Bound variable pronoun


A bound variable pronoun is a pronoun that has a quantified determiner phrase – such as every, some, or who – as its antecedent.
An example of a bound variable pronoun in English is given in.
Each manager exploits the secretary who works for him.

In, the quantified DP is each manager, and the bound variable pronoun is him. It is a bound variable pronoun because it does not refer to one single entity in the world. Rather, its reference varies depending on which entities are encompassed by the phrase each manager. For example, if each manager encompasses both John and Adam, then him will refer variably to both John and Adam. The meaning of this sentence in this case would then be:
John1 exploits the secretary who works for him1, and Adam2 exploits the secretary who works for him2.

where him first refers to John, and then to Adam.
In linguistics, the occurrence of bound variable pronouns is important for the study of the syntax and semantics of pronouns. Semantic analyses focus on the interpretation of the quantifiers. Syntactic analyses focus on issues relating to co-indexation, binding domain, and c-command.

Semantics: quantifier interpretation

is the branch of linguistics that examines the meaning of natural language, the notion of reference and denotation, and the concept of possible worlds. One concept used in the study of semantics is predicate logic, which is a system that uses symbols and alphabet letters to represent the overall meaning of a sentence. Quantifiers in semantics – such as the quantifier in the antecedent of a bound variable pronoun – can be expressed in two ways. There is an existential quantifier, ∃, meaning some. There is also a universal quantifier, ∀, meaning every, each, or all. Ambiguity arises when there are multiple quantifiers in one sentence.
An example of the use of quantifiers is shown in.
Every man thinks he is intelligent.
= ∀x: x thinks x is intelligent.
= For every man x, x thinks x is intelligent.
≠ Every man thinks every man is intelligent.

In this example, the quantified determiner phrase every man can be expressed in predicate logic as a universal quantifier. Because of this, he refers universally and variably to each man, rather than to a single specific man.

Syntax

is the branch of linguistics that deals with the formation and structure of a sentence in natural language. It is descriptive, meaning it is concerned with how language is actually used, spoken, or written by its users, unlike prescriptive grammar/prescription which is concerned with teaching people the "correct way" to speak.
There are three main aspects of syntax that are important to the study of bound variable pronouns. These are:
  1. Co-indexation of a pronoun and its antecedent
  2. C-command relationship between the antecedent and the pronoun
  3. Binding domain of the pronoun

    Co-indexation

According to the indexing theory, each phrase in a sentence can be given a unique index, which is a number that identifies that phrase as picking out a particular entity in the world. It is possible to modify the indexes on these phrases so that two or more phrases have the same index. This is called co-indexation. If co-indexation occurs, the phrases with the same-numbered index will all refer to the same entity. This phenomenon is called co-reference.
An example of co-indexation and co-reference is shown in.
*i likes j
i likes i

In, the determiner phrases – Mary and herself – are each given a unique index. In, co-indexing takes place, and herself changes its index to be the same as the one that Mary has. Because of this, Mary and herself now refer to the same entity in this sentence.

C-command

If a pronoun has a quantified expression as antecedent, the pronoun must be c-commanded by this antecedent. An antecedent c-commands a pronoun if, when observing the structure of the sentence, a sister of the antecedent dominates the pronoun.
The c-command relationship can be shown by drawing a tree for the sentence. Take, for instance, the following tree diagram for example.
File:Syntax tree for "Mary likes herself" adapted from Sportiche et al..png|thumb|left|"Mary likes herself", adapted from Sportiche et al., 2014: 161, drawn using
Here, Mary is the antecedent, and herself is the pronoun. The sister of Mary is the T' node, and this node dominates herself. So Mary c-commands herself in this instance.
When discussing bound variable pronouns, the pronoun is said to be bound if it is c-commanded by the quantified determiner phrase that is its antecedent.

