buIn this sentence you have two verbs and so you must have two clauses. It is fairly evident that "The girl that I love lives in London" is the main clause.
In "That I love", "that" has to be a relative pronoun because "to love" is a transitive verb, and so it must have an object in the clause; there is no other possibility than "that" for an object. Moreover this clause is a restrictive clause ("not any girl but the girl I love"); it is well known that only "whom", "that" and "zero pronoun" have this property as objective case pronouns. It would be complete nonsense to make of subordinators the object of the verb.
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ADDITION
The text that is contested in CGEL has been copied in full below for those who would like to know exactly what is being criticized (CGEL's demoting "that" from the rank of relative pronouns).
3.5.6 That as a subordinator (not a relative pronoun)
Traditional grammar analyses the that which introduces relative clauses as a relative
pronoun, comparable to which and who, but we believe that there is a good case for identifying it with the subordinator that which introduces declarative content clauses.
(a) Wide range of antecedent types and relativised elements
If that were a pronoun, or pro-form, its use would be much wider than that of the uncontroversial relative pronouns, or indeed of any pro-form at all in the language. Compare:
[71]
i They gave the prize to the girl [that spoke first]. [who]
ii Have you seen the book [that she was reading]? [which]
iii He was due to leave the day [that she arrived]. [when]
iv He followed her to every town [that she went]. [where]
v That’s not the reason [that she resigned]. [why]
vi I was impressed by the way [that she controlled the crowd]. [ ** how]
vii It wasn’t to you [that I was referring]. [no wh form]
viii She seems to be the happiest [that she has ever been].
[no wh form]
It would not only cover the ground of all the simple ‘wh’ words put together, as shown in [i–v]: it would also appear in a variety of constructions where no ‘wh’ word could replace it, as in [vi–viii]. Particularly important here is the cleft construction shown in [vii], and in [70iii–iv] above. Note that, leaving aside the disputed case of the relative construction, there is no pro-form in English that takes as antecedent such complements and adjuncts as to you (in the sense it has in [71vii]), with considerable misgivings, in order to avoid this kind of misunderstanding, and the like. Instead of postulating a pro-form with such an exceptional range of use, we are saying that that relatives do not contain any overt pro-form linked to the
antecedent: they simply have an anaphoric gap, like bare relatives.
(b) Lack of upward percolation
There are no that relatives matching wh relatives with a complex relative phrase:
[72]
i a. the woman [whose turn it was] b. the woman[that’s turn it was]
ii a. the knife [with which he cut it] b. the knife [with that he cut it]
If that were a pronoun we would have to stipulate that it has no genitive form, and that it never occurs as complement of a preposition – or rather that when it is complement of a preposition the latter must be stranded, for the knife that he cut it with is quite grammati-
cal. The severe restrictions here stand in sharp contrast to the remarkable versatility of the putative pronoun that illustrated in (a). In the analysis where that is a subordinator the un-
grammaticality of [72ib/iib] is predictable. Subordinators do not inflect and must occupy initial position; there is no relative word and hence no possibility of the relative feature percolating upwards into a larger constituent.9
(c) Finiteness
That relatives are always finite, as are the declarative content clauses introduced by that. Note, then, that we cannot insert that into non-wh relative infinitivals like a knife to cut it with – cf.
*a knife that to cut it with. If that were a pronoun this would be a special fact needing explanation, but under the subordinator analysis it is exactly what we would expect, given
that that is a finite clause subordinator.10
(d) Omissibility
As we have noted, that can be regarded as very largely omissible in relative clauses in the same way as in declarative content clauses. The conditions under which omission is prohibited are not the same in the two cases (those for content clauses are given in Ch. 11, §3.1), but in both they have it in common that they are related to the need to mark explicitly the beginning of a subordinate clause under certain structural conditions. And in both cases, moreover, that is more readily omitted in simple structures than in complex ones. There is no pro-form in English that is systematically omissible under remotely similar conditions.
9 There are non-standard regional dialects of English in which that’s does occur, as in the man that’s leg was
broken. We do not believe that such examples necessitate a pronoun analysis for the dialects concerned, and
certainly they do not establish this analysis as valid for all dialects.
10 The force of this argument is diminished by the fact that which can’t occur here either: we have a knife with
which to cut it, not a knife which to cut it with. The absence of *a knife with that to cut it is then already covered under point (b). Nevertheless, the analysis of that as a finite clause subordinator does provide a very general account of why the only type of bare relative that can’t be expanded by means of that should be the infinitival
one.
(a), (b), (c), and (d) are considered below in some detail, but very probably, more remains to be said.
I (Wide range of antecedent types and relativised elements)
This first argument does not hold.
The wide range of relativized elements is due to a phenomenon which is a tendancy towards simplification, in particular as manifest among the less proficient users of the language. The word has become an easy "filler" to be used when dealing with all sorts of relativized elements that leave a fuzzy picture in the mind before the day comes where one is perfectly familiar with them; necessarily the role it plays in doing so is the role of the more specific word (or combination of words) . That does not confer to the word a universal nature. There is no reasoning showing that.
