is because the Torah used the plural, and not the singular, like we expound "u'Sfartem Lachem" to teach that it is a Mitzvah for every individual to count the Omer (Menachos 65b). If so, after the first day, one Min should suffice, and one person can take on behalf of everyone! Rather, outside the Mikdash the Mitzvah on the first day is mid'Oraisa. On other days it is mid'Rabanan, to commemorate the Mikdash. The primary Mitzvah, i.e. everyone takes four Minim for himself, was enacted similarly on other days. Also Hadar was enacted on other days, to beautify the Mitzvah. They did not enact about Chaser or borrowed.

Poskim

Shulchan Aruch (OC 651:12): These four Minim are Me'akev each other. If one is missing, he does not bless on the rest, but he takes them for a mere commemoration.

Beis Yosef (DH v'Chosav ha'Rosh): The Rosh holds that in pressed circumstances, we bless on a dry Lulav. Therefore, he rejected the Ra'avad. There is no proof whether he holds that we take another species when one is not available, so the Tur omitted this. Maharik (41) says that Ba'al ha'Hashlamah says to take the species he has and bless on them. Maharik was astounded. Rather, he takes the species he has, lest the Mitzvah be forgotten, but does not bless on them.

Teshuvas Maharshal (8): I say that one should not bless, but the Shali'ach Tzibur waves normally during Hallel, lest the Mitzvah of Lulav be forgotten. We are not concerned lest disaster result, for all know that the Torah commands to take four Minim, like the Ra'avad says. The Rosh argues only about blessing on an old Lulav in pressed circumstances. He agreed with the reasoning that there is no concern lest disaster result when taking three Minim. Even if you will say that individuals praying without a Minyan do not wave, this is because there is no reason to. It is proper to wave b'Tzibur. Perhaps this praises Hash-m even with three Minim. I say that others should not even take three Minim. Perhaps another year they will bless on them! The Shali'ach Tzibur should wave them (also) before Hallel, at the time one normally blesses.

Taz (15): It seems to me that since it is a mere commemoration, he need not wave them.

Mishnah Berurah (51): Eliyahu Rabah says that everyone should take and wave, lest next year they rely on the Shali'ach Tzibur and not take at all! The Shulchan Aruch connotes like this.

Gra (DH Aval): 'He does not bring a quince or pomegranate' connotes that he does take the other three Minim.

Bi'ur Halachah (DH Aval): The Taz (17) says that he must not intend for the Mitzvah, lest he transgress Bal Tigra (detracting from a Mitzvah). The Sifri explicitly says so. The Taz adds that even after fulfilling the Mitzvah, if he wants to take the Minim again for dearness of the Mitzvah, he may not take only three Minim due to Bal Tigra. All the Acharonim say that once one fulfilled the Mitzvah, Bal Tosif and Bal Tigra do not apply.

Rema: This is on the first day, and also on other days.

Beis Yosef (DH Kasuv b'Hagahos): Hagahos Ashri says in the name of Or Zaru'a that they are Me'akev each other only on the first day. Lekichah Tamah disqualifies Chaser (incomplete Minim), and teaches that they are Me'akev each other. Just like Chaser is Pasul only on the first day, also they are Me'akev each other only on the first day, but not on Yom Tov Sheni. The Mordechai (749) says so in the name of R. Tam of Orleans, and he agreed to act on this, even though Avi ha'Ezri disagreed. The Poskim do not distinguish the first day from other days. The Maharil (112) says that one who does not have an Esrog takes another species, but does not bless on it. The Rosh refuted the Or Zaru'a's distinction between the first day and later days. One does not bless even on Yom Tov Sheni. The Rosh said that whatever is Pasul on Yom Tov Rishon is Pasul for us on Yom Tov Sheni. If so, since this is Me'akev on Yom Tov Rishon, we should be concerned lest it disgrace Yom Tov Sheni, just like we do not take the Lulav on Shemini Atzeres (due to Safek, lest it is really the seventh day of Sukos), for this would disgrace Yom Tov (to show that it is Safek Chol). Even so, I say that one should take it on Yom Tov Sheni, just like we take Sha'ul and Chaser. Disgrace does not apply here. It applies only to Shemini Atzeres, for a Lulav is Muktzah. Here, even if it is Yom Tov Rishon, it is not Muktzah, for there is a Mitzvah of the four Minim. Even though he does not have all of them, they are useful for one who has an Esrog.

Shulchan Aruch (13): If one is missing one of the Minim, he does not bring another species in place of it.