Disambiguation of pronoun reference has been an important issue for both theoretical and computational linguists. While linguistic theories on binding conditions eliminate impossible readings to a certain extent, many inter-sentential anaphora remain ambiguous. Nishiguchi (2011, 2012a,b, 2014, 2016a,b) consider pronoun resolution as a social choice among discourse participants which obeys Arrow's Impossibility Theorem (Arrow 1963). This paper further discusses discourse update of Social Welfare Function which provides updated variable assignment.
(1) Frances: ...Not while Emma’s not here. You know Emma Billy: Mm.
Frances: she’s, she was walking with Lisa and I weren’t there and her Mum sh– jus– , like she muc– , she mucks about a lot and she told Leigh that if he don’t serve her he’s gonna die, she’s gonna punch him right! However, proximity does not always resolve referential ambiguity of pronouns. Him in (2a) unambiguously means someone other than the closest John—some discourse-salient entity. In (2b), the pronoun is ambiguous.
b. Johni said hefi=jg likes himselffi=jg.
In linguistic binding theory
However, (3) is ambiguous in four ways and can have either one of the following interpretations: i) John broke John’s leg, ii) John broke Bill’s leg, iii) Bill broke Bill’s leg, or iv) Bill broke John’s leg. He and his can be bound by either John or Bill. The binding theories have no way of disambiguating these pronouns since there is no way of knowing speaker intention. Proximity does not predict the different readings in (3) either. (3) Anna: Billj is a good goalkeeper.
Kim: Johni said hei=j broke hisi=j leg recently. 1
Although Social Choice Theory
Axiom 1 states that the relation R is
connected—every candidate is related to each
other. Relations that satisfy Axiom 2 are
transitive. In (4), N, a finite set of individuals or voters,
consists of five individuals and , a nonempty set
of alternatives or candidates, has three members.
Let L( ) denote the set of all linear orders on
. A profile R is a vector of linear orders, or
preferences. Ri is a vector of preferences of an
individual i. NxR>y denotes the set of individuals
that prefer the candidate x to y. Supposing R the
profile given in this model, NoR>c is a set of people
who prefers Obama to Clinton, that are, Anna,
Heather and George
A social welfare function (SWF) F is a function which takes individual’s preferences and returns collective preference. Arrow demonstrated that any SWF for three or more alternatives the following conditions must be a dictatorship. Condition 2 states that the relative ranking of two candidates remains unchanged regardless of other candidates. Theorem 1 (General Possibility Theorem (Impossibility Theorem)). If there are at least three alternatives which the members of the society are free to order in any way, then every social welfare function satisfying Conditions 1 and 2 and yielding a social ordering satisfying Axioms 1 and 2 must be either imposed or dictatorial.
Condition 1 (Pareto condition). A SWF F satisfies the Pareto condition if, whenever all individuals rank x above y, then so does society: NxR>y = N implies xF(R)y Condition 2 (Independence of irrelevant alternatives (IIA)). A SWF F satisfies IIA if the relative social ranking of two alternatives only depends on their relative individual rankings: NxR>y = NxR>0y implies xF(R)y , xF(R’)y Condition 3 (Nondictatorship). There is no individual i such that for every element in the domain of rule f, 8x, y 2 X: xPiy ! xPy (Sen 1979)
SWF for pronoun resolution satisfies Arrow’s Impossibility Theorem, or General Possibility Theorem, by satisfying Axioms 1, 2, Pareto Condition and IIA but demonstrating dictatorship. Pronoun resolution is compared with voting by multiple voters, discourse participants. The candidates or choices would be different interpretation of the sentence. In (5), the referent of he is ambiguous. Chris meant he to be Bob, while Naomi interpreted him to be John. As the disagreement on pronominal reference is consolidated in the discourse, pronoun resolution is certainly a social choice and Social Choice Function (SCF) decides the antecedent. (5) Chris: John said he broke his leg.
Naomi: Did he? John looked fine when I saw him this morning.
Chris: It is Bob who broke his leg.
Naomi: I thought you were talking about John.
When individuals I = fc, ng, candidates = fj, bg, Chris and Naomi’s ordering is jRcb ^ bRnj, denote the set of linear orders on by L( ). Preferences (or ballots) are taken to be elements of L( ). A profile R 2 L( )I is a vector of preferences. SCF or voting rule is a function F : L( )I ! 2 n; mapping a given profile to a nonempty set of winners; e.g., a singleton set fbg for (5). SWF is a function F : L( )I ! L( ) mapping any given profile to a (single) collective preference order. Although the preferences between the candidates vary between the individuals, SWF returns a single preference order and ambiguities are resolved during the conversation.
