I have two options. Which one is correct?
a) I think him to be about 50.
b) I think he is about 50.
If both are correct, should I avoid one or the other?
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It explains in this book that while "believe him to be" is a phrase commonly used in English (as is "consider him to be"), "think him to be" is not, and, further, that there is no apparent logic for why this should be so.
So the correct answer is (b), but if you're learning English as a second language, you shouldn't feel bad for not getting this right.
The vast majority of native speakers do not say *I think him to be about 50. This isn't some arbitrary idiomatic quirk, as has been suggested elsewhere - it's a subtle distinction based on the precise range of meanings covered by words such as think, believe, assume, judge, know, etc., and the implications of (pro)noun with "to be" + adjective constructions such as to be about 50.
I think you are beautiful gets 11,200 hits in Google Books, whereas I believe you are beautiful gets only 7 hits (and some of those seem to be from non-native speakers). Contrast this with I think you are honest (8240 hits), and I believe you are honest (3740).
GB "estimated results" behave very oddly with I think you to be honest - the first page says there are over 5000 results, but scrolling through you find there are only 11 (less than half-a-dozen, once you ignore the duplicates). By contrast, I believe you to be honest really does have over 2000.
Superficially, to think [sth] and to believe [sth] seem equivalent, but as the above examples clearly show, there's something else involved. Specifically, you can cause [sth] to be "true" (from your point of view), simply by "thinking" it. In the case of belief, [sth] is already objectively true or false, and you're either right or wrong depending on whether what you believe corresponds with reality. If you have an esoteric/solipsistic world-view I suppose you might think God [in]to be[ing] real, but in normal English you believe God to be real (or not, as the case may be). And your belief itself has no effect on His existence or lack thereof.
Why such a massive shift in verb preference simply dependent on whether the proposition being considered is phrased as you are honest, or you to be honest? We've already seen how think can imply a level of "interaction" with the "thought" that doesn't arise with believe. Now consider the different meanings of I expect you to be honest and I expect you are honest. Although the latter is relatively uncommon phrasing, I'm sure most people will understand it to be an "objective" statement of belief, whereas the former says what you would like to be true.
Since we commonly use the same format with "I want/would like/require/etc. you to be honest", obviously (pro)noun + "to be" + adjective associates strongly with constructions where the fact of [noun] being [adjective] is to some extent affected by our own attitude. But using the verb to think here is inherently problematic, because it highlights that uncomfortable split between personally bestowed/interactive assessment and objective belief.
I know this is already a long answer, but here's just one more set of GB results to ponder. I think they are explained by the fact that because "it" is more "impersonal" than "you", the potential for any implied "subjective interaction" with its "rightness" doesn't bother us so much.
TL;DR: Native speakers don't say they think [noun] to be [adjective] because this creates a "disjunct" between the verbs & phrasings we use for "wish fulfilment" and those we use for "objective assessment and statement of likelihood". If we intend the former, we say we think [noun] is [adjective]. If the latter, we believe [noun] to be [adjective].
There's a dedicated chat room if you want to see more of this issue being kicked around, or (please) contribute to the ongoing discussion and help find a clearer definition of why we avoid this usage.
It seems odd to be posting a second answer to a question on which I've offered a bounty, but in the circumstances I think this is justified.
My starting position is that unquestionably, the construction "I think him to be stupid" is avoided by native speakers who don't avoid "I believe him to be stupid" (contrastingly though, "I think he is stupid" is somewhat more common than "I believe him stupid").
It now seems clear that if there's a "rule" involved here, it's subtle and/or complicated - otherwise someone would have posted a clear authoritative answer by now. I don't retract the somewhat fuzzy explanation given in my other answer - my purpose here is simply to refute the proposition that this is an "idiomatic" avoidance, "learnt" by noticing that other people avoid it.
We easily learn to copy what others do. It's much harder to learn not to do what others don't do, particularly in a case like this where others aren't likely to say you made a "mistake" (as we now see, they'd be flummoxed if you asked why what you said was wrong.
For me, the question has therefore morphed into "Why do people adopt this avoidance pattern?", rather than "What is 'wrong' with the construction?". I've come to the conclusion each individual speaker is making his own decision. It may even not matter much if people have different reasons.
As mentioned above, another thing I gleaned from searching Google Books is that people actually write "I think he is stupid" more often than "I believe he is stupid" - and I'd be prepared to bet that preference is even more marked in speech than in writing. Taking these utterances...
We reckon it's okay
I guess you're right
She says he's ugly
...I think it's clear you could easily replace every verb there by think with no significant change in meaning. But would anyone recast them using "to be"?
?We reckon it to be okay
*I guess you to be right
*She says him to be ugly
My feeling is in actual speech patterns (as opposed to dictionary definitions) the usage associations of "to think" with verbs like the above are stronger than any links with "to believe". The avoidance pattern thus stems from the fact that we definitely don't use "to be" with closely-associated verbs.
I have no idea whether there's a "rule" debarring "I say him to be stupid", or what it might be. But since "everybody knows" we don't say it, it's easy to see how we extrapolate this to "to think".
Because the sentence is a response, its objective is communicating the crux: 50. Therefore, the more directly "I" (the answerer) is brought (or thought) to "50" (the answer), the more trustworthy it is. The second sentence is the closest we can forge the thought: I think he = 50. To dabble in that transit by elongating it with a mystifying infinitive does not help clarify our point but dilutes it so that one becomes suspicious that the response has ulterior influence. I think the discrepancy here is a very live one: hard to discern on written text.