Is one allowed to study other religions? If not, how is one supposed to know that Judaism is correct? If yes, is such study encouraged?
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Mishne Tora, Avoda Zara 2:2 says the study of books on how to worship idols, written by their worshipers, is forbidden. Beyond that I don't know.
The premise here seems to be that one cannot know Judaism is correct except by eliminating other religions. But that can't be, as no one has the time to eliminate every other (existing or possible) religion! Rather, it must be we know Judaism is correct because of internal, as opposed to comparative, reasons.
The Rambam (Avodah Zarah 2:2) prohibits studying from the works of Avodah Zara (and other forms of heresy), and this is codified in the Shulchan Aruch (Yoreh Deah 139). Rav Moshe Feinstein (Yoreh Deah 2:53) says that one is allowed to study religions that no longer exist, as in ancient mythologies, because everyone today knows how foolish they are.
I've heard based on an interpretation of the Sefer Hachinuch (Introduction and Mitzvah 213) that the reason for this prohibition is that despite the fact that the Torah is absolute truth, we in out foolishness might come to be swayed by erroneous claims of heretics. Therefore, God did us the favor, so to speak, of prohibiting these works so that we don't waste our time trying to figure out whether their claims have any merit.
The Rambam in Hilchos Avodas Kochavim (פרק ב׳- הלכות א,ג) says:
The Rambam forbids the study of Avoda Zarah due to the prohibition of אל תפנו, this would imply that if the study was meant not to follow the ways of Avoda Zarah but rather to understand for specific purposes it would be permitted.
Indeed, this idea has precedent in the Gemara Rosh Hashana 24b:
Rashi on the passuk mentions this Gemara as well:
Not learn to do. But you learn to understand and instruct, that is, to understand how their actions are destructive, and instruct your children not to do so, as it is the ways of the nations.
Rav Hirsch also echoed such a distinction:
It is forbidden to study the ways of idolatry to emulate them, but you can learn them for the purpose of theoretical knowledge to understand and appreciate their emptiness. Moreover, except under certain conditions such study is necessary for members of the Rabbinical Court to discuss court cases properly that are brought before them.
In summation: If the studying of the other religions will aide in your Torah study, or it will teach you something beneficial, there are opinions that it is permitted.
This is not to be relied upon as practical halacha. Consult your Local Orthodox Rabbi
Rabbi Moshe Feinstein instituted a ban on the study of evolution.
presumably because it could lead people who rely on their limited views astray
The mishna in Megila ch.4 calls heretics "Chitzonim", which the bartenura defines as those who trust their own intellect over those of the sages.
You ask how can one know whether his religion is the truth. But how much of Judaism have you really understood? the Moray Nevuchim alone with the shem tov commentary can take a lifetime to properly understand (and this requires proper guidance).
I recommend you go speak to some big wise Rabbi about your doubts in emuna.