Natural Language Quantification

Now let’s turn to ordinary language quantification. What are we to say about a sen- tence like:

My thought is that we treat this as relational. That is, ’every’ stands for a relation between the denotation of (the logical form of) its noun phrase sister and the denotation of (the logical form of) its nuclear scope. In particular, it takes a superplurally referring expression S (in this case ’man’) , and a plurally referring expression P (in this case ’swims’), and returns true provided a plurality which S superplurally refers to has in it some stages which are in P, which is to say if everything which S superplurally refers to is P7_.

Since on the way we’re doing things, relations stand for ordered pairs, its denota- tion would then be:

• _JEveryK=ιxx. ∀xxx.∀uu. if (for all ww if ww«

2 _{xxx then∃zz zz«ww and zz«uu)}

thenιx.x=<xxx,uu> « xx.

In English this says that for any superplurality (such as that referred to by ’man’) and plurality (such as the denotation of ’swims’), if an object is in the superplurality and some of its stages are in the plurality, then the ordered pair containing them is one of the things denoted by ’every’. Here are the entries for some other determiners:

• _JsomeK=ιxx.∀xxx.∀uu. if there’s some ww«

2_{xxx and∃zz zz<ww and zz«uu then} ιx.x=<xxx,uu> « xx.

• _JnoK=ιxx. ∀xxx.∀uu. if there’s no ww such that ww«

2 _{xxx and∃zz zz«ww and}

zz«uu thenιx.x=<xxx,uu> « xx

The truth conditions of our sentence will be:

• Every man swims iffJEveryKJ(man,swims)K

• Every man swims iff =Jman, swimsK«JEveryK.

• Every man swims iff <JmanK,JswimsK> «JEveryK

And that last will hold just in case for all xx«2 JmanK, i.e. for all men, there’s some

zz«xx that are in JswimsK, i.e. if all men have some swimming stages, which is the

correct result.

However, we need to consider what the logical form will look like. A first thought, taking our lead from above, would be as so:

• Every (man(xx),swims(xx))

As it stands, the arguments provided to ’Every’ aren’t respectively a superplurality and a plurality, but a pair of open sentences. So we really need to convert these open sentences by binding operators. In the case of the second argument, things are easy:

• Every (man(xx),ιxx.swims(xx))

7_{Obviously, this raises questions about sentences like ’Every man is a human’, where it’s arguable that}

the nuclear scope stands for a superplurality. There are a couple of possible ways to go: either take this as evidence we should treat common nouns and verbs as on a par, and amend the semantics accordingly (which would be straightforward) or say that the contribution of ’is a human’ is just an expression refer- ring plurally to all the human stages, by assuming the function of the copula is, as it were, to dearticulate a superplurally referring expression into a plurally referring one.

However, if we were to take the same tack for the first argument, we’d get the wrong result. For we want the first argument to be a superplurally referring expression. That is, we want it to be such that there are some xxs which are «2 _{the denotation of the}

first argument. But our simple iota binder would merely give us ιxx. man(xx), i.e.

ιxx.xx«JmanK, that is to say all the men stages. I think the solution is to introduce a

new iota operator, call it double iota, which looks as so: ιιwith the following rule for trees containing it:

Double Iota Rule.Ifαis of the form [ιιxx [vpF(xx)] thenJιιxx F(xx)K=ιxxx.

∀xx: xx«2xxx iff F(xx).

What the double iota does is take an open sentence and converts it into a superplurally referring expression such that anything which is «2 _{it satisfies the open sentence. We}

then have a final logical form like so:

• Every (ιιxx.man(xx),ιxx.swims(xx))

Let’s go on to consider cases of multiple generality, as evinced by: (157) Every man loves some woman

One might initially have fears about how to analyse this: it’ll seem somehow to involve a relation inside another relation, and one’s brain might start to hurt at the thought. But in fact a little reflection reveals that the problem is not really any harder than the traditional problem, and doesn’t require (overly) different resources.

In particular recall that for Heim and Kratzer we account for such constructions by means of traces, movement, and lambda binders, so that our sentence would look as so:

• [s[dpEvery man] [vpλ1. Some woman.λ2. 1 loves 2.K

I can more or less take this over, with one change: I make the lambda binders into iota operators (and change the variables from singular to plural, of course). That is, as a first stab:

• Every (man,ιxx. Some woman.ιyy. loves(xx,yy)) This isn’t quite right, since, as we’ve seen, we actually have:

• Every (man(xx),ιxx. Some (woman(yy),ιyy. loves(xx,yy)))

Or rather, since we need to bind the variables in the determiners’ sisters, we get:

• Every(ιιman(xx),ιxx. Some (ιιyy woman(yy),ιyy. loves(xx,yy)))

I grant the reader this isn’t the most pulcritudinous logical form ever proposed, but, for example, the massively complicated functions posited by the variable free semanticists are hardly exactly pretty either. The wide scope reading will be derived in the same way as normal, but permuting the order of the quantifiers at LF. Let’s run through a derivation of the narrow scope existential reading to keep us honest. The second argument of the lower quantifier will be computed as so:

• _Jloves(xx,yy)K

g_{=1 iffιx.x=<g(xx),g(yy)> «}

• _Jιyy. loves(xx,yy)K=1 iffιyy.ιx. x=<g(xx),yy> «JlovesK

The same procedure will apply to the first argument. We can then say:

• _J(ιιyy. woman(yy),ιyy. loves(xx,yy)K=ιx.x=<ιxxx.∀yy. yy«

2_{xxx iff woman(yy), {} ιyy.ιx. x=<g(xx),yy> «JlovesK}>

That is to say, the bracketed phrase will stand for the ordered pair the first element of which is the superplurality consisting of the women, and the second of which is the yys such that yy are loved by g(xx). Next, applying ’some’:

• _JSome(ιιyy. woman(yy), ιyy. loves(xx,yy)))K

g_{=1 iffιx.x=<ιxxx.∀zz. zz«}2 _{xxx iff}

woman(zz), {ιyy. ιx. x=<g(xx),yy> and x «JlovesK}> «JsomeK

That is to say, we have a sentence true relative to g provided the ordered pair consisting of the woman superplurality and the plurality of things loved by g(xx) belong to the denotation of ’some’. Next, we bind the currently free xx variable with the iota binder, to turn the whole second argument of the quantifier into a referring expression:

• _Jιxx.Some(ιιyy. woman(yy),ιyy. loves(xx,yy))K

g_{=ιxx. [ιx.x=<ιxxx.∀zz. zz«}2 _xxx

iff woman(zz), {ιyy.ιx. x=<xx,yy>. «JlovesK}>] «JsomeK

Although very complicated to parse, what this refers to is the xxs, for which the follow- ing condition obtains: there is an ordered pair the first element of which is the women and the second of which is the stages which are loved by the xxs, and this ordered pair is in the denotation of some. And for that to be the case, some woman must have some stages among her that are loved by the xxs. And that’s to say that the xxs refer to the stages loved by some woman. And that is what we want, intuitively speaking, it to denote.

The remainder of the derivation works in the same way: we look at the men, and providedallof them have some stages which are in the referring expression just parsed, i.e. all of them have some stages loved by some woman, we get a true sentence.

While obviously there is more to be done (for example: the treatment of English language plural quantifiers, and of pronouns), I hope this sketch suffices to show that my Stage Semantics has the resources to capture quantification, while both respecting the core of predicativism, and the existence of stages.