NEGATION AND GENERALIZED QUANTIFIERS Frans Zwarts ...

NEGATION AND GENERALIZED QUANTIFIERS 

Frans Zwarts 

Centre for Behavioural, Cognitive and Neuro-sciences 

Rijksuniversiteit Groningen 

1. Introduction 

Linguistic research has clearly shown that natural languages exhibit different forms of 

negation. Foremost among these are sentence negation and predicate negation. In this 

study, the relationship between both will be discussed. In particular, it wiu be argued 

that we can give an accurate description of the logical connections between sentence 

negation and predicate negation by reference to the semantical nature of the subject 

noun phrase. To this end, two important classes of generalized quantifiers will be 

distinguished, having to do with the familiar logical notions of consistency and 

completeness. Next, it will be shown that the distribution of negative adverbs in 

languages as Dutch and German forces us to make a distinction between predicate 

negation and verb negation as well. The logical relationship between these two forms 

of negation appears to be the same as that between sentence negation and predicate 

negation. Finally, it will be demonstrated that the difference between predicate negation 

and verb negation also shows up in languages such as English, particularly in co~ection 

with monotonicity patterns. 

2. Consistency 

It is well-known that there are considerable differences between sentential negation, an 

the one hand, and predicate negation, on the other. Less known is the fact that the 

logical relationship between both forms of negation depends entirely on the semantical 

nature of the subject. In order to convince ourselves, we do well to take the following 

two examples into consideration. 

(1) a At least two willows do not flower + 

It is not the case that at least two willows flower 

b Most weeping willows do not flower 4 

It is not the case that most weeping willows flower 

One sees immediately that the conditional in (la) can not be regarded as valid. For if 

the state of affairs is such that of the seven willows only four happen to flower, then the 

antecedent is true, but the consequent false. On the other hand, it is evident that the 

conditional in (lb) must be accepted as valid. If we are willing to accept the truth of the 

statement Most weeping willows do not flower, then we shall also have to acknowledge 

that it is not the case that the majority of weeping willows flowers. This entails that the 

quantifier which is associated with noun phrases of the form most N is invariably 

consistent in nature - consistent in the sense that it cannot contain a given set of

individuals as well as the complement of that set. For the sake of clarity we give the 

following definition. 

(2) Definition 

Let B be a Boolean algebra. A quantifier Q on B is said to be consistent iff 

for each element X of the algebra B: if -X E Q, then X # Q. 

Noun phrases which invariably receive a consistent quantifier as their semantic value, 

will accordingly be referred to as consistent noun phrases. 

The property of consistency can of course be formulated in more than one way. Indeed, 

it is readily established that the law of contraposition allows us to replace the definition 

in (2) by the alternative characterization in (3). 


Let B be a Boolean algebra. A quantifier Q on B is said to be consistent iff 

for each element X of the algebra B: if X E Q, then -X 4 Q. 

In spite of this equivalence, we prefer the definition in (2), primarily because it affords 

us a handy way of establishing the corollary below. 

(4) Corollary 

A noun phrase is consistent iff the following schema is logically valid: 

(1) NP (NEG VP) -, NEG (NP VP) 

This simple result is important, because it clearly shows that with a consistent noun 

phrase as subject the use of predicate negation invariably entails sentence negation. 

From the fact that the implication in (lb) is valid, it follows immediately that noun 

phrases of the form most N are consistent. Do not suppose that this exhausts the matter, 

for one easily proves that proper names and expressions of the forms both N, the n N, 

the NG and the N must also be regarded as belonging to the class of consistent noun 

phrases. In the &flowing examples, the use of predicate negation invariably entails 

sentence negation. 

(5) a Themistocles does not mourn 

It is not the case that Thernistocles mourns 

b Both feet are not ulcerated 

It is not the case that both feet are ulcerated 

c The seventeen donkeys do not bray 

It is not the case that the seventeen donkeys bray 

d The scientists do not drink coffee 

It is not the case that the scientist drink coffee 

e The shopkeeper does not waste time 

It is not the case that the shopkeeper wastes time

Similarly, one easily shows that expressions of the forms neither N, none of the n N and 

none of the N are also consistent in nature. To this end, it is enough to take the 

conditionals in (6) into consideration. 

(6) a None of the six donkeys does not bray 

It is not the case that none of the six donkeys brays 

b Neither foot is not ulcerated 

It is not the case that neither foot is ulcerated 

c None of the scientists does not waste time 

It is not the case that none of the scientists wastes time 

That each of these entailments is valid, follows from the definitions of the quantifiers 

corresponding to the noun phrases in question. To give an example, if none of the 

scientists does not waste time, then it follows that they all waste time, which in turn 

means that it is not the case that none of them wastes time. Consequently, the 

conditional in (6c) must be accepted as valid. 

