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共同研究：ことばと論理Ⅳ

The Square of Opposition with “most” and “many”1)

Shin-ichi SHIMIZU*“All” (and some of its analogues like “every”) and “some” are traditionally well-known§１

and well-established quantifiers : and it might as well be reasonable to count “most”, “few”, and“many” as another set of quantifiers as well. Logical traditions, formal or symbolic, have been enjoying their own “squares of opposition” on the basis of categorical propositions. We claim that
another “square of opposition” is formed with respect to a pair of “most” and “many” as well, and
it is incorporated into a traditional “square of opposition”, which produces a larger “square of
opposition”. In what follows some of the argumentations will be presented from a logical andmodern semantical perspective.
§２ Traditional formal logic recognizes four categoricals ; universals, existentials, negative universals, and negative existentials. As shown in Figure 1, four logical relations among thosecategoricals are established ; contradiction, contrary, subcontrary, and subalternation.
What modern symbolic logic counts is the “square of opposition” with nothing but contradiction.
Take affirmative universal (1a) for example. With modern symbolic logic in mind, this general
proposition is conventionally translated into (1b).
(1) a.

All humans are mortal.

b.

(x)[H(x)→M(x)]This move makes it possible to hold (1a) true even when the subject term designates an empty
set because of material implication in the scope. This, however, gives rise to a problem. (1b)renders it hard to treat syntactic form and logical representation in an isomorphic fashion, andmaintain compositional perspective on semantic interpretation. Most remarkably, the conditionalin the logical form (1b) is not to be found in syntactic form (1a) at all.
To solve this problem, Barwise and Cooper (1981) [henceforth B & C] proposes that NP in(2) is to be treated as qauntifier.
(2)[S [NP Q N] [VP ....]]*文学部

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contraries
AllsubalternationNo

contradictories

Some

Some-notsubcontraries(Figure 1)

This means that a quantifier Q is regarded as a set of sets, and is captured to be a binary relation
on set A and set B, so that QE A B and QE 
(E)) (where E is a domain of objects or entitiesand (E) is a power set of E : the quantifier so characterized is labeled as generalized quantifier
QE.)
Eijck (1984) and 







(1985) elaborate upon characteristics of QE in the spirit of B &C. It has a property of Extension (






uses a different terminology Constancy) ; given
the domain E which includes subsets A and B, and E’ which in turn includes E, the effect of binary
relation between A and B on E is equivalent to the one on E’. This means that if the domain E
extends, say, C, whose property is distinct from those of A and B, is added to E, the binary relation between A and B is not affected, nor is it on the contrary if the domain E’ is reduced to E.
In other words, QE in question does not count the size of the domain.
The second property of QE is conservativity. Observe the following.
(3) a.
b.

Every man runs.
Every man is man who runs.

When we see if (3) is true or not in a given model, we focus only on entities which have a property of being man and those which have a property of running, that is, specific subsets in the domain E. Other entities in E are not involved in evaluation of (3). (3a) corresponds to (3b) ininterpretation. The following general condition is thus imposed on the determiner of NP.
(4)

conservativity : If A, BE then DE A B  DE A(AB)
E
M

R

men who run (MR)(Figure 2)

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This allows us to show some relations between subsets on E. (3b) or (3a) is shown as a EVERY
relation between M and (MR) ; i.e. MR’ does not contain any member (where R’ stands for thecomplement of R).
Now combination of extension and conservativity makes it possible to do away with parameter
E, and instead “live on” only A (B & C) ; i.e. nothing but a subset A can be provided to “set a
stage” for QE (Eijck). The truth of QE A B is evaluated only on the basis of A and the intersection
AB, and with cardinality of the set in mind, only on A and AB . This property is calledstrong conservativity. This idea might be more intuitively captured via the following Venn diagram Figure 3.
A

E
B
A
AB

AB

(Figure 3)

If so, our traditional square of opposition is represented as in Figure 4, with the “stage” set interms of only A.
A

E

A―B

stands for the domain which is empty.

A

AB
I

O
*
*

(Figure 4)

The superset and subset of Figure 3 make it easier for us to have a better grasp of monotonicity. Now observe the following.
(5) a.
b.

All men walked.
All men walked slowly.

Entailing relations between them are as follows :(6) a.

All men [VP1 walked] / All men [VP2 walked slowly]

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b.

