Incomplete Descriptions and Qualified Definiteness
Bartosz Więckowski
TL;DR
The paper addresses the problem of incomplete descriptions by introducing qualified definiteness, which replaces the classical identity condition in Russell’s uniqueness with a notion of sameness across a chosen set of predicates $\mathcal{Q}$. It develops a formal, proof-theoretic framework by extending the bipredicational language to $\mathcal{L}\iota$ and building subatomic systems with positive/negative qualified identity, then adds $\iota_{\mathcal{Q}}$ rules to capture degrees of definiteness. A canonical, proof-theoretic semantics is formulated, linking the meaning of terms and formulas to canonical derivations and avoiding model-theoretic commitments; normalization and the subformula property are established for the resulting systems. The approach yields three gradations of reading for incomplete descriptions—maximal, restricted, and minimal—depending on the selected $\mathcal{Q}$, enabling fine-grained analyses of readings like strict uniqueness, restricted indiscernibility, and generic definiteness. This work provides a robust, non-semanticist account of incomplete descriptions suitable for non-denoting terms and sets the stage for applications to a broader class of definiteness phenomena and related linguistic constructions.
Abstract
According to Russell, strict uses of the definite article 'the' in a definite description 'the F' involve uniqueness; in case there is more than one F, 'the F' is used somewhat loosely, and an indefinite description 'an F' should be preferred. We give an account of constructions of the form 'the F is G' in which the definite article is used loosely (and in which 'the F' is, therefore, incomplete), essentially by replacing the usual notion of identity in Russell's uniqueness clause with the notion of qualified identity, i.e., 'a is the same as b in all Q-respects', where Q is a subset of the set of predicates P. This modification gives us qualified notions of uniqueness and definiteness. A qualified definiteness statement 'the Q-unique F is G' is strict in case Q = P and loose in case Q is a proper subset of P. The account is made formally precise in terms of proof theory and proof-theoretic semantics.