The foregoing observations by no means imply that every noun phrase is consistent. The 

invalid implication in (1) clearly shows that expressions of the form at least n N do not 

belong to this class. In an analogous manner, one easily proves that noun phrases of the 

forms all N and no N can also not be regarded as consistent,, The next two examples, 

if interpreted as cases of regular negation, are both invalid. 

(7) a All students do not complain 

It is not the case that all students complain 

b No child does not complain 

It is not the case that no child complains 

By way of an illustration, it should be noted that, when the universe does not contain 

any child, the statements No child complains and No child does not complain are both 

true, which means that the denial It is not the case that no child complains must be 

regarded as false. Similarly, if the universe happens to lack students, the two statements 

All students complain and All students do not complain must both be accepted as true, 

and hence the denial It is not the case that all students complain must be rejected as 

false. 

In this connection, the behaviour of partitive noun phrases is rather interesting. It 

requires no lengthy reflection to see that expressions of the form at least n of the k N 

are consistent iff n > k/2. As a special case of the general pattern we have the valid 

conditional in (8). 

(8) At least three of the four children do not complain 3 

It is not the case that at least three of the four children complain 

On the other hand, if we consider expressions of the form (exactly) n of the k N, then 

the property of consistency appears to manifest itself just in case n f k/2. That is to say: 

(9) a Six of the eight children do not complain 

It is not the case that six of the eight children complain 

b One of the eight children does not complain

It is not the case that one of the eight children complains 

Finally, it should be easy to see that noun phrases of the form at most n of the k N are 

consistent only if n < k/2. In other words: 

(10) At most one of the four children does not complain + 

It is not the case that at most one of the four children complains 

The logical behaviour of partitive expressions thus appears to show some regularities. 

These find expression in three general laws concerning the phenomenon of consistency. 

(11) 

Laws of consistency for partitive' expressions 

(1) Expressions of the form at least n of the k N are consistent iff n > k/2. 

(2) Expressions of the form (exact&) n of the k N are consistent iff n # k/2. 

(3) Expressions of the form at most n of the k N are consistent iff n < k/2. 

We must therefore conclude that, with a substantial number of partitive subject phrases, 

the use of predicate negation implies sentence negation. 

With the aid of the foregoing test we can usually arrive at rather trustworthy judgments 

when it comes to deciding whether a given noun phrase does or does not enjoy the 

property of consistency. For the sake of clarity the outcomes of the test have been 

collected in table 1. The eighteen classes of noun phrases mentioned there must all be 

regarded as consistent. Such a catalogue, though at first sight merely of encyclopedic 

value, is important because we shall soon see that it leads to a coherent and complete 

account of the relationship between sentence negation and predicate negation. 

It should be pointed out in this connection that the property of consistency shows a 

striking resemblance to the logical theorem known as the law of contradiction. Indeed, 

the principle in question is meant to exclude the possibility that two contradictory 

propositions are both accepted as true. For that reason it is usually stated as '-(p A 

-p)'. One sees immediately that the property of consistency is similar to the logical 

theorem in that it excludes that two sets X and -X both belong to the quantifier. It will 

become apparent that this state of affairs has far-reaching consequences for our views 

on the different forms of negation. 

3. Completeness 

It requires no lengthy reflection to realize that the property of consistency has a 

counterpart. In order to convince ourselves, we take the following two examples into 

account. 

(12) a It is not the case that most tulips flower 

Most tulips do not flower 

b It is not the case that Seneca plays chess 

Seneca does not play chess

Table 1: Eighteen classes of consistent noun phrases. 

most N 

the majority of the N 

at least n of the k N (n > k/2) 

(exactly) n of the k N (n f k/2) 

at most n of the k N (n < k/2) 

both N 

the n N (n =. 0) 

the more than n N (n > 0) 

the no more than n N (n > 0) 

the Npl 

the Nsg 

neither N 

none of the n N (n > 0) 

none of the more than n N (n > 0) 

none of the no more than n N (n > 0) 

none of the N 

proper names 

negated proper names 

Clearly, the conditionaI in (124 cannot be regarded as valid. For if the state of affairs 

in the universe is such that half of all tulips flowers, then the antecedent is true, but the 

consequent false. On the other hand, the conditional sentence in (12b) surely belongs 

to the class of valid statements. If we accept the truth of the statement It zk not the case 

that Senecaplays chess, then we will also have to accept that Seneca does not play chess. 

Tbis entails that the quantifiers which are associated with proper names invariably are 

complete in nature - complete in the sense that it cannot be that neither the 

complement of a given set nor that set itself is a member of the quantifier in question. 

For the sake of accuracy we record this in the form of a definition. 

(1 3) Definition 

Let B be a Boolean algebra. A quantifier Q on B is said to be complete iff for 

each element X of the algebra B: if X & Q, then -X E Q. 

