Sloppy versus strict identity in ellipsis

Statement

The distinction between free and bound variables provides a powerful explanation for certain ambiguities that arise under VP-ellipsis. Consider the following sentence:

John loves his mother, and Bill does too.

This sentence has two distinct interpretations:

1. Strict identity: Bill loves John's mother.
2. Sloppy identity: Bill loves Bill's mother.

This ambiguity can be explained by the status of the pronoun his in the first clause.

• If his is treated as a free variable referring to John, the elided (or "missing") verb phrase is interpreted as "loves John's mother". When this is applied to Bill, the result is the strict reading.
• If his is treated as a bound variable bound by the subject of its clause (i.e., John), the verb phrase is interpreted as a lambda-abstracted property: λx.x loves x's mother. When this property is applied to Bill, the result is the sloppy reading.

The existence of the sloppy identity reading is considered strong evidence for the psychological reality of bound variable interpretations in the grammar of natural languages.

Thus, the distribution and interpretation of pronouns and other referring expressions in natural languages are not random but are governed by a sophisticated syntactic and semantic system.

The distinction between free and bound variables is a cornerstone of modern linguistic theory, providing the analytical tools necessary to account for coreference, quantification, question formation, and ellipsis.