Are we allowed to teach Sifrei Mussar (for instance Orchos Tzadikim, Mesilas Yesharim, Chovos Halevavos) to non-Jews?
What about the Gra on Mishlei?
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R' Yechiel Perr, RY in Far Rockaway was asked by Dr Alan Morinis about this. Morinis's Mussar Institute teaches a lot of people whose spouse is Jewish, or who self-identify as Jewish but are far from being Jews halachically. I was present when the question was asked.
Rabbi Perr said that Mussar -- development of one's middos -- is also obligatory for non-Jews no less than Jews. So, like teaching them the laws incumbent on non-Jews as terms of the covenant with Noah, teaching Mussar is a mitzvah, not merely permissable.
(Even outside the institute's context of needing to teach to some non-Jews in order to reach a primarily Jewish population.)
Any errors are in my memory, and R' Perr (as a scion of Novhardok, this is his turf) should be consulted if one is asking for a practical halachic ruling.
The most common distinction made is between Written and Oral Torah. Mussar is not part of the first one but I am unclear if it is formally part of the second (which is traditionally meant to include Mishna and Gemara). Teaching the Oral Torah is more restrictive than the Written Torah which is open to nearly all.
In any case a number of poskim allow teaching Written and Oral Torah to non-Jews with the goal that they observe the seven Noachide commandments (to which they are obligated) and if it is not meant to disgrace Judaism or pass themselves for Jews.
R Shlomo Brody for instance writes here
Rabbi Menahem Hameiri connected this edict to a similar prohibition in the same talmudic passage which prohibits gentiles from observing Shabbat. He believed the prohibition stemmed from concerns that “insider knowledge” might allow the gentile to pass himself off as a Jew and undermine ritual behaviors. Yet gentiles genuinely searching for wisdom (or to observe other commandments) toward personal growth may study without impunity. Hameiri, preceded by Maimonides, asserted that gentiles who perform commandments, even beyond the Noahide laws, deserve reward for their virtuous behavior and therefore allowed for Torah study toward those goals.
Particularly notable was Rabbi Yehiel Weinberg, who himself taught at a pre-World War II German university. He contended that the proscription only banned gentile study intended to form competing religious ideals and rituals. It remained perfectly permissible, however, to teach even an exclusively non- Jewish audience if the goal was simply to spread Jewish wisdom.
I have once personally attended a number of Talmud classes given in a French university (to a mixed audience) by the talmid muvhak of R Weinberg (the Sridei Eish), now a rosh yeshiva in Jerusalem.
It seems to this writer that while there exists no obligation to volunteer information (although it may well be laudable to do so), there is an obligation to respond to a request for information. Jews are commanded to disseminate Torah as widely as possible among their fellow Jews, but there is no obligation to seize the initiative in teaching the seven commandments to Noachides. Nevertheless, when information or advice is solicited there is a definite obligation to respond. When the non-Jew takes the initiative in posing a query, the Jew must respond to the best of his ability.
But since some are more restrictive (e.g., on passing over the secrets of the Torah, see Maharsha and Meiri on Hagiga 13a), it is important to CYLOR in practical cases.