Spectral theory of ordinary differential equations

A donkey pronoun is a pronoun that is bound in semantics but not syntax. Some writers prefer the term donkey anaphora, since it is the referential aspects and discourse or syntactic context that are of interest to researchers (see anaphora). The terms d-type or e-type pronoun are also used, mutually exclusively, dependent on theoretical approach to interpretation. A sentence containing a donkey pronoun is sometimes called a donkey sentence.

The term donkey pronoun was coined from a counterexample provided by Peter Geach (1962) to Richard Montague's proposal for a generalized formal representation of quantification in natural language. The example was reused by David Lewis (1975), Gareth Evans (1977) and many others, and is still quoted in recent publications. The original donkey sentence is as follows.

• Every farmer who owns a donkey beats it. — Peter Geach, Reference and Generality

This sentence is significant because it represents a class of well-formed natural language sentences that defy straightforward attempts to generate their formal language equivalents. The difficulty is with understanding how English speakers parse the scope of quantification in such sentences. Additionally, the indefinite article 'a' is normally understood as an existential quantifier, but the most natural reading of the donkey sentence requires it to be understood as a nested universal quantifier. There are other features of the sentence that require careful consideration for adequate description. (Notice, however, how reading "each" in place of "every" simplifies the formal analysis.)

The donkey pronoun in the example sentence is the word it.

There is nothing wrong with donkey sentences; they are grammatically correct, they are well-formed, their syntax is regular. They are also logically meaningful, they have well-defined truth conditions, their semantics are unambiguous. It is precisely this that makes them interesting. The difficulty is with explaining how syntactic elements give rise to the semantic result and in a way that generalizes consistently with all other language use. Then, for example, we could program a computer to accurately translate natural language forms into logical form. The question is, how are natural language users, apparently effortlessly, agreeing on the meaning of sentences like these?

There may be several equivalent ways of describing this process. In fact, Hans Kamp (1981) and Irene Heim (1982) independently proposed very similar accounts in different terminology, which they called discourse representation theory (DRT) and file change semantics (FCS) respectively.

In 2007, Adrian Brasoveanu published studies of donkey pronoun analogs in Hindi, and analysis of complex and modal versions of donkey pronouns in English.

Discourse Representation Theory

Donkey sentences became a major force in advancing semantic research in the 1980s, with the introduction of discourse representation theory (DRT). During that time, an effort was made to settle the inconsistencies which arose from the attempts to translate donkey sentences into first-order logic.

Donkey sentences present the following problem, when represented in first-order logic: The systematic translation of every existential expression in the sentence into existential quantifiers produces an incorrect representation of the sentence, since it leaves a free occurrence of the variable y in BEAT(x.y):

$\forall x\,({\text{FARMER}}(x)\land \exists y\,({\text{DONKEY}}(y)\land {\text{OWNS}}(x,y))\rightarrow {\text{BEAT}}(x,y))$

Trying to extend the scope of the existential quantifier also does not solve the problem:

$\forall x\,\exists y\,({\text{FARMER}}(x)\land {\text{DONKEY}}(y)\land {\text{OWNS}}(x,y)\rightarrow {\text{BEAT}}(x,y))$

In this case, the logical translation fails to give correct truth conditions to donkey sentences: Imagine a situation where there is a farmer owning a donkey and a pig, and not beating any of them. The formula will be true in that situation, because for each farmer we need to find at least one object that either is not a donkey owned by this farmer, or is beaten by the farmer. Hence, if this object denotes the pig, the sentence will be true in that situation.

A correct translation into first-order logic for the donkey sentence seems to be:

$\forall x\,\forall y\,({\text{FARMER}}(x)\land {\text{DONKEY}}(y)\land {\text{OWNS}}(x,y)\rightarrow {\text{BEAT}}(x,y))$

Unfortunately, this translation leads to a serious problem of inconsistency. Indefinites must sometimes be interpreted as existential quantifiers, and other times as universal quantifiers, without any apparent regularity.

The solution that DRT provides for the donkey sentence problem can be roughly outlined as follows: The common semantic function of non-anaphoric noun phrases is the introduction of a new discourse referent, which is in turn available for the binding of anaphoric expressions. No quantifiers are introduced into the representation, thus overcoming the scope problem that the logical translations had.

See also

• Epsilon calculus
• Generic antecedent
• Lambda calculus
• Montague grammar
• Singular they
• Donkey sentence