It is evident that the notion of completeness just introduced is the reversal of the notion 

of consistency mentioned before. This means that there are alternative characterizations 

of the property in question. Indeed, it is easily established that the law of contraposition 

allows us to replace the conditional in (13) by the equivalent statement 'if -X # Q, then 

X E Q'. Yet, we shall stick to the original definition, primarily because of the following 

corollary. 

(14) Corollary 

A noun phrase is complete if£ the following schema is logically valid: 

(1) NEG (NP VP) -* NP (NEG VP3 

This elementary result is important because it expresses in a lucid way that with a 

complete noun phrase as subject the use of sentence negation invariably implies 

predicate negation. 

From the fact that the implication in (12b) is valid, it follows immediately that proper 

names are complete. In an analogous way, one easily shows that negated proper names

are also complete in nature. This does not exhaust the stock, for a short search produces 

several other cases of completeness. The next two examples serve as an illustration. 

(15) a It is not the case that at least half of all tulips flowers -3 

At least half of all tulips does not flower 

b It is not the case that at most half of all cows has died 

At most half of all cows has not died 

There can be no doubt that both conditionals are valid. Indeed, if we accept the truth 

of the statement It is not the case that at least half of all tulips flowers, then we shall also 

have to accept that at least half of all tulips does not flower. Similarly, it is easily 

established that anyone who accepts the statement It is not the case that at most half of 

all cows has died as true will also be committed to the truth of At most half of all cows 

has not died. Consequently, we must conclude that noun phrases of the forms at least 

half of all N and at most half of all N both belong to the class of complete expressions. 

For the sake of clarity, these results have been collected in table 2. One sees 

immediately that two of the four classes of noun phrases mentioned are also consistent, 

namely proper names and their negations. This is important, because it follows from the 

relevant definitions that with a consistent and complete noun phrase as subject the use 

of sentence negation is equivalent to predicate negation. For that reason, the 

biconditional in (16) must be regarded as valid. 

(16) 

It is not the case that Themistocles mourns 

Themistocles does not mourn 

Indeed, it follows from the completeness of the expression Themistocles that the use of 

sentence negation entails predicate negation. Conversely, the consistent nature of the 

element in question guarantees that the use of predicate negation implies sentence 

negation. In this way, we can give a semantic explanation of the at first sight rather 

intricate relationship between both forms of negation. 

4. Sentence negation and predicate negation 

'I therefore hold it more suitable to regard negation as marking the content of apossible 

judgment." In this way, Frege expresses in the fourth paragraph of Begn~sschrifr his 

conviction that the different forms of negation must all be regarded as sentence 

negation. Of course, such a view entails that the use of predicate negation must also be 

reduced to sentence negation. In the posthumous fragment 'Logic', which dates from 

1897, he states this very clearly: 'In the German language we usually indicate that a 

thought is false by inserting the word 'not7 into the predicate.'2 To be sure, there are 

moments at which Frege expresses himself more carefully, as in the unpublished 

fragment 'Introduction to Logic' (1906), where we read: 

Frege (1980: 4). 

Frege (1979: 149).

-- - 

Table 2: Four classes of complete noun phrases. 

at least half of all N at most half of all N 

proper names negated proper names 

'To think is to grasp a thought. Once we have grasped a thought, we can recognize it as 

true (make a judgement) and give expression to our recognition of its truth (make an 

assertion). Assertoric force is to be dissociated fiom negation too. To each thought there 

corresponds an opposite, so that rejecting one of them is accepting the other. One can 

say that to make a judgment is to make a choice between opposites. Rejecting the one 

and accepting the other is one and the same act. Therefore there is no need of a special 

name, or special sign, for rejecting a thought. We may speak of the negation of a 

thought before we have made any distinction of parts within it. To argue whether 

negation belongs to the whole thought or to the predicative part is every bit as unfruitful 

as to argue whether a coat clothes a man who is already clothed or whether it belongs 

together with the rest of his clothing. Since a coat covers a man who is already clothed, 

it automatically becomes part and parcel with the rest of his apparel. We may, 

metaphorically speaking, regard the predicative component of a thought as a covering 

for the subject-component. If further coverings are added, these automatically become 

one with those already there." 

