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I was curious about the difference between "to inculcate" and "to indoctrinate." What distinguishes these synonyms?

Maybe the two can mean the same thing but they aquired different connotations.

From Merriam-Webster:

indoctrinate verb : to teach (someone) to fully accept the ideas,
opinions, and beliefs of a particular group and to not consider other
ideas, opinions, and beliefs

This conveys a non-neutral, negative meaning. We usually hear that people got indoctrinated and joined a sect or a terrorist group for example.

inculcate verb : to cause (something) to be learned by (someone) by repeating it again and again

This can convey a more neutral, or better yet, positive meaning. Example: "dedicated teachers inculcating young minds with a love of learning."

James Fernald, English Synonyms, Antonyms and Prepositions (1899) lists both inculcate and indoctrinate as synonyms of teach, although it doesn't discuss either term individually. The updated edition of Fernald, Funk & Wagnalls Standard Handbook of Synonyms, Antonyms & Prepositions (1947) repeats this treatment.

But S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968) takes a different approach, listing indoctrinate under teach (where it appears alongside educate, instruct, school, and tutor) but moving inculcate under implant (where it appears alongside imbue, infuse, ingrain, inseminate, and instill). Here are Hayakawa's remarks about the two words:

Inculcate limits th[e] range of possibilities in instill to one situation, that in which both teacher and student are well aware of the learning process under way; a further limiting exists in that the word [inculcate] refers specifically to an ingraining of facts, ideas, or attitudes by the technique of laborious repetition: a generation of students that had not inculcated with the rules of grammar. {English spelling cannot be reasoned out; it must be inculcated, example by insufferable example.} The word more recently has taken on a disapproving tone to refer to the deliberate ingraining of propaganda or flagrant falsehoods in unsuspecting subjects: inculcating the doctrines of race hatred in innocent children.

...

Indoctrinate alone of these words [in the teach category], suggests the inculcation of propaganda or prejudices rather than unbiased knowledge: parents who indoctrinate their children with race hatred; schools that unconsciously indoctrinate their students with middle-class values.

Merriam-Webster's Dictionary of Synonyms (1984) similarly lists inculcate (along with instill) under the entry for implant:

implant, inculcate, instill are comparable when they mean to introduce into the mind. ... Inculcate implies persistent or repeated endeavor with the intent to impress firmly on the mind [examples omitted]

Merriam-Webster's synonym dictionary doesn't list or discuss indoctrinate at all—a surprise since it seems so similar to inculcate in Hayakawa's descriptions. Like the definitions cited in Mina's answer, Hayakawa's discussion makes inculcate seem less malign than indoctrinate. Inculcate can refer to teaching something undesirable, but the central notion of the word is "ingraining in the mind through repetition," which is also how one learns things as practical and unbiased as the multiplication tables. Indoctrinate, in Hayakawa's telling, involves specifically propagandistic and prejudicial teaching-by-ingraining. It reminds me of the term reeducate, used (euphemistically) in Vietnam after the communist victory there to describe the inculcation or indoctrination of former supporters of the South Vietnamese government with orthodox communist thought.

A step beyond either inculcate or indoctrinate is brainwash, which suggests not reeducation but a kind of hypnosis-based alteration of someone's mind that dispenses with repetition or indeed any attempt to engage with a person's reasoning faculties and instead take over the person's mental control room, as in The Manchurian Candidate.

So, if I understand correctly, inculcation is to indoctrination as education is to propaganda?