All men [VP2 walked slowly]  All men[VP1 walked]

The definition of monotone quantifiers is provided in B & C as follows :A quantifier Q is monotone increasing (mon↑) if XQ and XYE implies YQ (i.e.

(7)

for any set XQ, Q also contains all the supersets of X.)
Q is monotone decreasing (mon↓) if XQ and YXE implies YQ (i.e. for any set(8)XQ, Q also contains all the subsets of X.)
Monotonicity of (6a, b) is thus represented as in Figure 5, where 
VP2(a set of things having
a property of walking slowly) is subordinate to 
VP1(a set of things having a property of walking), and it shows us that universal affirmatives are of upward entailment, i.e. monotonic increasing.
(6a)

(6b)


VP1

VP2

VP1


VP2

(Figure 5)(9) a.
b.

No girl drinks heavily.
No girl drinks.

(10)a.

No girl [VP1 drinks]  No girl [VP2 drinks heavily]b.

No girl [VP2 drinks heavily] / No girl [VP1 drinks]“Drink heavily” is subordinate to “drink”. Hence we get the diagram for (9a, b) such as :
(10a)

VP1

VP2


(10b)

VP1

VP2

(Figure 6)

This indicates that universal negatives are of downward entailment, i.e. monotonic decreasing.
§3

Judgment of entailment of (6) and (10) depends upon only our semantic knowledge. We

will see if we can have a more formal test of monotonicity. Again take up B & C’s definition (7),and see what it has to say. It says that a necessary and sufficient condition for upward mono-

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tonicity is “if XQ and XY, then YQ”. This means that “if XYQ, then XQ and Y
Q.”
Q : {X, Y}YXXY(Figure 7)

And assume that Y＝VP1and X＝VP2, and this gives us (11a).
(11)a.

NP (VP1 and VP2) NP VP1 and NP VP2b.

(NP VP1 or NP VP2)  NP (VP1 or VP2)

We can also have (11b) for the test. Why (11b)? (11b) holds true iff (12a), a slightly simplified
version of B & C’s (7), is equivalent to (12b).
(12)a.
b.

Q is monotonic increasing iff for all X, YE : if XQ and XY, then YQ.
Q is monotonic increasing iff for all X, YE : if XQ or YQ, then XYQ.

Assume that Q satisfies the condition that “if XQ and XY, then YQ” with respect to (12a).
And select X or Y such that XQ and YQ. Then does “XYQ” hold? It is evident that
Figure 7 will provide us a positive answer. This means that the implication (12a)→(12b) holds.
Now assume that Q satisfies “if XQ or YQ, then XYQ” with respect to (12b) ; and selectX and Y such that XQ and XY : Then, does “YQ” hold? It does, indeed. We will get the implication (12b)→(12a) ; hence a mutual implication. (12b) does hold true. This means that wehave (11b) if X and Y are substituted for VP1and VP2respectively. Now see if (11a, b) correctly predict upward monotonicity of, say, “all”.
(13)a.

All students were singing and dancing.
 All students were singing and all students were dancing.

b.

All students were singing or all students were dancing.
 All students were singing or dancing.

(14)a.

Some girls were smoking and drinking.
 Some girls were smoking and some girls were drinking.

b.

Some girls were smoking or some girls were drinking.
 Some girls were smoking or drinking.

(15)a.

Not a single cat was singing and dancing.
/ Not a single cat was singing and not a single cat was dancing.

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b.

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Not a single cat was singing or dancing.
/ Not a single cat was singing or not a single cat was dancing.

Entailment of each pair of (13) and (14) shows that quantifiers “all” and “some” are of upwardentailment ; whereas, from (14), a quantifier “not a single CN” is found to be of downward entailment. Similarly, from (8), we will have (16a, b) as for downward monotonicity.
(16)a.
b.

NP (VP1 or VP2)  NP VP1 and NP VP2(NP VP1 or NP VP2)  NP (VP1 and VP2)(16) predicts that “no” is of downward entailment. Observe (17), which shows a sharp distinction from (18) where none reveals valid entailment.
(17)a.

No girl prayed or knelt.
 No girl prayed and no girl knelt.

b.

No girl prayed or no girl knelt.
 No girl prayed and knelt.

(18)a.

All girls prayed or knelt.
/ All girls prayed and all girls knelt.

b.

All girls prayed or all girls knelt.
/ All girls prayed and knelt.