Also, the extent to which "that" is used has nothing to do with its role in the clause; it is probably the case that it appeared ulteriorly as a pronoun (the ngrams collected below tend to support this contention (blue curve almost always above zero at the start) and in most cases did not gained more currency than the wh-pronouns, one of the possible reasons explaining that being a want in specificity whereas the greater "uniqueness" of the role of the wh-pronoun gives them that added specificity ("that" is a demonstrative, a pronoun, a conjunction (now called "complementizer")); the fact that there is a modern trend to choose the more specific pronoun (blue curve dominates often at the end) shows that simplification has less appeal (possibly due to an increase in literacy in recent times).
It is said as an argument towards proving the non-pronoun nature of "that" that on top of serving as a "wh"-word, it is also used in constructions where it has nothing to do with those words; but, more generally, "that" as a demonstrative has nothing to do with wh-words ; is that a sufficient reason to say that it must not be a demonstrative? I don't think so. The reasoning adopted in H&P is entirely similar and therefore fallacious.
The fact that there is no grammatical word used to refer to "to you" proves nothing. In the more usual and explicit "It wasn't you (that) I was referring to." the usual relations are clear and they appear neatly in the plain corresponding sentence "Who(m) I was referring to was not you.". The fact that "that" took on a new role in cleft sentences, where wh-words fail to render the needed effect, is a proof of nothing. In "It was the dog to which I gave the water.", "to which" keeps its relative restrictive role, and therefore cannot be a marker of the cleft-construction. It becomes evident that the simplifying use of "that" in this sentence would blur the meaning, although in speech, main stress on "dog" tends to clear up the ambiguity.
the girl who/that spoke first
the book which/that she read
the day when/that she arrived
the reason why/that he resigned
the reason why/that she was
II (Lack of upward percolation)
The second argument fares no better.
As seen in "I", "that" has very probably been introduced or used consistently enough after the more usual pronouns had been established, possibly at first as simplifications that suited less proficient users of the language. This is usage; why should the locutors have introduced different forms of "that" made to correspond to each one of the wh-words? Why this redundany in the first place? If the context was that of simplification and relaxed language, there was certainly no interest in systemization, and the lack of these forms in the language is quite natural; that does not take anything away tfrom the initial nature of pronoun of "that". The other contention, namely, the impossibility of combining "that" with certain prepositions in the way that is possible for the wh-word would reinforce the belief that "that" is not a pronoun, does not hold water either. Here again, we are dealing with usage. "That", let's not forget, is originally essentially "the" (Etymonline); it is clear that several shifts took place from article, to demonstrative, and finally to relative and conjunction. We can speculate as to what caused combinations such as "with that", "to that" and "that's" to have remained inexistant when "that" is a relative; perhaps this is so for the first two because they occur in plenty when "that" is the demonstrative ("to that", "with that"), or because they appeared to be too complicated (if simplification was the main idea, this is even more likely)—and as concerns "that's" see note "9" above in H&P's text (proof that usage is king)—, but it is clear that if they begin to be used nowadays, no principle can keep people from doing that, because there are none; these new combinations will be legitimate new constructions of the language. Therefore this is merely a question of usage.
III (Finiteness)
The question of what verbal forms are usable in a relative clause introduced by "that" yields no more light on the nature of "that" than the preceding argument (II); the reasoning is similar. Why it is not acceptable to say "That's the knife with that to cut the vegetables" is a matter of usage, strictly. The day the user's mind can get to the point of apprehending easily that in this verbal context "with that" is not a PP and that "that" can't have but the role of relative pronoun, there is a new legitimate usage in the language and "that" is certainly a pronoun in it. The fact that both finite and nonfinite verbs are used with the wh-pronouns is inconsequential: it is not unlikely that in most languages this would have been achieved in two steps, first the finite verbs, and later the nonfinite.
IV (Omissibility)
The possible omission of "that" is certainly a matter of simplification, and it has so far been considered to be an informal practice; it is in harmony with the idea that what prompted this word into the rank of the pronouns could have been simplification. There is again nothing inherent in the various possibilities of omission that point to a particular grammatical nature of this word.
CRITERION
(CoGEL) The chain relationship is an 'and' relationship, whereas the choice relationship is an 'or' relationship.
Thus if two units X and Y occur one after the other in a larger unit, they are in a chain relationship, X + Y. But if X and Y can be substituted for one another in a larger unit, they are in a choice relationship, X/Y.
(Substitution here means 'commutability', ie acceptable replacement in terms of the structure of the sentence, not necessarily in terms of meaning.)
Given the sentences
- "They told the only person who was left in the room to leave right away.", and
- "They told the only person that was left in the room to leave right away.",
it can be said that "who" and "that" are commutable; moreover the meaning is the same in both sentences. This is a garantee that "that" is a pronoun.
Similarly, "They told a person whom they knew as being a relative." and "They told a person that they knew as being a relative." show that "that" is the accusative relative pronoun. However, "They told a person whom they knew." and "They told a person that they knew." does not allow to say anything about "that", except that it is either pronoun or complementizer.