Monotone-Decreasing Quantifiers

A monotone increasing quantifier, like ``most'', is ``monotone increasing'' beacuse when the predicate in the body of the quantified expression is made less restrictive, the truth value is preserved. Thus,

Most men work hard.

entails

(1)	Most men work,

By contrast, for monotone decreasing quantifiers, when the predicate in the body of the quantified expression is made less restrictive, the truth value is not necessarily preserved. Quite the opposite. It is preserved when the body is made more restrictive.

(2)	Few men work.

entails

Few men work hard.

Since ``x works hard'' entails ``x works'', a flat, scope-free representation for ``few men work hard'' runs into problems, because it would seem to allow the incorrect inference ``few men work''.

Virtually every utterance describes a situation in a more general fashion than the speaker actually means to convey. If I say, ``I went to Tokyo,'' you are likely to interpret this as saying that I flew to Tokyo, even though I did not specify the means of transportation, and I would expect you to interpret it in this way. Indexicality is one example of this phenomenon. If I say ``He went to Tokyo,'' I am saying that a male person went to Tokyo, but my listener will generally use contextual information to arrive at a more specific interpretation. This observation is at the core of the IA framework. To interpret a sentence is to find the ``best'' proof of its logical form, together with the selectional constraints that predicates impose on their arguments, allowing for coercions to handle metonymy, making assumptions where necessary. In brief, we must find the best set of specific facts and assumptions that imply the generalities conveyed explicitly by the utterance.

The parts of the logical form that we are able to prove constitute the given information that provides the referential anchor for the sentence. The assumptions that we must make in order to interpret a sentence constitute the new information; this is what the sentence is asserting. Typically, information in the main verb is what is asserted, and information that is grammatically subordinated is given, or presupposed. But this is not necessarily the case. In

An innocent man was convicted today.

the listener may already know that someone was convicted, and the new, asserted information is that the man was innocent. Similarly, in

I have a sore throat.

you know I have a throat. The new information is that it is sore. Reinterpreting what is asserted by the sentence will be a key move in dealing with monotone decreasing quantifiers.

The solution to the problem of monotone decreasing quantifiers that I propose consists of three steps.

1. We first generate the logical form of the sentence exactly as we would for other quantifiers. For sentence (2), the logical form would be analogous to (14), namely,

(25)	$(\exists \,s_{1},s_{2},x,y,e_{1},e_{2}) few(s_{2},s_{1}) \& dset(s_{1},x,e_{1})$
	$\& man'(e_{1},x) \& typelt(y,s_{2}) \& work'(e_{2},y)$
	$\& Rexist(e_{2})$

That is, there is a set s₁ defined by the property e₁ of its typical element x being a man, there is a set s₂ which is few of s₁ and has y as its typical element, and the eventuality e₂ of y's working exists in the real world. Note that all of this is true, as far as it goes; there is a set consisting of few men, and the members of this set work. It just doesn't go far enough, because it does not rule out a much larger set.

2. The next step is to specialize or strengthen the predication typelt(y,s₂) to the more specific $dset(s_{2},y,e_{2}\+e_{3})$, by back-chaining on Axiom (9), and instantiating the defining eventuality to the conjunction of the two eventualities we see in the sentence, or rather, in place of the eventuality e₁ the ``substitution'' eventuality e₃ such that

Subst(x,e₁,y,e₃)

That is, we have further specified the set s₂ to be not just some subset of s₁ that has few elements, but the subset defined by the conjunction of conditions e₃ and e₂, where

$man'(e_{3},y) \& work'(e_{2},y)$

which, by (24), is the set of men who work.

It is Axiom (9) that places this interpretation in the space of possible interpretations, but nothing so far guarantees that this is the interpretation that will be selected. I would like to suggest one way this could happen, without, however, denying other possible accounts.

To promote this particular strengthening, we can associate as a selectional constraint on the arguments of few the requirement that its first argument be a set with a defining property.

(26)	few(s2,s1): $(\exists \,y,e) dset(s_{2},y,e)$

This requirement then becomes something that has to be proved in addition to the logical form to arrive at an interpretation. It forces us to look for an eventuality e that defines the set s₂. The three most readily available eventualities are those explicit in the sentence itself--e₁ (or rather, e₃), e₂ and the conjunction of the two. e₃ (being a man) is impossible as a defining condition for s₂since it is the defining condition for s₁, of which s₂ is a proper subset. e₂ (working) is also impossible as a defining condition, since the members of s₂ are men, and more than just men work. That leaves the conjunction $e_{2}\+e_{3}$. The set s₂ is the set of men who work.

3. The proposition few(s₂,s₁) is taken to be the assertion of the sentence, rather than work(y). That is, the sentence would be interpreted as saying

The men who work are few.