But that does not alter the fact that Sommers is absolutely right when he claims that 

Frege's way of portraying the matter rests on 'the negatability criterion for genuine 

subjecthood'? According to this criterion, the status of logical subject can only be 

assigned to a given phrase if attachment of two contradictory predicates to the 

expression in question will give us two contradictory propositions. In other words, should 

we wish to speak of a logical subject, then the use of predicate negation must in Frege's 

opinion invariably be equivalent to sentence negation. It is evident that such a 

requirement entails that proper names be assigned a separate place, for on the basis of 

the fact that Aristotle is not a philosopher is the denial of Aristotle is a philosopher, Frege 

labels the occurrence of Aristotle in both sentences as the logical subject. On the other 

hand, he refuses to regard the occurrence of all mammals in All mammals we land- 

dwellers as the logical subject, because the statement All mammals are not land-dwellers 

clearly cannot be classified as the denial of All mammals are land-dwellers. Indeed, his 

remarks in 'On Concept and Object' show that Frege is not even willing to analyze the 

expression all marnmak as a logical unit3 In this way, he withholds logical 

Frege (1979: 185). 

Sommers (1982: 27). 

Frege (1980: 47-48) offers us the following explanation: 'We may say in brief, taking 'subject and 

'predicate' in the linguistic sense: A concept is the reference of a predicate; an object is something that 

can never be the whole reference of a predicate, but can be the reference of a subject. It must here be 

remarked that the words 'all', 'any', 'no', 'some', are prefixed to concept-words. In universal and particular 

affirmative and negative sentences, we are expressing relations between concepts: we use these words to 

indicate the special kind of relation. They are thus, logically speaking, not to be more closely associated

constituenthood from phrases that grammar considers to be unsuspected units, and 

creates an unbridgeable gap between logical form, on the one hand, and grammatical 

form, on the other. Whether Frege was really content with this solution is not clear, for 

at the end of the posthumous fragment 'Logic' he concludes that the use of the word not 

'in ordinary language is a purely external criterion and an unreliable one at that7, and 

complains about the fact that 'we are tangling with some thorny problems hereY.' 

The difficulties that Frege experiences can be attributed to the fact that he sticks to a 

first order analysis of sentences. If we give up this restriction and regard noun phrases 

as second order expressions, then any doubt a. to the relationship between sentence 

negation and predicate negation must disappear. For it can be described completely in 

terms of the earlier mentioned properties of consistency and completeness. To forestall 

any misunderstandings, we do well to express this in the form of two general laws 

concerning the use of both forms of negation. 

(17) 

Laws of negation (preliminary version) 

(1) The use of sentence negation implies predicate negation just in case the 

subject of the sentence is complete in nature. 

(2) The use of predicate negation implies sentence negation just in case the 

subject of the sentence is consistent in nature. 

It goess without saying that, in the presence of a subject which is complete and 

consistent, the use of sentence negation is equivalent to predicate negation. 

4. The theory of quaternality 

There are other formulations of the laws of negation which are perhaps less obvious, but 

completely equivalent to the original formulations. To this end, we shall introduce three 

operations which, applied to given quantifiers, produce new quantifiers. Starting from 

a quantifier Q on a Boolean algebra B, the first enables us to form the complement of 

Q, that is to say, the quantifier consisting of all elements of the algebra B which do not 

belong to Q. This new quantifier will be written as -Q. With the aid of the second 

operation we form the so-called contradual of Q, represented by the symbol -Q. This 

is the quantifier which consists of all elements X of the algebra B such that -X E Q. The 

third operation, finally, serves to create a new quantifier -Q- which we call the dual of 

with the concept-words that follow them, but are to be related to the sentence as a whole. It is easy to 

see this in the case of negation. If in the sentence 

'all mammals are land-dwellers' 

the phrase 'all mammals' expressed the logical subject of the predicate are land-dwellers, then in order 

to negate the whole sentence we should have to negate the predicate: 'are not land-dwellers'. Instead, we 

must put the 'not' in front of 'all'; from which it follows that 'all' logically belongs with the predicate.' The 

reader should note that Frege (1979: 87-117) contains a preliminary draft of the article. 

' Frege (1979: 150).

Q. It contains each element X of the algebra B such that -X $ Q. For the sake of 

transparency we record this in the form of a definition.' 


Let B be a Boolean algebra and let Q be a quantifier on B. Then: 

(1) The complement of Q is the quantifier -Q = {X E B: X 4 Q). 

(2) The contradual of Q is the quantifier Q- = {X E B: -X E Q). 

(3) The dual of Q is the quantifier -Q- = {X E B: -X 4 Q). 

One sees immediately that it makes no difference whether we regard the dual -Q- as the 

complement of the contradual of Q or as the contradual of the complement of Q. For 

this reason we write -Q- instead of -(Q-) or (-Q)-. It is also readily established that the 

complement of the complement of Q, the contradual of the contradual of Q, and the 

dual of the dual of Q are all identical to Q. In symbols: --Q = Q-- = --Q--= Q. 