So far we have dealt with “all”, “some”, and “no” ; i.e. universal affirmative and negative, andparticular affirmative. The question in point is what monotonicity particular negative, or Oproposition entertains. We will return to this question later.
When it comes to the square of opposition, negation plays a crucial role in scrutinizing the nature of categoricals. QE we have been discussing is symbolized as :(19)

QE＝{XE
XQ}(19) claims that QE is a quantifier which is a set of X which constitutes a subset of the universe
E of objects such that the X is a member of Q : QE is a set of sets, after all. Now according to B
& C’s definition, we define the external negation (20) on the basis of (19).
(20)

Q＝{XEXQ}

In other words, Q is equivalent to a difference (E)−Q. The second type of negation is internal negation which corresponds a negative particular, and it is defined as in (21).
(21)

Q＝{XE(E−X)Q}(21) says that the internal negation is a set of X such that XCQ (where XC stands for the complement of X).
Now let us ask what are the external and internal negations of all and some, respectively ; toput it another way, what are  all and all , and  some and some . First we consider
all CN . Given (20), this reduces to {XX
all CN }. This gives us a situation like ;NII‑Electronic

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E
X
E―X

CN
N
(E−X)＝
N
(Figure 8)

This diagram shows that thinking in terms of CN
X is to claim that if you take any elementout of CN
, it never belongs to the set X. This means that CN
X ; and therefore CN
X

CN
, which stands for some. So we have a series of equivalences like ;all CN＝{XX
all CN}(by (20))

CNX
CN}＝{X
＝someNow what about the internal negation all. We get by (21) :
all CN＝{X
(E−X)

all CN}Let us think what the underlined description of Q, all CN, represents. This gives us a binary
relation Q like :
all CN＝{CN
, E−X}
X

E―X

Q

CN
(E−X)＝
N
N
(Figure 9)

Figure 9 is trivially reduced to Figure 10 with CN (E−X). Given this, it is evident thatCN
X＝; this is exactly what no CNrefers to. This train of reasoning can be thus summarized like ;
all CN
＝{X(E−X)

all CN}CN
(E−X)＝
CN
}
＝{X

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X

Q
CN
E―X
(Figure 10)

＝{X

CN
X＝}
no CN
＝
Similarly, we can be provided with the following train of reasoning as far as some is concerned.
some CN＝{X
X
some CN}(by (20))


CNX＝}＝{X＝no CN
Finally the internal negation some is found to be the same relational effect as all.

some CN
＝{X(E−X) 
some CN}(by (21))CN(E−X)}
＝{X
CN
X
CN
}
＝{X
＝all
It should be recognized that we have established relations of negation, external and internal,among the categoricals A, E, I, and O.
CNXCNN X



CN
X＝

all
A

all
E

I
some

O
all

some

some

CN
X
CN
N X

(Figure 11)

Now let us think of expressing some in terms of all ; that is to say, how to use negations to

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express some. Notice that there is one more possibility of negating a given Q ; we could externally negate an internally negated quantified proposition, or vice versa. B & C gives us a formal
definition like ;(22)

The dual of a quantifier Q on E is the quantifier Q* defined by Q*＝{XE(E−X)

(Q}, i.e., Q*＝〜(Q〜)＝( 〜Q)〜.2)
It follows from this that we are in a position to place some as a dual of all, i.e. (all) or (all).
Nature of negations is very closely related to the directionality of monotonicity. Given the upward entailing nature of Q, the externally and internally negated Q’s both have the reverse directionality ; and it is the case as well that given the downward entailing nature of Q, they both areof upward entailment. Assume that Q＝{XXQ}, and that it is upward entailing. Then what
could we say about the monotonic directionality of Q? As is shown already, by definition (20),Q＝{ XXQ}. Select X and Y such that XQ and YX. If we assume that YQ since Y
X and Q is upward entailing, then XQ, which is inconsistent with the assumption of XQ aswith Q＝{XXQ} ; and therefore XQ and YQ. This means that YQ, and so Q is
downward entailing. The internal negation is in order now ; and by (21), Q＝{X(E−X)Q}.
Let us assume that Q is upward entailing, i.e. XQYX. Given this, (E−X) Q. Since
YX, it is the case that (E−X)  (E−Y) ; hence (E−Y)Q since Q is assumed to be upwardentailing. This means that YQ ; and Q is of downward entailment.
From all above, we can make a claim that the monotonic directionality of some, i.e. all, isthe same as all, which is upward entailing, (mon↑). The internal negation all is then downward entailing (mon↓), and the external negation of all makes us a dual of all (all),
which should be in turn (mon↑).
(mon↑)(mon↓)

all
A

all
E

I
all

O
all

(mon↑)(Figure 12)

