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Summary According to Russell's classic semantics for definite descriptions, descriptions like "the F" are satisfied only if there is a unique F.  Other accounts of definite descriptions opt to make this uniqueness requirement a presupposition rather than a part of the strict semantic content expressed by the description.  Either way, such a requirement seems at odds with how we commonly use descriptions: to talk about objects that neither are, nor are presupposed to be, uniquely F.  This puts the predictions of many popular theories of definite descriptions at odds with at least one aspect of common usage.  Some theorists have proposed to deal with this "problem of incompleteness" by allowing that the context can effectively supplement the available descriptive material.  Others have proposed to deal with this problem by appealing to a supplemental pragmatic theory.
Key works Strawson 1950 first appealed to incomplete descriptions as an objection to Russell 1905's theory of definite descriptions.  Kripke 1977 allows that the problem of incomplete descriptions might constitute a sufficient reason to accept that there are semantically-significant referential uses of definite descriptions.  Neale 1990 argues, conversely, that this is not so.  More recently, the literature on incomplete descriptions has become bound up in the literature on quantifier-domain restriction.  Since the Russellian analysis has it that definite descriptions are to be treated as a certain sort of quantifier, it stands to reason that solutions to this larger problem should port over to the problem of incomplete descriptions.  This strategy is pursued explicitly in Stanley & Szabó 2000.
Introductions Ludlow 2008
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  1. Barbara Abbott (2006). Definite and Indefinite. In Keith Brown (ed.), Encyclopedia of Language and Linguistics. Elsevier. 3--392.
  2. Alan Brinton (1977). Definite Descriptions and Context-Dependence. Noûs 11 (4):397-407.
  3. Ray Buchanan (2010). A Puzzle About Meaning and Communication. Noûs 44 (2):340-371.
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  4. Ray Buchanan & Gary Ostertag (2005). Has the Problem of Incompleteness Rested on a Mistake? Mind 114 (456):889-913.
    A common objection to Russell's theory of descriptions concerns incomplete definite descriptions: uses of (for example) ‘the book is overdue’ in contexts where there is clearly more than one book. Many contemporary Russellians hold that such utterances will invariably convey a contextually determined complete proposition, for example, that the book in your briefcase is overdue. But according to the objection this gets things wrong: typically, when a speaker utters such a sentence, no facts about the context or the speaker's communicative (...)
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  5. Delia Graff Fara (forthcoming). Names Are Predicates. Philosophical Review.
    Tyler Burge convinced us that names are predicates in at least some of their occurrences: -/- There are relatively few Alfreds in Princeton. -/- Names, when predicates, satisfy the being-called condition: schematically, a name "N" is true of a thing just in case that thing is called N. This paper defends the unified view that names are predicates in all of their occurrences. I follow Clarence Sloat, Paul Elbourne, and Ora Matushansky in saying that when a name seems to occur (...)
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  6. Laurence Goldstein (2002). The Indefinability of “One”. Journal of Philosophical Logic 31 (1):29 - 42.
    Logicism is one of the great reductionist projects. Numbers and the relationships in which they stand may seem to possess suspect ontological credentials - to be entia non grata - and, further, to be beyond the reach of knowledge. In seeking to reduce mathematics to a small set of principles that form the logical basis of all reasoning, logicism holds out the prospect of ontological economy and epistemological security. This paper attempts to show that a fundamental logicist project, that of (...)
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  7. Dale Jacquette (ed.) (2002). A Companion to Philosophical Logic. Wiley-Blackwell.
    ... and new questions in philosophical logic arose, when Russell introduced his ... whether Scott is the author of Waverley without wishing to know whether ..
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  8. Ernest Lepore & Kirk Ludwig (2000). The Semantics and Pragmatics of Complex Demonstratives. Mind 109 (434):199-240.
    Complex demonstratives, expressions of the form 'That F', 'These Fs', etc., have traditionally been taken to be referring terms. Yet they exhibit many of the features of quantified noun phrases. This has led some philosophers to suggest that demonstrative determiners are a special kind of quantifier, which can be paraphrased using a context sensitive definite description. Both these views contain elements of the truth, though each is mistaken. We advance a novel account of the semantic form of complex demonstratives that (...)
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  9. Friederike Moltmann (1997). Parts and Wholes in Semantics (TOC). Oxford University Press.
    This book present a unified semantic theory of expressions involving the notions of part and whole. It develops a theory of part structures which differs from traditional (extensional) mereological theories in that the notion of an integrated whole plays a central role and in that the part structure of an entity is allowed to vary across different situations, perspectives, and dimensions. The book presents a great range of empirical generalizations involving plurals, mass nouns, adnominal and adverbial modifiers such as 'whole', (...)
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  10. Stephen Neale (1990). Descriptions. Mit Press.
    When philosophers talk about descriptions, usually they have in mind singular definite descriptions such as ‘the finest Greek poet’ or ‘the positive square root of nine’, phrases formed with the definite article ‘the’. English also contains indefinite descriptions such as ‘a fine Greek poet’ or ‘a square root of nine’, phrases formed with the indefinite article ‘a’ (or ‘an’); and demonstrative descriptions (also known as complex demonstratives) such as ‘this Greek poet’ and ‘that tall woman’, formed with the demonstrative articles (...)
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  11. Gary Ostertag (2002). Descriptions and Logical Form. In Dale Jacquette (ed.), A Companion to Philosophical Logic.
  12. Gary Ostertag (1999). A Scorekeeping Error. Philosophical Studies 96 (2):123-146.
  13. Marga Reimer (1998). Quantification and Context. Linguistics and Philosophy 21 (1):95-115.
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  14. Jason Stanley & Zoltán Gendler Szabó (2000). On Quantifier Domain Restriction. Mind and Language 15 (2&3):219--61.
  15. P. F. Strawson (1950). On Referring. Mind 59 (235):320-344.
  16. Zsófia Zvolenszky (2007). Incomplete Descriptions, Incomplete Quantified Expressions (Part of the Dissertation Portfolio Modality, Names and Descriptions). Dissertation, New York University
    This paper offers a unified, quantificational treatment of incomplete descriptions like ‘the table’. An incomplete quantified expression like ‘every bottle’ (as in “Every bottle is empty”) can feature in true utterances despite the fact that the world contains nonempty bottles. Positing a contextual restriction on the bottles being talked about is a straightforward solution. It is argued that the same strategy can be extended to incomplete definite descriptions across the board. ncorporating the contextual restrictions into semantics involves meeting a complex (...)
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