Likewise, one easily shows that the contradual of the dud of Q is equal to the 

complement of Q, and that the complement of the dual of Q is equal to the contradual 

of Q. That is to say: -Q-- = -Q and --Q- = Q-. 

duals + 

complements 

-Q- 

Figure 1 

contraduals 

contraduals 

complements 

-, duals 

With the aid of these equations, we can visualize the mutual relationships between Q, - 

Q, Q- and -Q- by means of the picture in figure 1. The square represented there shows 

a clear resemblance with the traditional square of opposition.2 As an illustration of this 

The model-theoretic notion of the dual of a quantifier is discussed in Barwise and Cooper (1981: 

197). Among other things, they call attention to the important subclass of self-dual quantifiers, defined 

by the equation Q = -Q-. 

Gottschalk (1953) points out that every involution in a logical or mathematical system gives rise 

to a theory of quaternality and that the square of quaternality, of which the classical squares of opposition 

are special cases, provides a diagrammatic representation for much of this theory. 

45 1

correspondence, we consider the noun phrases all weasels (Q), not all weasels (-Q), no 

weasel (Q-), and at least one weasel (-Q-). By reference to the square it is readily 

established that all weasels and at least one weasel are each other's duals, just as not all 

weasels and no weasel are. Similarly, we can show that not all weasels is the contradual 

of at least one weasel and that no weasel is the contradual of all weasels. The expressions 

at least one weasel and no weasel, finally, must be regarded as each other's complements. 

Indeed, it requires no lengthy reflection to realize that every noun phrase determines a 

square of opposition. For this reason we are entitled to say that an expression as neither 

ox is the contradual of both oxen. 

The significance of such squares of opposition lies in the fact that they enable us to give 

an alternative characterization of the properties of consistency and completeness. The 

next two lemmata provide the necessary details. 

(19) Lemma 

(20) Lemma 

Let B be a Boolean algebra. The following two statements about a quantifier 

Q on B are equivalent: 

(1) Q is consistent. 

(2) Q c -Q-. 

Proof. Suppose that Q is consistent and that X E Q. Then -X # Q, and so, by 

the definition of duality, X E -Q-. Hence, Q c -Q-. Conversely, if Q c -Q- and 

X E Q, then X E -Q-. SO, by the definition of duality, -X # Q. II 



(1) Q is complete. 

(2) -Q- r Q. 

Proof. Suppose that Q is complete and that X E -Q-. Then -X # Q, by the 

definition of duality, and so X G Q, by the completeness of Q. Hence, -Q- r 

Q. Conversely, if X # Q, then X # -Q-, because of (2), and so, by the definition 

of duality, -X E Q. II 

From this it follows that a quantifier which is both consistent and complete is self-dual. 

That is to say: 




(1) Q is consistent and complete. 

(2) Q = -Q-. 

The above result provides us with two tests. For it is easily established that quantifiers 

which are consistent and complete have the property that -X belongs to Q iff X does not 

belong to Q. But from the fact that Q is self-dual in such cases we may also infer that

X belongs to Q iff -X does not belong to Q. As a consequence, we have the following 

corollary: 


A noun phrase is consistent and complete iff the following schemata are 

logically valid: 

(1) NEG (NP VP) - NP (NEG VP) 

(2) NP VP NEG (NP (NEG VP)) 

That proper names, regarded as consistent and complete noun phrases, satisfy the first 

requirement has already been established by means of the valid equivalence in (16). 

That they also satisfy the second requirement is shown by the validity of the 

biconditional in (23). 

(23) Themistocles mourns + 

It is not the case that Themistocles does not mourn 

Consequently, we may conclude that proper names belong to the class of expressions 

which are self-dual. 

In view of these findings, the earlier formulations of the laws of negation may be given 

a different form. Instead of saying that in the presence of a consistent subject the use 

of predicate negation implies sentence negation, it is possible to say that the subject 

should be contained in its dual if the use of predicate negation is to imply sentence 

negation. Conversely, it is necessary that the subject contains its dual if the use of 

sentence negation is to imply predicate negation. This is expressed in the following law 

concerning equivalent forms of negation. 

(24) Equivalent forms of negation 

The use of sentence negation is equivalent to predicate negation just in case 

the subject is consistent and complete, that is to say, just in case the subject 

is self-dual. 

It will be shown that this rather surprising relationship between negation and duality is 

significant in other respects as well. 

5. Laws of negation 

With the aid of the theory of quaternality, we can establish a number of simple facts 

concerning the mutual relationships between Q, -Q, Q- and -Q-. 

(25) Theorem 

Let B be a Boolean algebra. The following statements about a quantifier Q on 

B are equivalent: 

(1) X # Q.

The importance of this elementary result lies in the fact that it enables us to provide 

several alternative characterizations of the use of sentence negation. The next corollary 

gives the relevant details. It should be emphasized that we write NpN for the 

complement of a noun phrase, NP' for the contradual of a noun phrase, and N P for ~ 

the dual of a noun phrase. 

