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I found the following sentence in a math book (rigorously) defining the interior points of a region (denoted by R) in a plane (xy-plane):

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A point (x0 , y0) in a region R in the xy-plane is an interior point of R if it is the center of a disk of positive radius that lies entirely in R.

For which I'm a little confused to identify the subject that the relative pronoun "that" is referring to in the sentence. The grammar lessons I've learnt all telling me that for the case where the relative pronoun doesn't refer to the whole sentence then what comes immediately before it is subject (or object) that it's refering to.

So "positive radius" should be the subject in this case since it's followed directly by "that". However, I believe it should be "a disk" in this case, because otherwise, based on the illustration above the definition, the boundary point is also interior point (but the book has another definition for boundary points that says interior points aren't not boundary points, they're different).

Thus, I'm wondering if the sentence is grammatically correct or there's a typo in it. If it's correct, is there any grammar resources mentioning this case?

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  • "That" is a subordinator, not a relative pronoun, so it is not anaphoric. The covert relativised element appears to have "disk of positive radius" as antecedent: "... the center of a disk of positive radius that ___ lies entirely in R". The gap notation '____ ' marks the position of the subject of the verb "lies" and this gap is linked directly to the nominal "disk of positive radius".
    – BillJ
    Nov 17, 2023 at 10:39
  • In such contexts that doesn't really mean anything - it's just a syntactic requirement of the Simple Present verb lies. There's no such requirement with a continuous participle: ...the center of a disk of positive radius lying entirely in R. Only logic (not grammar) tells us that that represents a disk of positive radius - if it made more sense for it to refer back further, to the center of [such a disk], that's the reading we would apply. But that doesn't make sense, because a "centre" is a "dimensionless point", so it always lies entirely where it is Nov 17, 2023 at 12:31

2 Answers 2

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The subject here is "disc of positive radius". There is a formal ambiguity in English, since it could be bracketed differently. However since "A disc of (positive radius that lies entirely in R)" doesn't make mathematical sense, this is rejected.

The desire to avoid such ambiguities inherent in natural language led Hilbert and others to develop formal languages for geometry and other branches of math, that lose all ambiguity (but are almost completely incomprehensible to he human mind).

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  • Thanks for your answer! One more minor question. In common sense, and particularly in your sense, in the sentence "He wanted to get some feedback from his manager which is pretty important for him", what should be the subject that "which" is referring to?
    – Tran Khanh
    Nov 17, 2023 at 6:50
  • I asked because the following sentence "They're the people at John's party which are very famous" seems very natural, but "They are the people at John's party which was held yesterday" was reported incorrect by some grammar checker.
    – Tran Khanh
    Nov 17, 2023 at 7:01
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    Make sure you are using "that", "which" (or "{comma) which") and "who" correctly. The answer is the same. These might be formally ambiguous, but there is only one interpretation that makes sense. The manager is important, the people are famous, the party was yesterday.
    – James K
    Nov 17, 2023 at 7:24
  • @TranKhanh Note also the number agreement. "which are" is plural, so it can only refer to "people", since "party" is singular. But "which was" is singular, so it refers to "party".
    – Barmar
    Nov 17, 2023 at 15:21
  • @Barmar thanks for your help!
    – Tran Khanh
    Nov 18, 2023 at 2:00
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When someone said, "what comes immediately before", they mean the "thing" that comes immediately before, not the word. In this case, "disk" is the relevant noun. "of positive radius" is an adjective phrase modifying "disk".

As James K says, it could be ambiguous. You have to read the context. Life if you said, "I read a book with a paper cover that was torn", I'd probably take that to mean that the paper cover was torn. But if you said, "I read a book with a paper cover that was written by Fred Smith", I'd take that to mean that the book was written by Fred Smith, not that the cover was written by Fred Smith. In each case just because that interpretation makes the most logical sense. It's POSSIBLE that the speaker meant that the cover was written by Mr Smith, as in, Smith wrote the text on the cover. But that would be a funny way to say it so I wouldn't consider that unless something else in context called for it.

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