Why can't we say "... of its"?

This pattern is a colloquial use of possessive pronouns, and typically refers to people in a non-specific sense.

John Lawler has a good point here: its as a possessive pronoun in this pattern lacks specificity. The less specific the the possessive pronoun is, the less clear the sentence becomes, and the more meanings the sentence may take on.

The following sentence fragment:

a friend of his

implies a male subject of some kind. While not absolutely specific, the word his provides more information than the word its. You can re-write the sentence with the word his and still have a vague sense of who you are describing.

But the following sentence fragment:

a feature of its

while correct with respect to the pattern, can have multiple meanings:

its feature

a feature of its [design, application, downfall, facade, etc.]

Which meaning is the correct one?

Colloquial patterns do not always make for good written or spoken grammar. I could argue you should always write his friend instead of a friend of his because it is more concise. But you won't always hear it that way in spoken English.

In CGEL's terminology, we are talking about the oblique genitive construction (pp. 468-469). According to CGEL, compared to the non-oblique genitive, 'the range of semantic relations between subordinate and matrix NPs [is] considerably narrower in the oblique construction'. For example, while Mary's green eyes is equivalent to the green eyes of Mary's, we could not rephrase the summer's heat as ^*the heat of the summer's. (The asterisk '^*' in front of something signifies that what follows is not acceptable English.)

In the case of the summer's heat, we could of course use the non-genitive of-construction the heat of the summer. But that's a separate subject.

On p. 474, CGEL provides no fewer than 23 examples of semantic relations between the genitive NP and the head. For each example, it provides a corresponding clause and a generalized schema. For example, the clause corresponding to Mary's green eyes is Mary has green eyes., and the generalized schema is [d has body part h]. Here 'd' stands for 'dependent' and 'h' for 'head', since in Mary's green eyes, eyes is the head, and Mary's a dependent. And for the summer's heat, the clause is The summer is hot., and the schema is [d has non-human property h]. The motivation for which examples to include was this (p. 473):

the semantic distinctions made are typical of those known cross-linguistically to have different structural realisations even though they are similarly expressed in English.

The list of examples is organized in such a way that the oblique construction is perfectly acceptable in the upper half, but either questionable or clearly unacceptable in the lower half (CGEL, p. 478). And the main thing that distinguishes the entries in the upper half from those in the lower half is that those in the upper half concern humans, while those in the lower half do not.

Here is the list itself:

[54]  i  Mary's green eyes             Mary has green eyes.                          [d has body part h]
         ii  Mary s younger sister      Mary has a younger sister.            [d has kin relation h]
        iii  Mary's husband                Mary has a husband.               [d has married relation h]
         iv  Mary's boss                       Mary has a boss.                                      [d has superior h]
          v  Mary's secretary               Mary has a secretary.                     [d has subordinate h]
         vi  Mary's friend                     Mary has a friend.                                        [d has equal h]
        vii  Mary's team                       Mary belongs to a team.                       [d is member of h]
       viii  Mary's debut                      Mary performs her debut.                 [d is performer of h]
         ix  Mary's book                        Mary writes a book.                                 [d is creator of h]
          x  Mary's new house              Mary owns a new house.                          [d is owner of h]
         xi  Mary's honour                    Mary is honourable.                 [d has human property h]
        xii  Mary's anger                      Mary feels angry.                                       [d has feeling h]
       xiii  Mary's letter                        Mary receives a letter.                         [d is recipient of h]
       xiv  Mary's obituary                  Mary is the topic of an obituary. [d is human topic of h]
        xv  Mary's surgery                     Mary undergoes surgery.                [d is undergoer of h]
       xvi  the room's Persian carpet  The room contains a Persian carpet. [d is location of h]
      xvii  this year's new fashions    This year is a time of new fashions.           [d is time of h]
     xviii  the sun's rays                       The sun emits rays.                      [d is natural source of h]
        xix  the cathedral's spire          The cathedral has a spire.              [d has inherent part h]
         xx  the war's ancient origins  The war has ancient origins.                      [d has cause h]
        xxi  the flood's consequences   The flood has consequences.                       [d has result h]
       xxii  the lock's key                       The lock has a key.                       [d has associated part h]
      xxiii  the summer's heat             The summer is hot.            [d has non-human property h]

But the human-nonhuman distinction is not enough to predict whether the oblique genitive construction is possible: sometimes it is impossible even though the dependent (in the non-oblique genitive) is human. For example, note that while green eyes of Mary's is fine, ^*the green eyes of my boss's is not, at least to my ear. On the other hand, the green eyes of my mother's is at least marginally acceptable, again, at least to my ears. CGEL doesn't analyze this kinds of distinction (but some other source might); it seems that only some human roles, but not others, allow the oblique genitive construction.