(26) Laws of negation I 

The following schemata are logically equivalent: 

(1) NEG (NP VP) 

(2) N P (NEG ~ VP) 

(3) NPN W 

(4) NEG (NP' (NEG W)) 

It should be obvious that, in the presence of a self-dual subject, the use of sentence 

negation is equivalent to predicate negation. If there is no such subject, the transition 

from sentence negation to predicate negation or from predicate negation to sentence 

negation must involve the replacement of the subject by its dual. As a special case of 

this general law, we have the logical equivalence of It is not the case that every cow moos, 

At least one cow doesn't moo, Not every cow moos, and It is not the case that no cow 

doesn't moo. 

Do not suppose that this exhausts the matter, for the square of opposition depicted in 

figure 1 also permits us to derive the logical equivalences in (27) - (29). 

(27) Laws of negation I1 


(1) NEG (NPD VP) 

(2) NPpG VP) 

(3) NP VP 

(4) NEG ( NP~ (NEG VP)) 

(28) Laws of negation 111 


(1) NEG (NPC VP) 

(2) N P (NEG ~ VP) 

(3) NPD VP 

(4) NEG (NP (NEG VP)) 

(29) Laws of negation TV 


(1) NEG (NP~ VP) 

(2) N P (NEG ~ W)

(3) NP VP 

(4) NEG (NP~ (NEG VP)) 

As an illustration of these laws, it is sufficient to observe that the proposition It is not 

the case that at least one cow moos is equivalent to Every cow doesn 't moo, No cow moos 

and It is not the case that not every cow doesn't moo by the four schemata in (27). This 

is important because it shows clearly that a sentence can be denied in several ways. The 

idea of a one-to-one correspondence between positive and negative sentences, once 

suggested by Kraak (1966: 81), must therefore be immediately discarded. 

5. Verb negation 

Thus far, we have assumed that there are only two types of negation - sentence negation 

and predicate negation. As should be evident to anyone who is familiar with the word 

order of Dutch and German subordinate clauses, there exists a third type as well - to 

wit, verb negation1 By way of an example, consider the Dutch sentence in (30). 

(30) 

Ik weet dat hij veel opgaven niet begrijpt. 

I know that he many problems not understands 

'I know that there are many problems which he doesn't understand.' 

Clearly, the scope of the negative adverb niet 'not' is the transitive verb begnypt 

'understands' alone. This contrasts sharply with the example in (31), where the scope of 

niet is either the direct object veel opgaven 'many problems' or the entire verb phrase 

veel opgaven begrijpt. 

(31) 

Ik weet dat hij niet veel opgaven begrijpt. 

I know that he not many problems understands 

'I know that he doesn't understand many problems.' 

At first sight, the possibility of verb negation seems to be restricted to languages such 

as Dutch and German. However, when one takes the distribution of some and any into 

consideration, it becomes clear that English has this type of negation as well. The next 

two conditionals may serve as an illustration. 

(32) John didn't see any of the paintings .-, 

John didn't see any of the modern paintings 

(33) 

John didn't see some of the modem paintings 

John didn't see some of the paintings 

The validity of the entailment in (32) shows that the object noun phrase is the argument 

' of a monotone decreasing function. This is not the case in (33), where the direction of 

the conditional inference is in fact the opposite of what we saw before. In other words, 

Brown (1991) argues that the Scots spoken in Hawick also exhibits verb negation. One of his 

examples is He's still no working ('It is the case that he is still out of work'), where the position of the 

negative after the adverb still unambiguously indicates that the scope is the main verb.

the object noun phrase in (33) must be the argument of a monotone increasing function. 

In what follows, we will show that this can be explained if we assume that the 

occurrence of some of the modem paintings in (33) is the argument of the negative 

transitive verb phrase didn't see. Before we address this matter in more detail, we do 

well to establish some simple facts concerning the logical relationship between predicate 

negation and verb negation. 

By means of the theory of quaternality, we can provide several alternative ways of 

characterizing predicate negation. To show this, we assume (1) that extensional transitive 

verbs are of type (e, (e, t)), being associated with two-place relations between 

individuals; (2) that object noun phrases, like subject noun phrases, are generalized 

quantifiers of type ((e, t), t); and (3) that the combination of an extensional transitive 

verb with denotation R and an object noun phrase with denotation Q is to be 

interpreted as the set {a E U: {b E U: R(a,b)) E Q), where U is the universe of 

discourse.' In terms of these principles and the result established in (25), we can prove 

that the set {a E U: {b E U: R(a,b)) + Q), which corresponds to predicates of the form 

NEG (V NP), is identical to the sets {a E U: {b E U: R(a,b)) E -Q-), {a E U: {b E U: 

R(a,b)) E -Q), and {a E U: {b E U: R(a,b)) 4 Q-). The following lemma, which is an 

obvious extension of the result obtained in (26), gives the necessary linguistic details. 

