Eiruvin, Chapter 9, Mishnah 4. The central principle underlying this Mishnah is the view of Rabbi Yehudah, who holds that according to Torah law two partitions - two walls - are enough to create a private domain. On the basis of this principle he permits certain things, as we will see below.
The text of the Mishnah:
"HaBoneh aliyah al gabei shnei batim, v'chein gesharim hamefulashin - metaltelin tachteihen b'Shabbos, divrei Rabbi Yehudah. V'chachamim osrin"
"HaBoneh aliyah al gabei shnei batim" - two houses standing on either side of a public domain, with an upper story built above them extending from one house to the other, over the public domain.
"V'chein gesharim hamefulashin" - a bridge with a public domain passing beneath it: there are walls on two sides, but the two ends of the public domain are open, and the bridge passes above them.
In both of these cases Rabbi Yehudah rules that one may carry underneath them on Shabbos - beneath the bridge, or beneath the upper story built between the two houses.
'Pi tikrah yoreid v'soseim':
Beyond the principle that the two sides constitute a private domain according to Torah law, Rabbi Yehudah needs a further principle for the two open sides, known as 'pi tikrah yoreid v'soseim'. 'Pi tikrah' means the edge of the ceiling or roof, and we view it as though it descends and extends downward, sealing off that side. Thus, even if we are dealing with a beam only a cubit wide at the edge of the roof, we regard it as though it descends all the way to the ground. Rabbi Yehudah applies this principle to the other two sides in both of these cases.
"V'chachamim osrin" - the Sages disagree and hold that since the sides are open, we do not apply 'pi tikrah yoreid v'soseim' on the two sides, and therefore one may not carry there underneath them.
"V'od amar Rabbi Yehudah: me'arvin l'mavoi hamefulash":
On the basis of that same principle - that two sides create a private domain according to Torah law - Rabbi Yehudah adds that one may make an eiruv, that is, a shituf mevo'os, even in an open alleyway. An ordinary alleyway is closed on three sides; an open alleyway is closed on only two parallel sides, with both of its ends open. In Rabbi Yehudah's view, just as in an alleyway closed on three sides one sets up a lechi or a beam on the open side, so too in an open alleyway one sets up a lechi or a beam on each of the two open sides, and that is enough to make a shituf mevo'os.
"V'chachamim osrin" - in the view of the Sages this does not help. What then does help, according to them, in an open alleyway that is open on both sides? That on one side he set up a tzuras hapesach - an actual doorway form - and on the other side, the fourth side, it is enough to have a lechi or a beam. After that one may make a shituf mevo'os and carry in the alleyway.
In summary: The Mishnah presents the view of Rabbi Yehudah, that two partitions suffice according to Torah law to create a private domain: therefore one may carry beneath an upper story built upon two houses and beneath open bridges, together with the rule of 'pi tikrah yoreid v'soseim' on the two open sides; and likewise one may make an eiruv for an open alleyway by setting up a lechi or a beam at both of its ends. The Sages forbid in both cases, and in an open alleyway they require a tzuras hapesach on one side and a lechi or a beam on the other.