Donkey sentence

A donkey sentence is a kind of sentence that occurs in natural language, where an anaphoric pronoun (like hó) refers to a quantified variable (like sá aqshe) even though it appears to be outside of that variable's scope.

It is named after the following prototypical example of such a sentence:

Tú haqpaoche, ꝡë bo hóa sá aqshe, nä kıaı hóa áqshe da.
“Every farmer who owns a donkey, cares for it.”

The English sentence is clearly valid English. But the straightforward translation into logic of such a sentence is not well-formed:

⚠️ ∀f : Farmer(f) ∧ ∃d [Donkey(d) ∧ Owns(f, d)] → Cares(f, d).

The underlined instance of variable d is illegal, because it is outside of the scope of the quantifier ∃d.

In 2021, Hoemaı remarked that "donkey anaphora can't really be allowed in Toaq, unless you use a separate set of pronouns or a donkey conversion mechanism. Otherwise you get ambiguity." But some of our scientists remain unconvinced.

So, is the above Toaq sentence valid? What does it mean?

Description approaches

Hoemaı considers the case of anaphoric reference to a binding from a previous sentence not so different, and calls such pronouns donkey as well. There is then a "crude handling of donkey pronouns" demonstrated in On quantifiers and variables:

Kaqgaı jí sá guobe da. Nuo gúobe da.
[∃G : guobe(G)] kaqgaı(J,G). [∀R : raı(R) ∧ guobe(R) ∧ kaqgaı(J,R)] nuo(R).
“I see some cows. They (the cows) are asleep.”
“I see some cows. All the cows that I see are asleep.”

Note the "surprising ∀ quantifier" in the logical translation of the second sentence. For as long as gúobe is logically out of scope, but not explicitly rebound, it refers to "all the cows that I see". “The point of it is to catch the same things as appeared in the previous sentence … instead of just any.”

Applying this same "crude handling" to quantifiers in subclauses, the meaning of the Toaq sentence above is: "All farmers who own some donkeys, care for all the donkeys that they own." This is often sufficient.

One idea is to explicitly anaphorically refer back to the construct containing the now-unbound variable and talk about its participants, as proposed by Hoemaı and by Hoaqgıo.

"If you have a pet, then the pet in the previously mentioned state of affairs does XYZ." — Hoemaı
"Tú [ë] kaqgaı jí sá kusera [nä] … the one who participates by seeing something, likes the thing that participates by being seen." — Hoaqgīo

This idea and Hoemaı's "crude handling of donkey pronouns" are very similar to approaches found in the literature known as "E-type" or "D-type" analyses. The general idea is that donkey pronouns create some sort of definite description.

Dynamic approaches

The entire premise for donkey pronouns being "illegal" is that a quantified variable can only bind things that are in the same clause (in technical terms, only those things which it C-commands). But there's plenty of evidence to the contrary. For example, "Someone left and she whistled" has a reading on which 'someone' binds 'she', even though they're in different clauses.

There are a variety of theories that allow bindings to outlive their parent clause in some way. Linguists call these theories dynamic. Some of them invent a new form of predicate logic that gives formal meaning to variables that are out of scope, while others make the bold claim that donkey pronouns are actually in scope, and find some novel way to automatically compose the right meaning. As of 2024, this last approach is the one our scientists see the most promise in.

External links

• Donkey sentence on Wikipedia
• All about poi and noi on the Lojban wiki. The PA da noi section demonstrates a similar "surprising ∀ quantifier".