For the sake of transparency, we present two versions - one for SVO-languages, the 

other for SOV-languages. 

(34) Laws of negation V 


S VO- languages SOV-languages 

(1) NP (NEG (V (1) NP (NP V)) 

(2) NP ((NEG V) NP (2) NP (NP (NEG V)) 

(3) NP (v NP*) (3) NP (NP~ V) 

(4) NP (NEG ((NEG V) NP~)) (4) NP (NEG (NP~ (NEG v))) 

It should be clear that, in the presence of a simple transitive verb and a self-dual object, 

the use of predicate negation is equivalent to verb negation. If there is no such object, 

the transition from predicate negation to verb negation or from verb negation to 

predicate negation must involve the replacement of the object by its dual. As a special 

case of this general law, we have the logical equivalence of John (didn't (solve every 

problem)) and John ((didn't solve) some problem (s)) . 

Van Benthem (1986) seems to have been one of the first to propose such a treatment. In defense 

of this approach, he points out that extensional transitive verbs are sensitive to the semantic structure of 

the object noun phrase. If the expression in question is monotone decreasing, as in John ate or drank 

nothing, we may legitimately pass to John ate nothing or John &mk nothing. On the other hand, if the 

object is monotone increasing, as in John heard and felt something, we may legitimately pass to John 

heard something or John felt something. A more elaborate discussion of this matter can be found in 

Hoeksema (1989, 1991a, 1991b) and van Benthem (1991). Keenan (1989) proposes a slightly different 

treatment of subject and object noun phrases, based on what he refers to as semantic case. His approach 

is adopted by Zwarts (1991), in connection with a unified treatment of reflexives and noun phrases.

This does not exhaust the matter, for the theory of quaternality also permits US to derive 

the lemmata in (35) - (37). 

(35) Laws of negation VI 


S VO-languages SO V- languages 

(1) NP (NEG (V wD)) (1) NP (NEG (NP~ V)) 

(2) NP ((NEG V) NP) (2) NP (NPJrnG V)) 

(3) NP (v NP~) (3) Np (Np V) 

(4) NP (NEG ((NEG V) NP~)) (4) NP (NEG ( NP~ (NEG v)) 

(36) Laws of negation VII 


v)) 

(1) NP (NEG (V NP'A) (1) Np PC 

(2) Np ((NEGDV) Np 1 (2) Np (5 (NEG V)) 

(3) NP (V NP ) (3) NP (Np v) 

(4) NP (NEG ((NEG V) NP)) (4) NP (NEG (NP (NEG V)) 

(37) Laws of negation VIII 


SVO-languages S 0 V- languages 

(1) NP (NEG (V NP~J) (1) NP (NEG ( NF~ v)) 

(2) NP ((NEG V) NP ) (2) NP (NPc (NEG V)) 

(3) NP (V NP) (3) NP (NP V) 

(4) NP (NEG ((NEG V) NP~)) (4) NP (NEG (NP* (NEG V)) 

As an illustration of these laws, it is sufficient to observe that the English proposition 

John (didn't (solve any of the problems)) is equivalent to John ((didn't solve) all of the 

problems) and John (solved none of the problems) by the SVO-schemata (I), (2) and (3) 

in (35).' The SOV-patterns can be exemplified by means of the logically equivalent 

Dutch propositions Ik weet dat zzj' (niet (alle opgaven heeft opgelostt)) 'I know that she 

(didn't (solve all of the problems))', Ik weet dat zij (mixstem 6dn opgnve (niet heeft 

opgelost)) 'I know that she ((didn't solve) some of the problems)', ik weet dat zij ((niet 

It is much more dicult to find an instance of the fourth schema in (351, due to the absence of 

sentences with more than one preverbal negation in standard E~ghsh. Brown (1991: 82, 84, 85) reports 

cases involving two or even three negations in Hawick Scots, among them He couldnae have been no 

working 'It is impossible that he has been out of work', He could no have been no working 'It is possible 

that he has not been out of work', and He couldnae have no been no working 'It is impossible that he 

has not been out of work'.

Figure 2 

PUF, > u~~ 

alle opgaven) heeff opgelost 'I know that she (solved not all of the problems)', and Ik weet 

dat zij (niet (geen enkele opgave (niet heeft opgelost))). It should be clear that the 

schemata in (26) - (29), on the one hand, and those in (34) - (37), on the other, can be 

combined in various ways. The proposition It is not the case that everyone solved some 

of the problems, for example, is equivalent to Someone (didn't (solve some of the 

problems)) and Someone ((didn't solve) all of the problems) by the schemata (1) and (2) 

in (26) and (34). 

