Gemara

(Beraisa - R. Yishmael): Two occurrences of 'Totafos' are written missing the 'Vov' (as if it says 'Totafas', singular). We only learn one from each of those, and two from "Totafos" with a 'Vov', teaching four boxes in the head Tefilin;

88b (Mishnah): If a Chacham says that they are five Parshiyos in Tefilin, to add to what Chachamim explained, he is liable (for Zaken Mamre).

(R. Elazar): He is liable only for a Mitzvah with an explicit source in the Torah, and is explained by Chachamim; it is possible (but forbidden) to add to the law, and by adding he invalidates it. The only example is Tefilin, according to R. Yehudah (who obligates for a Torah law explained by Chachamim).

Question: Also Lulav has an explicit source in the Torah, Chachamim explained it, and if one adds other species to the four, he invalidates the Mitzvah!

Answer: If the species need not be tied, holding additional species at the same time has no consequence. If the species must be tied, the bundle (with the extra species) was already invalid (from when he tied it.)

Question: Also Tzitzis has an explicit source in Torah, it is explained by Chachamim, and if one adds additional strings he invalidates the Mitzvah!

Answer: If the Torah does not require a knot (after the strings were inserted through the corner), extra strings do not disqualify the Mitzvah. If the Torah requires a knot, it was Pasul from the start (due to the extra strings)!

Question: Likewise, if he made four boxes and put a fifth next to them, the Tefilin are Kosher. If he initially made five boxes, it was always Pasul!

Answer: (Really, he made four, and put a fifth next to them.) R. Zeira taught that if the outer boxes are not exposed to the air, the Mitzvah is invalid.

Menachos 34b (Rav Chananel): If the order of the Parshiyos of Tefilin was changed, they are Pesulim.

Eruvin 95a (Mishnah): If one finds Tefilin (on Shabbos in a field, where they are prone to be disgraced) he brings them in one pair at a time;

R. Gamliel says, he brings two pairs at a time.

95b (Rava): The first Tana permits saving on Shabbos only one pair of Tefilin (at a time), like he wears during the week.

Rav Shmuel bar Rav Yitzchak): R. Gamliel permits two pairs at a time, for there is room on the head (and arm) to wear two Tefilin at a time.

96a: The first Tana holds that one transgresses Bal Tosif (adding to Mitzvos) without intent. R. Gamliel holds that one transgresses only with intent.

Rejection: No. All hold that there is no Mitzvah of Tefilin on Shabbos. The first Tana holds that one transgresses Bal Tosif when the Mitzvah does not apply even without intent; R. Gamliel holds that he transgresses only with intent.

Rejection: If so, the first Tana should not allow even one pair! Rather, we must answer like above.

Rishonim

Rosh (Hilchos Tefilin (after Menachos) Siman 5): Since there is room on the head for two Tefilin, and also on the arm, one with Yir'as Shamayim will make two pairs of Tefilin (one according to Rashim and one according to R. Tam) and wear both together. He intends to be Yotzei with the Kosher pair; the other pair is merely straps. Bal Tosif is only if one makes five boxes.

Ma'adanei Yom Tov (20): If one made four boxes, and put a fifth next to it, this is not Bal Tosif. Since it is not connected, it is by itself (not part of the Mitzvah), like it says in Sanhedrin. However, in Eruvin we say that one transgresses even for another pair! The Halachah follows Chachamim, and we conclude that they forbid more than one pair due to Bal Tosif. The Rosh cannot say that we conclude that they argue about whether there is room to wear two Tefilin at once (and Chachamim say no), for the Rosh rules that there is room.

Levush (cited in Divrei Chamudos 19): I am unsure whether Bal Tosif applies to wearing two pairs of Tefilin. Perhaps it is like Tzitzis. One may wear many garments with Tzitzis on them! The Gemara found Bal Tosif only if he attached a fifth box. Or, perhaps Tzitzis are different, for every garment is obligated. Perhaps the Gemara mentioned five boxes just for similarity to five species including the Lulav.

Divrei Chamudos (19): I brought in Ma'adanei Yom Tov that Sugyos argue about this. All the Poskim, including the Levush (in Siman 301) rule like the Sugya in Eruvin regarding saving Tefilin on Shabbos. To resolve the Sugyos, it seems that in Sanhedrin we discuss only Zaken Mamrei. In truth, one who wears two pairs transgresses, like it says in Eruvin. The Mizrachi in Parshas va'Eschanan says that Bal Tosif applies to an addition, even if it is not tied, However, if so, what is the Rosh's proof from Sanhedrin?

