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When we are referring more than one proper noun, then do we need to capitalize the first alphabet of it. For example: which one of the following is correct?

1) By Theorems 4.2 and 4.3, we can prove the following.

2) By theorems 4.2 and 4.3, we can prove the following.

  • Most people write "Presidents Roosevelt and Truman" rather than "presidents Roosevelt and Truman". See Ngram. I think this answers your question. (I was actually quite surprised by how large the minority who lowercase "presidents" here is, when almost nobody lowercases "president Truman".) – Peter Shor Dec 1 '14 at 11:42
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"theorem" is not a proper noun. Capitalization is not needed.

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    But "Theorem 4.2" is a proper noun, as is "Theorem 4.3". See the comments under this related question, for example. – Dan Bron Dec 1 '14 at 8:49
  • According to you. "theorem 4.2" and "theorem 4.3" happen to be capitalized, probably because they were capitalized when used as a section heading in a textbook. That does not make "theorem" a proper noun. It should not have been capitalized in this or the "related" question or answers. One could refer to, say, the second chapter of a book as Chapter Two, but that does not make "chapter" a proper noun. You are correct that it does not need "the", but that is because it already carries its own qualifier, namely the number. – Brian Hitchcock Dec 1 '14 at 11:19
  • @Brian: by your reasoning, since "president" is not a proper noun, we should write "presidents Roosevelt and Truman" and not "Presidents Roosevelt and Truman". And since "ocean" is not a proper noun, we should write "Pacific ocean"? Do I correctly understand what you're saying? – Peter Shor Dec 1 '14 at 11:33
  • You are correct; honorifics are typically capitalized. – Brian Hitchcock Dec 1 '14 at 11:44
  • ... And in math textbooks, references like "Theorem 4.2" are typically capitalized, even in constructions like "by Theorem 4.2, there is a ...". – Peter Shor Dec 1 '14 at 11:45

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