6. The semantics of extensional transitive verbs 

In their Boolean Semantics for Natural Language (1985), Keenan and Faltz describe a 

way of endowing extensional verbs and verb phrases with a third-order Boolean 

structure. The approach advocated there contrasts in interesting ways with the first- 

order treatment of the same expressions in Keenan (1989). In this section, their 

proposals are summarized and related to the analysis presented in Partee and Rooth 

(1983). 

For Keenan and Faltz, as for many others, generalized quantifiers are sets of sets of 

individuals, belonging to what is commonly referred to as the universe of NP 

denotations, written UNP. The Boolean operations in this type are of course union, 

intersection and complementation. Keenan and Faltz give a special status to NP 

denotations corresponding to individuals. If the individual i is an element of the universe 

U, the corresponding quantifier is the set {X c U: i E U}, known as the principal 

ultrafilter generated by i. The set {PUF(i): i E U) of principal ultrafilters corresponding 

to individuals is denoted PUFU.

As Partee and Rooth (1983) show, PUF has a useful property: it is a set of so-called 

free generators for the Boolean algebra &.' This means that each element of UNp can 

be represented as a Boolean combination of elements of PUFu and that any function 

g from Uw to an arbitrary Boolean algebra B can be extended uniquely to a 

homomorphism h from UNp to B. Pictorially, the situation can be represented as in 

figure 2, where i is the inclusion map of PUFu into UNP. 

Following in essence Partee and Rooth (1983), we can now describe the higer-order 

Boolean structure that Keenan and Faltz assign to the space of extensional transitive 

verbs and verb phrases, which they take to be Ho%(UW Uv), :he s;t of Boolean 

homomorphisms from U to the algebra Uvp = 2 . First, a ijec ion etween Horn 

(Urn, Uw) and uVU rasserted to exist. Then this bijection is used to transfer the 

Boolean structure whch uVpU has as an algebra of sets to Horn (Urn, U,). Let f be 

a function from U to Uw and let I be the function which maps each element of U to 

the corresponding principal ultrafilter. Because PUFU is a set of free generators for the 

Boolean algebra Urn, f o I-' extends uniquely to a homomorphism h: Um + UVP. 

Consequently, we can define a map M: Uw U 

+ Horn (%, UW) which carries f to 

the corresponding homomorphism h. This can be depicted as in figure 3. 

PUF, 

Figure 3 

i 

M is a function, because the extension of M is unique. M is an injection one-to-one), 

because if fi # % f1 0 I-' f f2 0 I-', and M(fi) and M(fi) extend fi 0 I and f2 0 I- 

l, respectively. Finally, M is a surjection (onto), because if k is in Horn (Urn, UVP), M(1 

i o k) = k, again because the extension is unique. Thus, M is a bijection between 

uWU and Horn (Urn, UVP). 

M can be used to define Boolean operations on Horn (UNP, UVP). That is to say: 

The notion of a free generator is discussed extensively in Sikorski (1969: 42-45) and Halmos (1963; 

4-46). 

\


It is easily established that Horn (UW, U*), with operations so defined, is a Boolean 

algebra. In practice, what this means IS that the result of applying negation, regarded as 

the operation of Boolean complementation, to a transitive verb is another transitive verb 

with the same homomorphic properties as the original one. Thus, an expression such as 

didn't see, analyzed as a negative transitive verb phrase, denotes a homomorphic, hence 

monotone increasing, function. 

7. Monotonicity analyses 

By means of the semantics just described, we can associate a sentence such as John 

didn't see some of the paintings with the categorial derivation in (39). It is easy to see 

that application of didn't to see gives us the monotone increasing (more precisely, 

homomorphic) transitive verb phrase didn't see. The next step in the derivation uses the 

operation of function composition to create the expression John didn't see. 

(39) John didn't see some of the paintings 

S/VP ((VP/NP)/(VP/NP)) VP/NP NP 

f G ? 

It should be clear that the noun phrase some of the paintings is the argument of a 

monotone increasing function - to wit, the Boolean homomorphism associated with the 

composite function John didn't see. This explains the validity of the conditional in (33). 

To understand the relationship between monotonicity properties and function 

composition, one does well to take the table in figure 4 into consideration. 

Figure 4

As an illustration of this monotonicity table, consider the categorial derivation in (40). 

Clearly, the negative transitive verb phrase didn't see is the result of composing the 

monotone decreasing function didn't and the monotone increasing function see. 

According to the table, this derived expression is monotone decreasing in nature. In a 

similar way, we can use composition to create the phrase John didn't see, which must 

also be regarded as a monotone decreasing expression. 

John didn't see some of the paintings 

VP/NP NP 

+ J, + 

It should now be clear that the expression some of the paintings in (40) occurs in a 

downward monotonic context. Hence, the validity of the conditional in (32). This brings 

our discussion of the relationship between various types of negation to an end. 

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NEGATION AND GENERALIZED QUANTIFIERS Frans Zwarts ...

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