# Donkey sentence

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Donkey sentences are sentences that contain a pronoun with clear meaning (it is bound semantically) but whose syntactical role in the sentence poses challenges to grammarians.[1] Such sentences defy straightforward attempts to generate their formal language equivalents. The difficulty is with understanding how English speakers parse such sentences.[2]

Barker and Shan define a donkey pronoun as "a pronoun that lies outside the restrictor of a quantifier or the antecedent of a conditional, yet covaries with some quantificational element inside it, usually an indefinite."[3] The pronoun in question is sometimes termed a donkey pronoun or donkey anaphora.

The following sentences are examples of donkey sentences.

• "Omne homo habens asinum videt illum." ("Every man who owns a donkey sees it") — Walter Burley (1328), De puritate artis logicae tractatus longior[4][5]
• "Every farmer who owns a donkey beats it."[6]
• "Every police officer who arrested a murderer insulted him."

## History

Walter Burley, a medieval scholastic philosopher, introduced donkey sentences in the context of the theory of suppositio, the medieval equivalent of reference theory.

Peter Geach reintroduced donkey sentences as a counterexample to Richard Montague's proposal for a generalized formal representation of quantification in natural language (see Geach 1962). His example was reused by David Lewis (1975), Gareth Evans (1977) and many others, and is still quoted in recent publications.

## Features

Features of the sentence, "Every farmer who owns a donkey beats it", require careful consideration for adequate description (though reading "each" in place of "every" does simplify the formal analysis). The donkey pronoun in this case is the word it. 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 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, and their semantics are unambiguous. However, it is difficult to explain how donkey sentences produce their semantic results, and how those results generalize consistently with all other language use. If such an analysis were successful, it might allow a computer program to accurately translate natural language forms into logical form.[7] 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.

## Theories of donkey anaphora

It is usual to distinguish two main kinds of theories about the semantics of donkey pronouns. The most classical proposals fall within the so-called description-theoretic approach, a label that is meant to encompass all the theories that treat the semantics of these pronouns as akin to, or derivative from, the semantics of definite descriptions. The second main family of proposals goes by the name dynamic theories, and they model donkey anaphora -and anaphora in general- on the assumption that the meaning of a sentence lies in its potential to change the context (understood as the information shared by the participants in a conversation).[8]

### Description-theoretic approaches

Description-theoretic approaches are theories of donkey pronouns in which definite descriptions play an important role. They were pioneered by Gareth Evans's E-type approach,[9] which holds that donkey pronouns can be understood as referring terms whose reference is fixed by description. Later authors have attributed an even larger role to definite descriptions, to the point of arguing that donkey pronouns have the semantics,[10][11] and even the syntax,[12] of definite descriptions. Approaches of the latter kind are usually called D-type.

### 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):

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

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

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

In this case, the logical translation fails to give correct truth conditions to donkey sentences: Imagine a farmer not beating his donkey. 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, or not owned by this farmer, or is beaten by the farmer. Hence, if this object denotes a pig he also owns, anything unrelated, or even the farmer himself, the sentence will be true in that situation.

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

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

indicating that indefinites must sometimes be interpreted as existential quantifiers, and other times as universal quantifiers.

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.

### Dynamic Predicate Logic

Dynamic Predicate Logic models pronouns as first-order logic variables, but allows quantifiers in a formula to bind variables in other formulae.[13]

## Notes

1. ^ Emar Maier describes donkey pronouns as "bound but not c-commanded" in a Linguist List review of Paul D. Elbourne's Situations and Individuals (MIT Press, 2006).
2. ^ David Lewis describes this as his motivation for considering the issue in the introduction to Papers in Philosophical Logic, a collection of reprints of his articles. "There was no satisfactory way to assign relative scopes to quantifier phrases." (CUP, 1998: 2.)
3. ^ Chris Barker and Chung-chieh Shan, 'Donkey Anaphora is Simply Binding' Archived May 15, 2008, at the Wayback Machine, colloquium presentation, Frankfurt, 2007.
4. ^ Gualterus Burlaeus (1988). De puritate artis logicae tractatus longior. Meiner Verlag. ISBN 9783787307173.
5. ^ Keith Allan (2010). Concise Encyclopedia of Semantics. Elsevier. ISBN 9780080959696.
6. ^ Peter Geach (1962). Reference and Generality.
7. ^ Alistair Knott, "An Algorithmic Framework for Specifying the Semantics of Discourse Relations", Computational Intelligence 16 (2000).
8. ^ Paul Elbourne (2005). Situations and individuals. MIT Press. ISBN 9780262550611.
9. ^ Evans, Gareth (September 1977). "Pronouns, Quantifiers, and Relative Clauses (I)". Canadian Journal of Philosophy. 7 (3): 467–536. doi:10.1080/00455091.1977.10717030.
10. ^ Robin Cooper (1979). "The interpretation of pronouns". In Frank Heny; Helmut Schnelle (eds.). Syntax and Semantics 10: Selections from the third Gröningen roundtable. Academic Press. ISBN 012613510X.
11. ^ Stephen Neale (1990). Descriptions. The MIT Press. ISBN 0262640317.
12. ^ Irene Heim; Angelika Kratzer (1998). Semantics in Generative Grammar. Blackwell. ISBN 0631197133.
13. ^ Groenendijk, Jeroen; Stokhof, Martin (1991). "Dynamic Predicate Logic". Linguistics & Philosophy. 14: 39–100. doi:10.1007/BF00628304.

## References

• Kamp, H. and Reyle, U. 1993. From Discourse to Logic. Kluwer, Dordrecht.
• Kadmon, N. 2001. Formal Pragmatics: Semantics, Pragmatics, Presupposition, and Focus. Oxford: Blackwell Publishers.