I am wondering if there is a way to describe this concept in Japanese, or an
equivalent phrase.

According to <http://en.wikipedia.org/wiki/Crisis_of_faith> it is a mostly
western concept (if you believe wikipedia anyway).

I am asking in relation to the definition in the first paragraph, not in the
religious sense.

This phrase relates to “deciding whether to make (or being forced to make) a
choice that goes against ones most basic beliefs.

And in the process, either retain that belief, or drop it in favor of another
one. It is this situation that causes the internal conflict.

I do not believe it is directly related to simple inability to make a
decision, not knowing what to believe, nor the process of self-discovery in
general.”
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meaning phrase-requests
Is there a Japanese phrase equivalent to "crisis of faith"
There are a few phrases that are commonly used.

  * 「分からなくなってきた」（e.g. 何を信じればいいのか分からなくなってきた、自分の気持ちが 分からなくなってきた、あなたのことが分からなくなってきた）

  * 「自分探し」 (e.g. やりたい事が見つからないので、自分探しにいってきます）

You could also use
葛藤、e.g.「自分がやってきたことが正しかったのか、すごく葛藤した時期」、「どちらの道に進むべきか分からず、葛藤し続けた」
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2013-03-05T03:30:40.050
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Depending on the context, you might also be able to use 疑心暗鬼（ぎしんあんき）, meaning
doubt begets doubt, which is commonly used to express a sort of negative
momentum in one's thinking.
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