Poskim

Shulchan Aruch (OC 34:2): One with Yir'as Shamayim will fulfill both opinions. He makes two pairs of Tefilin and wears both. He intends to be Yotzei with the Kosher pair; the other pair is merely straps.

Bach (3): If one transgresses for five boxes, all the more so he should transgress for two pairs of Tefilin, which have eight boxes! Indeed, the Rosh means that one transgresses for five boxes only if he intends to add to the Mitzvah. Here, he intends to fulfill the Mitzvah only once. Really, the Rosh should have said that one does not transgress without intent to add. He mentioned five boxes, for the Gemara discussed this.

Taz (2): The Tur says that Bal Tosif is only if one makes five boxes, but not if he wears two pairs. The Levush was unsure. He overlooked Eruvin 95b. I do not understand the distinction of Divrei Chamudos. Tosfos in Sanhedrin (88b DH Iy) asked similarly, and answered that Eruvin is a stringency mid'Rabanan. Why did the Rosh distinguish between five boxes and wearing R. Tam Tefilin with those of Rashi? He should have distinguished that here, one intends that the second pair is mere straps! He did not make the latter distinction, for we concluded (Rosh Hashanah 28b) that Bal Tosif applies to even without intent. Rather, he taught that mid'Oraisa Bal Tosif is only when there are five boxes. Mid'Rabanan two pairs of Tefilin are forbidden, but only if he intended that both are for the Mitzvah. Surely Chachamim did not decree if e wants to be Yotzei only through the Kosher pair! Even though in some stages of the Sugya in Eruvin we say that the Tana'im argue about whether one transgresses Bal Tosif without Kavanah, not all of them are according to the Halachah. We conclude that they argue about whether Shabbos is Zman Tefilin even mid'Rabanan. Even though without intent it is forbidden only mid'Rabanan, we are not lenient to allow one to save two pairs at once, even though he intends only to save (and not for the Mitzvah). This should be permitted, like wearing R. Tam Tefilin with Rashi Tefilin! Regarding saving Tefilin there is an Isur even without Bal Tosif of carrying on Shabbos (if only one pair is a Mitzvah), therefore, we forbid even without intent. What the Bach said is wrong.

Magen Avraham (3): If one wore two pairs of Kosher Tefilin, he transgresses Bal Tosif, like it says in Eruvin. The Beis Yosef (651 DH Aval) says so in the name of the Magid Mishneh (Hilchos Lulav 7:7). The Levush and Divrei Chamudos were unsure about this. They overlooked the Gemara and the Magid Mishneh. There is no proof from Sanhedrin, which discusses one who wrote another Parshah that has no connection to Tefilin.

Gra (5): Because one of them is Pasul, there is no Bal Tosif. Even if one is Pasul, if it were tied to the Kosher one, Bal Tosif would apply to, like it says in Sukah (31a) and Sanhedrin. This explains the Rosh and Tur.

Mishnah Berurah (7): If he would intend for Tefilin for both of them, he would transgress Bal Tosif at least mid'Rabanan.

Bi'ur Halachah (DH va'Yechaven): The Rosh holds that Mitzvos require intent. Stam intent does not suffice. Perhaps he would not be Yotzei with intent for the Mitzvah with both, since one is Pasul. Therefore, he must stipulate to intend for the Mitzvah only with the Kosher pair. Additionally, this way one does not transgress Bal Tosif. He must say that the other pair is like mere straps so it will not be an interruption between the hand and head Tefilin.

Chasdei David (Korbanos 8 DH v'Adayin p.135): Even if one who wears two pairs if Tefilin needs intent to transgress Bal Tosif, mid'Rabanan it is forbidden even Stam, for it looks like he transgresses. Since the Tur says that one transgresses only for five boxes, why must one intend that the other is mere straps? Also, the Gemara says that one transgresses for two pairs! Rather, since the pairs are not tied to the other, since we hold that Mitzvos need Kavanah, one transgresses only with intent. One must intend that the other pair is straps to permit even mid'Rabanan.

Ben Ish Chai (1 Yayera 21): Both Tefilin are Kosher. One should wear both at the same time.

Halichot Olam Vayera 5. The Shulchan Aruch and some Mekubalim say that it is a Safek which is Kosher. One should intend that the other is mere straps, lest he transgress Bal Tosif.

Ohr l'Tziyon (2:3:8) One should stipulate once 'if the Kabalah is correct, I wear both l'Shem Mitzvah. If not, I wear l'Shem Mitzvah the pair that is like the Halachah, and the other is mere straps.