There are three possible antecedents for she in (1)—Emma (e), Lisa (l) and Lisa’s mother (m). Let us say that Billy (b) prefers e to l, and also l to m to be the antecedent. On the other hand, the speaker Francis (f) prefers m to l, and l to e according to the proximity. All three candidates are ordered in accordance with Axiom 1, i.e., eRbl ^ lRbm and mRf l ^ lRf e. Transitivity also holds for pronoun antecedent preferences. Each of them implies eRblRbm and mRf lRf e. SWF for pronoun resolution also meets Pareto condition. When the interpretation of the addressees agrees with the one of the speaker, the decision of the society follows. It is unlikely that pronouns refer to someone else other than speaker’s intention and hearer’s interpretation. A SWF F satisfies IIA if the relative social ranking of two alternatives only depends on their relative individual rankings. Let us say that the preference relations are denoted by R and R’. Assume that IIA does not hold and consider a dialogue in (7) where the relative rankings between Bob and John is affected by irrelevant candidate Victor’s ranking. The social decision differs from the relative ranking between John and Bob of speaker and hearer, which does not happen, in (8). (6) Chris: Bob is a good skier. But John said he broke his leg.
Naomi: Did he? Poor Bob! (7) Chris: Victor is a good skier and so is Bob.
But John said he broke his leg.
Naomi: Did he? Poor Bob! (8) bR’cvR’cj ^ bR’njR’nv 6! jF(R)b
Then, NbR>j = NbR>0 j implies bF(R)j , bF(R’)j The speaker’s decision on pronominal reference dictates the social preference. Even when there is disagreement or misunderstanding, the speaker corrects unifies interpretation in general, as in (9). Pronoun resolution is dominated, or dictated, by the speaker’s meaning. (9) Chris: Bob is a good skier. But John said he broke his leg.
Naomi: Did he? Poor Bob!
Chris: No. I mean John broke his leg. (10) xPcy ! xPy Proof. Suppose: xPcy ! xPy, that is, xPcy ! yRx, where R is weak preference. However, the dialogue normally proceeds jPcb ! jPb as in (10). Contradiction.
Lemma 1. The social welfare function for pronoun resolution is IIA and Pareto but is dictatorial. 3
In linguistic literature, a variable assignment function g has been assumed to assign the referent to indices indexed to pronouns. For example, g may assign John to the variable x: g(x) = John. Now, g can be considered to be SCF which selects a referent for a pronoun socially. Let us define g and the space as in (11). The assignment function g is updated throughout the discourse as in (12). (11) a. g = f<x, i>: x refers to ig b. Information state consists of Social Welfare Function F, Social Choice Function g for variable assignment, individual’s preferences R, individuals in the discourse X, a set of indices such as i, a set of discourse participants V, and relation between decisions B.
= < F, G, R, X, I, V, B> (12) 1 There were ooh’s and aah’s when hex1 finished, and some unbridled laughter. Aileena was looking dubiously at hery1 husbandh but hex2 was in no mood to disapprove.
2 Hex3 winked at the Duked and called across to himx4, ‘What a grand thing, your Honour, to have a wedding without a minister!’ The Duked did hisx5 stately bow at that and then Donaldm was calling for another song.
3 Some of the veteransv were on the point of giving tongue but young Donald McCullochm was on hisx6 feet and moving into the middle of the ring, hex7 was full of himselfx8, sparkling with mischief but with an undertow of ardour.
4 ‘Duncan Ban MacIntyreb wrote a song for hisx9 wife Maryr.
5 I do not know if Alexl used it to court his10 M aryr – hex11 must have used something —‘The joke was unconscious but crowing laughter came from the young menn beside the whisky jar. (BNC A0N1311-1315, King Cameron) (13) a. g1= f<y1, a>, <x2, h>g
I = fa, rg (a: author, r: reader)
S = fa, hg b. g2 = f<x3, h>, <x4, d>g
S = fa, h, d, mg c. g3 = f <x6, m>, <x7, m>, <x8, m>g
S = fa, h, d, v, mg d. g4 = f<x9, b>g
S = fa, h, d, v, m, b, rg e. g5 = f<x10, l>, <x11, l>g
S =fa, h, d, v, m, b, r, l, ng f. [[hery]]g1 = a
G is regarded as SCF. Also, the set of best
elements S’ can be called its choice set of the whole
set of alternaties, and is denoted g(S’, R)
As the author’s dynamic preferences change in the discourse as in (15a), g is updated throughout the discourse by means of a relation B. (15) a. 2: hRad for hex3 ^ dRah for hex4 ^ dRahIam for hex5 (aIxb: x is indifferent between a and b, ^: dynamic conjunction) b. Social Decision: hRd ^ dRh ^ dRhIm c. B(gn,gn+1) (16) Dynamic Social Welfare Function:
FnBFn+1BFn+2,... 4
Dynamic Predicate Logic
In order to implement Dynamic Social Choice for pronoun disambiguation, speaker’s intention needs to be detected from the text. The phrases such as “I mean” are used to resolve ambiguity of pronominal reference in the discourse. as in (17). (17) ‘...And Sarah Morgan likes the idea of Angela marrying someone in the government.’ McLeish considered this cold and rational assessment. ‘When did you last see her? Miss
Out of 18 instances of “I mean PNP” (PNP stands for proper name” ) found with the query “I mean N” in BNC, 7 instances had a preceding pronoun, the caraphor.