Increasing the plausibility of this part of the analysis is the fact that it is hard to unstress the word ``few'' when it is functioning as a monotone decreasing quantifier, and high stress is an indication that the information conveyed by the morpheme is new.

To demonstrate that this approach goes through, under this formulation, I need to show that from ``Few men work'' we can indeed conclude ``Few men work hard,'' (assuming anyone works hard) once these two sentences have been interpreted as in Steps 1-3, and assuming, for the sake of this paper, that we have an axiom

(27)	$(\forall \,x) work$- $hard(x) \, \supset \,work(x)$

The logical form of ``Few men work'', generated in Step 1, is given in (25). By Step 2, typelt(y,s₂) is strengthened to $dset(s_{2},y,e_{3}\+e_{2})$. Since this sentence is the premise, we assume the strengthened logical form is all true.

The logical form of the sentence ``Few men work hard'' is

$(\exists \,s_{1},s_{4},x,z,e_{1},e_{4}) few(s_{4},s_{1}) \& dset(s_{1},x,e_{1})$
$\& man'(e_{1},x) \& typelt(z,s_{4}) \& work$- hard'(e₄,z)
$\& Rexist(e_{4})$

By Step 2, we strengthen typelt(z,s₄) to $dset(s_{4},z,e_{5}\+e_{4})$, where

Subst(x,e₁,z,e₅).

Step 3 tells us that what is asserted is few(s₄,s₁), while the rest is presupposed. Thus, we assume the rest is true, and we must demonstrate few(s₄,s₁).

The conclusion few(s₄,s₁) will follow from Axiom (13) if we can demonstrate few(s₂,s₁) and subset(s₄,s₂). But few(s₂,s₁) is part of the premise assumed above. To demonstrate that subset(s₄,s₂) holds, we need to show that any member v of s₄ is also a member of s₂. We do this in three steps. First we show, using the premises $dset(s_{4},z,e_{5}\+e_{4})$, man'(e₅,z) and work- hard'(e₄,z), together with a rightward use of the inner biconditional in Axiom (24), that any member v of s₄ is a man and works hard. We then use Axiom (27) to conclude that v works. We then use Axiom (24) in a leftward direction to show that v is in the set s₂. This establishes that the monotone decreasing property of the word ``few'' is preserved in this formulation.

It would be good if the predicate few used in ``Few men work'' captured the same notion of few-ness that is expressed in ``A few men work.'' I will only sketch a possible account in which this would be the case. Consider

A few men work.

The word ``a'' expresses a relationship between the entity referred to by the noun phrase and the description it provides; it says roughly that the entity is not uniquely identifiable in context solely on the basis of that description. The logical form of this sentence would be almost the same as (25). But we need first to introduce the eventuality e<sub>0</sub>corresponding to the few-ness relation between s<sub>2</sub> and s<sub>1</sub>-- few'(e<sub>0</sub>,s<sub>2</sub>,s<sub>1</sub>). Then to express the relation conveyed by the determiner ``a'' we add the predication $a(y,e_{0}\+e_{3})$, saying that y is not uniquely identifiable in context on the basis of the properties e₀ and e₃. If we were to proceed in Step 2 as before and specialize typelt(y,s₂) to $dset(s_{2},y,e_{2}\+e_{3})$, then we would have a contradiction, for the properties e₂ and e₃ would uniquely identify y as the typical element of the set defined by these properties (assuming sets have a unique typical element). The word ``a'' thus blocks this strengthening of ``few'', the eventuality e in (26) remains unresolved, and we are left with only the few relation between the sets s₂ and s₁.

It is often argued that one way of drawing the line between compositional semantics (Step 1) and pragmatics (Step 2) is to say that the results of compositional semantics are not defeasible whereas the results of pragmatics are. This would appear to be an argument against the approach suggested here, since the interpretation of ``Few men work'' as ``The men who work are few'' does not seem to be defeasible. But another force that strongly constrains likely interpretations is conventionalization. The IA account of discourse comprehension traces out a space of possible interpretations and provides a graded mechanism for choosing among them, given a context. But conventionalization picks out among the possible interpretations a particular interpretation of a given word, phrase, or grammatical structure. It collapses the space of possible interpretations to only the conventional interpretation. It thus eliminates the defeasibility one ordinarily associates with pragmatic processing.

An example of this, unrelated to quantifiers, involves ``let's''. This is a contraction of ``let us''. But the sentence ``Let us go'' could be said by two victims to a kidnapper, whereas the sentence ``Let's go'' would not be. The general meaning of ``Let us go''--

Don't cause us not to go.

is, for the contraction, conventionally specialized to

Don't cause us (inclusive) not to go by not going yourself.

The favored interpretations of ``few men'' and ``a few men'' are no doubt conventionalized, even though they can be derived de novo according to the accounts given above.

``Only'' could be viewed as a determiner, and as such it would be monotone decreasing. Its interpretation would be derived very much as that of ``few'', but differing in one crucial respect.