Notes

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Further reading

• Abbott, Barbara. 'Donkey Demonstratives'. Natural Language Semantics 10 (2002): 285–298.
• Barker, Chris. 'Individuation and Quantification'. Linguistic Inquiry 30 (1999): 683–691.
• Barker, Chris. 'Presuppositions for Proportional Quantifiers'. Natural Language Semantics 4 (1996): 237–259.
• Brasoveanu, Adrian. Structured Nominal and Modal Reference. Rutgers University PhD dissertation, 2007.
• Burgess, John P. E Pluribus Unum: Plural Logic and Set Theory', Philosophia Mathematica 12' (2004): 193–221.
• Cheng, Lisa LS and C-T James Huang. 'Two Types of Donkey Sentences'. Natural Language Semantics 4 (1996): 121–163.
• Cohen, Ariel. Think Generic! Stanford, California: CSLI Publications, 1999.
• Conway, L. and S. Crain. 'Donkey Anaphora in Child Grammar'. In Proceedings of the North East Linguistics Society (NELS) 25. University of Massachusetts Amherst, 1995.
• Evans, Gareth. 'Pronouns'. Linguistic Inquiry 11 (1980): 337–362.
• Geach Peter. Reference and Generality: An Examination of Some Medieval and Modern Theories. Ithaca, New York: Cornell University Press, 1962.
• Geurts, Bart. Presuppositions and Pronouns. Oxford: Elsevier, 1999.
• Harman, Gilbert. 'Anaphoric Pronouns as Bound Variables: Syntax or Semantics?' Language 52 (1976): 78–81.
• Heim, Irene. 'E-Type Pronouns and Donkey Anaphora'. Linguistics and Philosophy 13 (1990): 137–177.
• Heim, Irene. The Semantics of Definite and Indefinite Noun Phrases. University of Massachusetts Amherst PhD dissertation, 1982.
• Just, MA. 'Comprehending Quantified Sentences: The Relation between Sentencepicture and Semantic Memory Verification'. Cognitive Psychology 6 (1974): 216–236.
• Just, MA and PA Carpenter. 'Comprehension of Negation with Quantification'. Journal of Verbal Learning and Verbal Behavior 10 (1971): 244–253.
• Kanazawa, Makoto. 'Singular Donkey Pronouns Are Semantically Singular'. Linguistics and Philosophy 24 (2001): 383–403.
• Kanazawa, Makoto. 'Weak vs. Strong Readings of Donkey Sentences and Monotonicity Inference in a Dynamic Setting'. Linguistics and Philosophy 17 (1994): 109–158.
• Krifka, Manfred. 'Pragmatic Strengthening in Plural Predications and Donkey Sentences'. In Proceedings from Semantics and Linguistic Theory (SALT) 6. Ithaca, New York: Cornell University, 1996. Pages 136–153.
• Lappin, Shalom. 'An Intensional Parametric Semantics for Vague Quantifiers'. Linguistics and Philosophy 23 (2000): 599–620.
• Lappin, Shalom Lappin and Nissim Francez. 'E-type Pronouns, i-Sums, and Donkey Anaphora'. Linguistics and Philosophy 17 (1994): 391–428.
• Lappin, Shalom. 'Donkey Pronouns Unbound'. Theoretical Linguistics 15 (1989): 263–286.
• Lewis, David. Parts of Classes, Oxford: Blackwell Publishing, 1991.
• Lewis, David. 'General Semantics'. Synthese 22 (1970): 18–27.
• Partee, Barbara H. 'Opacity, Coreference, and Pronouns'. Synthese 21 (1970): 359–385.
• Montague, Richard. 'Universal Grammar'. Theoria 26 (1970): 373–398.
• Neale, Stephen. Descriptions. Cambridge: MIT Press, 1990.
• Neale, Stephen. 'Descriptive Pronouns and Donkey Anaphora'. Journal of Philosophy 87 (1990): 113-150.
• Quine, Willard Van Orman. Word and Object. Cambridge, Massachusetts: MIT Press, 1970.
• Rooij, Robert van. 'Free Choice Counterfactual Donkeys'. Journal of Semantics 23 (2006): 383–402.
• Yoon, Y-E. Weak and Strong Interpretations of Quantifiers and Definite NPs in English and Korean. University of Texas at Austin PhD dissertation, 1994.
• Kamp, Hans. and Reyle, U. 1993. From Discourse to Logic. Kluwer, Dordrecht.
• Kadmon, N. 2001. Formal Pragmatics: Semantics, Pragmatics, Presupposition, and Focus. Oxford: Blackwell Publishers.

External links

• The Handbook of Philosophical Logic
• Discourse Representation Theory
• Introduction to Discourse Representation Theory
• SEP Entry
• Archive of CSI 5386 Donkey Sentence Discussion
• Barker, Chris. 'A Presuppositional Account of Proportional Ambiguity'. In Proceedings of Semantic and Linguistic Theory (SALT) 3. Ithaca, New York: Cornell University, 1993. Pages 1–18.
• Brasoveanu, Adrian. 'Donkey Pluralities: Plural Information States vs. Non-Atomic Individuals'. In Proceedings of Sinn und Bedeutung 11. Edited by E. Puig-Waldmüller. Barcelona: Pompeu Fabra University, 2007. Pages 106–120.
• Evans, Gareth. 'Pronouns, Quantifiers, and Relative Clauses (I)'. Canadian Journal of Philosophy 7 (1977): 467–536.
• Geurts, Bart. 'Donkey Business'. Linguistics and Philosophy 25 (2002): 129–156.
• Huang, C-T James. 'Logical Form'. Chapter 3 in Government and Binding Theory and the Minimalist Program: Principles and Parameters in Syntactic Theory edited by Gert Webelhuth. Oxford and Cambridge: Blackwell Publishing, 1995. Pages 127–177.
• Kamp, Hans. 'A Theory of Truth and Semantic Representation'. In J. Groenendijk and others (eds.). Formal Methods in the Study of Language. Amsterdam: Mathematics Center, 1981.
• Kitagawa, Yoshihishi. 'Copying Variables'. Chapter 2 in Functional Structure(s), Form and Interpretation: Perspectives from East Asian Languages. Edited by Yen-hui Audrey Li and others. Routledge, 2003. Pages 28–64.
• Lewis, David. 'Adverbs of Quantification'. In Formal Semantics of Natural Language. Edited by Edward L Keenan. Cambridge: Cambridge University Press, 1975. Pages 3–15.
• Montague, Richard. 'The Proper Treatment of Quantification in Ordinary English'. In KJJ Hintikka and others (eds). Proceedings of the 1970 Stanford Workshop on Grammar and Semantics. Dordrecht: Reidel, 1973. Pages 212–242.