# Donkeys and Natural Language

In logic and in the philosophy of language, the meanings of the parts of a sentence are often appealed to in order to explain the meaning of the whole sentence. In logic, this involves building a syntax (a set of symbols) that in turn matches the semantics (truth-conditional meanings) of a natural language. Thus, the word ‘a’ followed by a noun, will give meaning regarding the quantity of the thing denoted by the noun. That quantity is often taken to be the equivalent of one or more of the thing in question.

For example, “A donkey lives at the zoo.” Here, ‘a’ signifies that at least one donkey lives at the zoo. Indeed, in basic logic courses, we are trained to use the existential quantifier almost exclusively for cases of such appearances of ‘a’. The existential quantifier looks like this ‘∃x’, where ‘x’ is a variable that stands for a thing, and as a whole it reads: there is at least one x such that . . . . For example, ‘There is at least one x such that x is a donkey and x lives at the zoo’ would be an apt translation of our example sentence above.

In many instances, the existential would appear to be too loose. For example, if I say “I see a donkey”, this would tend to mean that it was just one donkey I saw, and not more than one. Similarly, a conversation between two people could readily go:

“I’ll have a hamburger”

“Just one?”

“Yeah, I’m not that hungry.”

And so, ‘a’ is often restricted to instances of only one occurrence of the noun. This is similar to how ‘the’ operates. ‘The’ often picks out a unique object. For example, “I saw the tiger named Tony”. The use of ‘a’ as picking out one thing still differs from ‘the’ in that ‘a’ tends to pick out a single object of a certain type (hamburger, donkey) while ‘the’ tends to pick out a particular –i.e. something not cast as being of a certain type; for example, there being only one Tony the tiger in this here town, as opposed to there being many things that would fit a description of being a Tony the tiger (as if it were green or soft or so forth). So sometimes ‘a’ is more like ‘the’, with some differences.

However, more than this, the appearance of ‘a’ followed by a noun is not always best translated by the existential quantifier at all. For example, “Everyone who bakes a casserole eats it”. ‘Everyone’ involves not the existential quantifier, but the universal quantifier, which looks like this ‘∀x’ where ‘x’ is a variable that stands for a thing, and the whole thing translates to “For any x, if x . . . then . . . .” For example, “Everyone who bakes is happy” would be translated as “For any x, if x bakes then x is happy”. Thus, to begin with “Everyone who bakes . . . ” we would have “For any x, if x bakes . . ., then”. We can symbolize this and our first attempt at translating ‘a casserole’ using the existential quantifier as follows:

∀x∃y((Bxy & Cy) –> Exy)

This says: “For anything x, there is at least one y such that if x bakes y and y is a casserole, then x eats y”.

This may sound like an apt translation at first. However, by the dictates of logic and its truth values, if our baker bakes just one casserole and eats it, the sentence comes out true, even if on any other casserole our baker gives it all to the cat. How to capture the meaning that for each and every casserole the baker bakes, it will be eaten by the baker? The answer is that we need to universally quantify over this case of ‘a’:

∀x∀y((Bxy & Cy) –> Exy)

This says: “For anything x and for anything y if x bakes y and y is a casserole, then x eats y”. This may not sound like much of a difference so let me explain. In order for this sentence to be true now, it is not enough that just one casserole be baked and eaten by our baker; now anything at all that is both a casserole and baked by our baker must result in our baker eating that thing. This was the intended meaning.

Why was this the intended meaning at all? That is, what can explain why this use of ‘a’ deviates from typical cases that would be translated with the existential quantifier? This is tough to answer for anyone who wants to pinpoint a syntactic feature that would map on to these deviant semantics. The answer is likely that there is no syntactical feature. How, then do we pickup on the semantics that deviate from standard forms so readily?