Free Variables and Bound Variables - Natural Language

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.

Lisa found her book.

In the sentence above, the possessive 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:

Lisa_i found her_i book. (interpretation #1: her = Lisa)

Lisa_i found her_j book. (interpretation #2: her = female that is not Lisa)

The distinction is not purely of academic interest, as some languages do actually have different forms for her_i and her_j: for example, Norwegian translates her_i as sin and her_j as hennes.

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:

Jane hurt herself.

the reflexive herself can only refer to the previously mentioned antecedent Jane. It can never refer to a different female person. In other words, the person being hurt and the person doing the hurting are both the same person, i.e. Jane. The semantics of this sentence is abstractly: JANE hurt JANE. And it cannot be the case that this sentence could mean JANE hurt LISA. The reflexive herself must refer and can only refer to the previously mentioned Jane. In this sense, 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 ungrammatical interpretation is indicated with an asterisk):

Jane_i hurt herself_i. (interpretation #1: herself = Jane)

*Jane_i hurt herself_j. (interpretation #2: herself = a female that is not Jane)

Note that 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

((λx(x hurt x))(Jane))

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

Ashley met her.

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 met herself. The grammatical and ungrammatical interpretations are:

*Ashley_i met her_i. (interpretation #1: her = Ashley)

Ashley_i met her_j. (interpretation #2: her = a female that is not Ashley)

The first interpretation is impossible, but the second interpretation is grammatical (and in this case, is the only interpretation).

Thus, it can be seen that reflexives and reciprocals are bound variables (known technically as anaphors) while true pronouns can be free variables in some grammatical structures or 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)).