Note also that an oblique paraphrase is normally not possible if the dependent in the non-oblique construction is an animal (the cat's owner but not ^*the owner of the cat's). Of course, this cat of Mary's is fine (but, again, ^*this cat of the owner's is not).

I should mention that, in many cases, one can use the non-genitive of-construction (either instead of or in addition to the oblique genitive), e.g. the owner of the cat. At the same time, there are certainly plenty of cases when the non-gentivie of-construction is not possible (e.g. ^*this cat of Mary). However, again, that's a different subject.

All this suggests that the problem with its—the reason why the genitive constructions in which its enters as a dependent cannot be turned into oblique genitives—is that its generically refers to non-human objects. More generally, the problem is that the semantic relations into which its enters as a dependent are not of the kind that can be expressible using the oblique genitive construction. While we don't have a clear necessary and sufficient condition for when the oblique construction is possible, we can say that it is at least necessary (though not sufficient) that the dependent in the non-genitive construction refer to a human—and its generically does not.

Further discussion on the semantic restrictions of the oblique genitive

In the book chapter 'The oblique genitive in English' by John Payne (in Morphosyntactic Categories and the Expression of Possession, K. Börjars, D. Denison, and A. Scott, eds, John Benjamins Publishing Company, Philadelphia, 2013), we find the following summary:

there is widespread agreement that the oblique genitive is considerably more semantically restricted than the subject-determiner genitive. Sometimes the constraint is expressed in terms of animacy. But opinions differ, e.g. the dependent must be human (Quirk et al. 1985: 1283), personal (Quirk et al. 1972: 203), or just animate (Lyons 1986; Taylor 1989). Sometimes the constraint is expressed in terms of semantic relations: Storto (2000) is most restrictive, claiming that the relationship in the oblique genitive must be one of “ownership or possession proper”. Taylor (1989) claims that the oblique genitive works best with “true” possession and various kinds of interpersonal relationship, but also “somewhat marginally” with what he calls authorship and attribution. Christianson (1997: 102) counterclaims, correctly we believe, that the authorship and attribution relations are far from marginal. For further discussion of the permitted semantic relations, see also Payne & Huddleston (2002: 478).

Examples of semantic relations occurring in the oblique genitive would be those in (7):

(7)  a.  that dog of yours          true possession
       b.  that son of theirs           interpersonal
       c.  that new book of his     authorship
       d.  no fault of mine            attribution

These relations can also be observed with the subject-determiner genitive:

(8)  a.  your dog          true possession
       b.  their son           interpersonal
       c.  his new book    authorship
       d.  my fault           attribution

However, some semantic relations which are possible for the subject-determiner genitive are claimed to be systematically excluded in the oblique genitive (examples adapted from Lyons 1986: 128–129, Christianson 1997: 101, Payne & Huddleston 2002: 478):

(9)  a.  ^*another destruction of the city’s    patient of event nominal
        b.   ^*that funnel of the ship’s                  proper part
        c.  ^*this weather of the summer’s         time
        d.  ^*that photograph of mine                theme

The term “theme” would here denote the person depicted on the photograph. Compare:

(10)  a.  the city’s destruction     patient of event nominal
         b.  the ship’s funnel              proper part
         c.  this summer’s weather   time
         d.  my photograph               theme

Examples such as those in (9a–c) and (10a–c) of course involve non-human dependents, and the unacceptable examples of the oblique genitive in (9a–c) would fall under the general ban on such. However, there are clearly also some relations, like the thematic relationship in (9d), which potentially involve human dependents and in which the oblique genitive is nevertheless excluded. Similarly, there is no improvement in examples illustrating the patient relationship when the patient is human: ^*another destruction of the Roman legion’s appears equally unacceptable, as does the patient interpretation in an example like ^*another rejection of Mary’s.

The consensus therefore appears to be that the oblique genitive occurs in a subset of the semantic relations permitted to the subject-determiner genitive. The dependent in the oblique genitive must be high on the animacy scale, and not in a thematic or patient-like relation with the head noun. Syntactically, a choice of determiner other than the dependent NP itself rules out the oblique genitive. However, the definiteness effect inherent in the subject-determiner construction does not totally rule out occurrences of the oblique genitive with the definite article.