Binding domain

The domain of a determiner phrase is defined as "the smallest XP with a subject that contains the DP". This domain is illustrated in the picture below.
The way a DP can bind given this domain depends on the kind of that DP that is being bound. Anaphors must be bound in their domain, meaning they must have a c-commanding antecedent in their domain. Pronouns must not be bound in their domain, meaning they cannot have a c-commanding antecedent in their domain. Finally, R-expressions must not be bound, meaning they must not have a c-commanding antecedent at all.
When determining the binding possibilities of a bound variable pronoun, in addition to the above conditions, the bound variable pronoun must also be c-commanded by the quantified determiner phrase that is its antecedent.

Theories

Higginbotham's (1980) indexing theory

One theory used to describe pronominal binding is to use index marking rules to determine possible bindings. Index marking rules are rules used to determine which parts of a sentence carry the same reference. Each element in a sentence is given an index, which is a unique identifier of that element. A set of rules can then be applied to modify the index of one element to be the same as the index of another. Those two elements will then share the same index, and so will refer to the same thing. This indexing theory was used as a way to describe pronominal binding by Noam Chomsky, and expanded upon by James Higginbotham. The theory holds that the binding of pronouns consists of three main parts.
  1. There are coindexing rules that assign unique indexes to the elements in a sentence.
  2. There are contraindexing rules, which create a list of indexes for which an element cannot hold a reference.
  3. There are deletion/reindexing rules, which are rules used to allow some previously prohibited references to occur, and which modify index numbers of certain elements to be the same as another element and allow these two elements to refer to the same entity.

    Co-indexing and contra-indexing

In the coindexing stage, each noun phrase is given a unique index, called a "referential index". In the contraindexing stage, each non-anaphoric noun phrase is given a set of "anaphoric indices". This set consists of the referential indices of all elements that c-command it. This set of anaphoric indices is used to determine whether coreference can occur between two noun phrases. In order for coreference to occur, neither noun phrase can contain the other's referential index in its set of anaphoric indices. For example, in sentence :
"John" has a referential index of i, but its anaphoric index is empty, since it is not c-commanded by anything. "Him" has a referential index of j, and its set of anaphoric indices contains only i, because "John" c-commands "him". Since the set of anaphoric indices for "him" contains i, "John" and "him" cannot be coreferenced, which is expected in this sentence.
Deletion rules must then be applied to account for sentences with permissible coreference such as :
The deletion rule, as broadly stated by Chomsky, can be focused to pronouns as Higginbotham describes:
If B is a pronoun that is free in the minimal X = S or NP containing B and B is either:
nominative; or,
in the domain of the subject of X,
then i deletes from its anaphoric index.

Where a "pronoun B is free in X iff it occurs in X and there is nothing in X with the referential index i that c-commands B".

Reindexing rules

Once the appropriate indices are determined, bound variable pronouns can be coreferenced with their antecedents, where possible, by applying a set of reindexing rules. During this process, when one element is reindexed, all other elements with the same initial referential index will also be reindexed. Reindexing can also occur between a pronoun and a trace or PRO element, as follows:
In a configuration:
... ei... pronounj
reindex j to i.

Where ei is a trace or PRO element.
This reindexing rule is constrained by what Higginbotham calls the "C-Constraint", which states that reindexing cannot cause the following pattern to occur in the logical form of the sentence:
...j...pronouni...ej...

For example, a sentence such as:
it2s climate is hated by 4]3

would have the logical form:
4 3 it2s climate is hated by e3

In the logical form, the noun phrase everybody in some city is one logical unit, and the noun phrase some city is another. These phrases are unfolded and brought to the front of the form, leaving their traces behind to show where they would appear in the sentence. Without the C-constraint proposed above, applying the reindexing rule to this logical form would allow it2 to be reindexed to it4, resulting in the form:
4 3 it4s climate is hated by e3

This sentence, when reindexed, is supposed to carry the same meaning as "everybody in some city hates its climate", but does not do so correctly. With the C-Constraint in place, it2 would not be allowed to reindex to it4 which, Higginbotham claims, is what speakers of English would expect.