grammatical and semantic difference of "嘘と思う" and "嘘だと思う"

Question

Before I move on to the topic, here is my analysis based on my grammar book:

これを嬉しいと思う。

これを親切だと思う。

The bolded part are quotation (引用), and the と behaves as a 格助詞 (quotation particle).

これを嬉しく思う。

これを親切に思う。

The bolded part are adverbial phrases.

Following this logic, I conclude that

これを嘘だと思う。　→quotation (引用)

これを嘘と思う。　　→adverbial phrases

Hopefully someone could comment on my analysis, and more importantly, the difference of their usage. I mean, the situation when the two sentences are not interchangeable. One Japanese told me that the two sentences have little difference and can be used interchangeably, but I wonder if there is any exception.

Feel free to add more/change words to make the sentence sound more natural, 違和感がない. For instance, 「それをうそと思うかはあなた次第だ」

"当然のことと思う"/"当然のことだと思う" is another pair where I have the same question.

Answer 1

This depends on the verb you're using. For many verbs, AをBとV and AをBだとV are completely interchangeable. (I may be wrong, but maybe だ tends to be dropped more often if it's in either highly-casual speech or a highly-stiff written document.)

• それを嘘と思う = それを嘘だと思う
• それを嘘と見なす = それを嘘だと見なす
• それを嘘と考える = それを嘘だと考える
• それを嘘と仮定する = それを嘘だと仮定する

But you cannot add だ at least for these verbs:

• 娘を花子と名付ける (×娘を花子だと名付ける)
• これをゼータ関数と定義する (×これをゼータ関数だと定義する)
• 人はそれを愛と呼ぶ (×人はそれを愛だと呼ぶ)

I can't explain well why. Recently @DariusJahandarie introduced me an article exactly regarding this topic, so I hope it helps you too:

Tsuyoshi Sawada, Da-Deletion: Classification of Clause-Final Elements in Japanese, Nanzan Linguistics: Special Issue 3, Vol. 2, 139–163 (PDF)

Section 8, On The Ambiguity of To, seems to be particularly related. But I haven't read them all, and note that some examples marked with * (incorrect) actually look okay to me. Particularly, the author says AをBと思った is wrong (see (66)), but I don't think so.