Insemantics, adonkey sentence is a sentence containing apronoun which is semanticallybound but syntactically free. They are a classic puzzle informal semantics andphilosophy of language because they are fullygrammatical and yet defy straightforward attempts to generate theirformal language equivalents. In order to explain how speakers are able to understand them, semanticists have proposed a variety of formalisms including systems ofdynamic semantics such asDiscourse representation theory. Their name comes from the example sentence "Every farmer who owns a donkey beats it", in which "it" acts asa donkey pronoun because it is semantically but not syntactically bound by the indefinitenoun phrase "a donkey". The phenomenon is known asdonkey anaphora.^[a]

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^[3][4]
• Every farmer who owns a donkey beats it.^[5]
• If a farmer owns a donkey, he beats it.
• Every police officer who arrested a murderer insulted him."

The goal offormal semantics is to show how sentences of anatural language such asEnglish could be translated into a formal logical language, and so would then be amenable to mathematical analysis. FollowingRussell, it is typical to translateindefinite noun phrases using anexistential quantifier,^[6]as in the following simple example from Burchardtet al:

"A woman smokes." is translated as${\displaystyle \mathrm{\exists }x(\text{WOMAN}(x)\wedge \text{SMOKES}(x))}$^[7]

The prototypical donkey sentence, "Every farmer who owns a donkey beats it.", requires 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 wordit. Correctly translating this sentence will require using auniversal quantifier for the indefinite noun phrase "a donkey", rather than the expected existential quantifier.

The naive first attempt at translation given below is not a well-formed sentence, since the variable${\displaystyle y}$ is leftfree in the predicate${\displaystyle \text{BEAT}(x,y)}$.^[8]

${\displaystyle \mathrm{\forall }x(\text{FARMER}(x)\wedge \mathrm{\exists }y(\text{DONKEY}(y)\wedge \text{OWNS}(x,y))\to \text{BEAT}(x,y))}$

It may be attempted to extend thescope of the existential quantifier to bind the free instance of${\displaystyle y}$, but it still does not give a correct translation.^[8]

${\displaystyle \mathrm{\forall }x\mathrm{\exists }y(\text{FARMER}(x)\wedge \text{DONKEY}(y)\wedge \text{OWNS}(x,y)\to \text{BEAT}(x,y))}$

This translation is incorrect since it is already true if there exists any object that is not a donkey: Given any object to be substituted for${\displaystyle x}$, substituting any non-donkey object for${\displaystyle y}$ makes thematerial conditional true (since itsantecedent is false), and so existential clause is true for every choice of${\displaystyle x}$.

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

${\displaystyle \mathrm{\forall }x\mathrm{\forall }y((\text{FARMER}(x)\wedge \text{DONKEY}(y)\wedge \text{OWNS}(x,y))\to \text{BEAT}(x,y))}$,

indicating that indefinites must sometimes be interpreted as existential quantifiers, and other times as universal quantifiers.^[8]

There is nothing wrong with donkey sentences: they are grammatically correct, they are well-formed and meaningful, and their syntax is regular. 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 intological form.^[9] It is unknown how natural language users agree – apparently effortlessly – on the meaning of sentences such as the examples.^{[citation needed]}

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

It is usual to distinguish two main kinds of theories about the semantics of donkey pronouns. The most classical proposals fall within the so-calleddescription-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 ofdefinite descriptions. The second main family of proposals goes by the namedynamic 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).^[10]

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Description-theoretic approaches are theories of donkey pronouns in which definite descriptions play an important role. They were pioneered byGareth Evans's E-type approach,^[11] which holds that donkey pronouns can be understood as referring terms whose reference is fixed by description.

For example, in "Every farmer who owns a donkey beats it.", the donkey pronoun "it" can be expanded as a definite description to yield "Every farmer who owns a donkey beats the donkey he/she owns." This expanded sentence can be interpreted along the lines of Russell'stheory of descriptions.^[12]

Later authors have attributed an even larger role to definite descriptions, to the point of arguing that donkey pronouns have the semantics,^[13][14] and even the syntax,^[15] of definite descriptions. Approaches of the latter kind are usually calledD-type.

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

The solution thatDRT provides for the donkey sentence problem can be roughly outlined as follows: The common semantic function of non-anaphoricnoun phrases is the introduction of a newdiscourse 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.

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Dynamic Predicate Logic models pronouns asfirst-order logicvariables, but allows quantifiers in aformula to bind variables in other formulae.^[16]

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

Peter Geach reintroduced donkey sentences as acounterexample toRichard Montague's proposal for a generalized formal representation ofquantification innatural language.^[5] His example was reused byDavid Lewis (1975),^[17]Gareth Evans (1977)^[11] and many others, and is still quoted in recent publications.

• Epsilon calculus – Extension of a formal language by the epsilon operator
• Garden-path sentence – Sentence that starts in a way that a reader's likely interpretation will be wrong
• Generic antecedent – Representatives of classes in a situation in which gender is typically unknown
• Lambda calculus – Mathematical-logic system based on functions
• Montague grammar – Approach to natural language semantics
• Singular they – Gender-neutral English pronoun

• 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'. InProceedings 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'. InProceedings of Sinn und Bedeutung11. Edited by E. Puig-Waldmüller. Barcelona:Pompeu Fabra University, 2007. Pages 106–120.
• Evans, Gareth. 'Pronouns, Quantifiers, and Relative Clauses (I)'.Canadian Journal of Philosophy7 (1977): 467–536.
• Geurts, Bart.'Donkey Business'.Linguistics and Philosophy25 (2002): 129–156.
• Huang, C-T James.'Logical Form'. Chapter 3 inGovernment 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, Yoshihisa.'Copying Variables'. Chapter 2 inFunctional 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'. InFormal 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.

• Pronouns
• Quantifier (logic)
• Semantics
• Formal semantics (natural language)

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