B’tzel HaChochma: Writing Bais-Hey (B”H) or Bais-Samech-Daled (BS”D) On a Paper
There is a Minhag to write on the top of every page you write the letters Bais-Hey (B”H) for Boruch Hashem or B’Ezras Hashem. Does that preclude you from ever throwing out that paper? If so is it better to refrain from writing it?
The B’tzel HaChochma (4:105) says that we see that most Gedolim did in fact write B”H. Not only on Chidushei Torah that wouldn’t get thrown out but also papers that did not contain any Torah in it. The only time writing a single letter of Shem Hashem comes into question is if you intention is to refer specifically to a letter from the Shem “Yud-Kei”. In the case of B”H the Hey is not referring to a letter from the Shen Hakadosh. Another problem raised by the Radvaz is if you write the first letter of Yud-Kei or Elokim with a line on top signifying for the reader to finish the word. In the case of B”H the Hey is for the word “Hashem” which is not a Shem Hashem.
He therefore concludes that you should write B”H on all your letters and notes and you need not be careful of where it ends up. If you wrote B”H and you want to be machmir it is enough for you not to throw it directly into the garbage but there is no need to to save it. Better yet he suggests using the letter Bais-Samech-Dalet