``Only'' is indeed monotone decreasing, since ``Only men work'' entails ``Only men work hard.'' But unlike ``few'' it is not conservative. The conservativity property can be illustrated as follows: The sentence ``Few men are men who work'' entails ``Few men work,'' and ``few'' is hence conservative. By contrast, ``Only men are men who work'' does not entail ``Only men work.'' In fact, the first is tautologically true, and the second is false. ``Only'' is hence nonconservative (cf. van Benthem, 1983). This means the process of interpreting ``only'' must differ at some point from the process of interpreting ``few''. In fact, it differs in Step 2.

The logical form of ``Only men work'' would parallel (25).

$(\exists \,s_{1},s_{2},x,y,e_{1},e_{2}) only(s_{2},s_{1}) \& dset(s_{1},x,e_{1})$
$\& man'(e_{1},x) \& typelt(y,s_{2}) \& work'(e_{2},y)$
$\& Rexist(e_{2})$

Under this analysis, as before, ``only'' will be taken to express a relation between a set s<sub>2</sub> and the set s<sub>1</sub> of all men, and the noun phrase ``only men'' will be taken to refer to the set s₂ in the sense that it is the members of s₂ who work. Thus, Step 1 in the analysis of ``only'' does not differ from Step 1 in the analysis of ``few''.

In Step 2, however, the set s₂ is not specialized to the set of men who work. Rather it is specialized to the set of workers. That is, typelt(y,s₂) is strengthened to dset(s₂,y,e₂). The relation that only expresses between s₂ and s₁ is then simply the subset relation. The set of workers is a subset of the set of men. That is, only men work.

Step 3 is the same as for ``few''. The predication only(s<sub>2</sub>,s<sub>1</sub>) is picked as the assertion of the sentence. That is, ``Only men work'' is interpreted as though it were ``The set of workers is a subset of the set of men,'' or ``All workers are men.''

This is a limited account of the interpretation of ``only'' as a determiner. In fact, a proper account would encompass adverbial uses as well. My real view is that only is a predicate of three arguments--an entity or eventuality x, a scale s that has x as its lowest element, and a property that is true of x but not of the other, higher elements of s. In ``John only walked'', x is John's walking, s is a scale of actions ordered, say, by energy requirements, and the property is the property of having John as an agent. When used as a determiner, x is the entity or set referred to by the NP, s is the set of subsets containing x and ordered by inclusion, and the property is the main predication of the sentence. In ``Only men work'', x is a set of men, s is the set of subsets of the relevant entities containing x, and the property is working. The sentence says that the members of x work, but the members of no larger set in s works. This implies that the workers are a subset of all men, the meaning of ``only'' assumed in the account above.

Coercion of the assertion of a sentence plays a key role in the treatment of monotone decreasing quantifiers in the present framework, as described in greater detail in Hobbs (1996). Consider the sentence

Few men work.

I propose that the syntactic component of the interpretation process generate as a logical form for this sentence the expression

$few'(e_{1},s_{2},s_{1}) \& dset(s_{1},x,e_{2}) \& man'(e_{2},x) \& plural'(e_{3},y,s_{2})$
$\& work'(e_{4},y)$

That is, there is a set s₁ defined by the property e₂ of its typical element x being a man, there is a set s₂ which is few of s₁ where this few-ness is property e₁, and s₂ has y as its typical element (property e₃), and the eventuality e₂ of y's working exists in the real world. Note that all of this is true, as far as it goes; there is a set consisting of few men, and the members of this set work. It just doesn't go far enough, because it does not rule out a much larger set.

This stronger interpretation is achieved in two steps. First, the predication plural'(e₃,y,s₂) is specialized or strengthened to the more specific $dset'(e_{3},s_{2},y,e_{2}\+e_{4})$. That is, the set s₂ is not just some subset of s₁ that has few elements, but the subset defined by the conjunction of conditions e₂ and e₄, where

$man'(e_{2},y) \& work'(e_{4},y)$

This is the set of men who work.

Finally, in a manner similar to the ``sore throat'' example, the property e₁ where few'(e₁,s₂,s₁) is taken to be the assertion of the sentence, rather than the property e₄ where work'(e₄,y). That is, the sentence would be interpreted as saying

The men who work are few.

The coercion relation rel(e₄,e₁), used in this example to coerce from the working to the few-ness, comes from the explicit content of the sentence, namely,

$few'(e_{1},s_{2},s_{1}) \& plural'(e_{3},y,s_{2}) \& work'(e_{4},y)$

These three predications provide the link from e₄ to y to s₂to e₁.

Increasing the plausibility of this analysis is the fact that it is hard to unstress the word ``few'' when it is functioning as a monotone decreasing quantifier, and high stress, as noted above, is an indication that the information conveyed by the morpheme is new.