structural:reformatting
Natural language

When analyzed in formal semantics, natural languages can be seen to have free and bound variables. In English, personal pronouns like he, she, they, etc. can act as free variables.

In the sentence above, the possessive pronoun her is a free variable. It may refer to the previously mentioned Lisa or to any other female. In other words, her book could be referring to Lisa's book (an instance of coreference) or to a book that belongs to a different female (e.g. Jane's book). Whoever the referent of her is can be established according to the situational (i.e. pragmatic) context. The identity of the referent can be shown using coindexing subscripts where i indicates one referent and j indicates a second referent (different from i). Thus, the sentence Lisa found her book has the following interpretations:

The distinction is not purely of academic interest, as some languages do actually have different forms for her and her: for example, Norwegian and Swedish translate coreferent her as sin and noncoreferent her as hennes.

English does allow specifying coreference, but it is optional, as both interpretations of the previous example are valid (the ungrammatical interpretation is indicated with an asterisk):

However, reflexive pronouns, such as himself, herself, themselves, etc., and reciprocal pronouns, such as each other, act as bound variables. In a sentence like the following:

the reflexive herself can only refer to the previously mentioned antecedent, in this case Jane, and can never refer to a different female person. In this example, the variable herself is bound to the noun Jane that occurs in subject position. Indicating the coindexation, the first interpretation with Jane and herself coindexed is permissible, but the other interpretation where they are not coindexed is ungrammatical:

The coreference binding can be represented using a lambda expression as mentioned in the previous Formal explanation section. The sentence with the reflexive could be represented as

in which Jane is the subject referent argument and λx.x hurt x is the predicate function (a lambda abstraction) with the lambda notation and x indicating both the semantic subject and the semantic object of sentence as being bound. This returns the semantic interpretation JANE hurt JANE with JANE being the same person.

Pronouns can also behave in a different way. In the sentence below

the pronoun her can only refer to a female that is not Ashley. This means that it can never have a reflexive meaning equivalent to Ashley hit herself. The grammatical and ungrammatical interpretations are:

The first interpretation is impossible. Only the second interpretation is permitted by the grammar.

Thus, it can be seen that reflexives and reciprocals are bound variables (known technically as anaphors) while true pronouns are free variables in some grammatical structures but variables that cannot be bound in other grammatical structures. The binding phenomena found in natural languages was particularly important to the syntactic government and binding theory (see also: Binding (linguistics)).