When Eragon stated that he and Saphira traveled around far too much to be lugging around a half-dozen blades, Fredric referred to the Curse of the Named Blade as the reason why a warrior like Eragon wasn't expected to have more than one weapon. He stated that "every great warrior wields a sword—it's usually a sword—that has a name." He went on to explain that either the warrior named the sword himself or the bards named it for him after some great battle. From that point on, people would expect to see it and might even insist on waiting to fight until the warrior fetched that famed blade.
Fredric predicted that as soon as Eragon fought Murtagh, or did anything else of note with the new sword that he gave him, the Varden would insist on naming it and would expect to see it from then on. On the contrary, the Varden did not give a name to the falchion. Most likely this was because he had not yet wielded it for a long enough time and because it was destroyed soon afterwards under Tronjheim in a fight with several dwarf assassins.