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It is true of right monotonicity that universal affirmative entails particular affirmative, in that they
enjoy the same directionality. Now the negation of all makes all (mon↓), which assures usthat universal negative entails particular negative because of their same directionality (mon↓).
From this it follows that the dual of Q shares the same directionality as the one of Q, which assures entailment from universal to particular.
§4. Now with “right-monotonicity” scrutinized for “standard” quantifiers, we are in the position
to see how to position “most” and “many” into the square of opposition. First, asking whether the
Figure 13 holds or not is in order.
most

most-notmany

many-not(Figure 13)

Observe the following pair :(23)a.
b.

Many philosophers are liberal.
Many philosophers are philosophers who are liberal.

(23a) is derived from (23b) via deletion of redundant “philosophers who are”. (23b) refers to P
(a set of things having a property of being philosophers) and PL (intersection of P and L, which
is a set of things having a property of being liberal), and if either one of the following two descriptions in terms of cardinality is satisfied, conservativity of the quantified expressions with “many”
remains true and monotonicity is thus guaranteed to hold good (cf. 







1985) :(24)a.
b.


PL
・
P(where is a constant and 
)

PL
(E
) (where E is a universe of a given model)

The latter guarantees that fewer than (E) is never “many”. Monotonic directionality of“many” is provided via the above-mentioned tests with the result of (mon↑).
(25)a.

Many MLB players run fast and hit hard.

b.

 Many MLB players run fast and many MLB players hit hard.

(26)a.

Many philosophers are liberal or many philosophers are stingy.

b.

 Many philosophers are liberal or stingy.

According to 







(1985), “most” meets logicality ; however, it is ambiguous between“more than half” interpretation and “almost all” interpretation. With the former “most”, (27a)has the interpretation (27b), and with the latter “most”, (27c).

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Most X’s are Y’s 





b.




, or

c.



・(where possibly 


)



With respect to monotonicity, “most” also shows us a monotonically increasing property as thefollowing tested sentences show :(28)a.
b.
(29)a.
b.

Most Americans are both pious and conservative.
 Most Americans are pious and most Americans are conservative.
Most Americans are pious or most Americans are conservative.
 Most Americans are pious or conservative.

Now that it turned out that “many” and “most” share the same monotonicity, they are both entitled to entailing relations ; and then the next question in order is which quantifier is to be sitting
on the “upper deck” or “lower deck” of the square of opposition. Our claim is that the Figure 13,
where “most” but not “many” sits on the upper deck, is correct.
Keynes (1906) tells us in passing that interestingly enough, we may begin with “few”, whichshould be counted as a member of the set of quantifiers, to shed fresh light upon the problemof where to position “most” in the square, relative to “many”. He notices that, in relation to“standard” quantifiers such as “all” and “some”, some logicians such as De Morgan and Hamilton
treat as quantified propositions “plurative” propositions like ;(30)a.

Most S’s are P’s.

b.

Few S’s are P’s.

and says that “most” is interpreted to have the meaning “at least one more than half ” and (30b)is equivalent to (31).
Most S’s are not P’s.

(31)

We could interpret his “most” to be either more or less the same as 







“most” with(27b) or the one with (27c) because of liberality of the meaning “at least one more than half ”.
Keynes himself is not definite on this difference. If the latter hold, there could be some proposal
in which a quantifier like “almost all” may be stacked up on a quantifier “most”, two decks up“many”.
Let us begin with examination of (32a) and (32b), an internal negation of (32a).
(32)a.
b.

Few philosophers are stingy.
Few philosophers are not stingy.

Because of exhaustiveness of “stingy”, (32a) provides us with P−S
PS, and (32b)with P−SPS; which means that there is never the chance that both (32a) and (32b)

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can not be true at the same time. This corresponds to our “contrariness” in the sense of traditional formal logic.
It turns out that “few” is monotonic decreasing. The following pair of sentences comes from
McCawley (1993), who refers to them for discussion of quantifier scope and applicability of Conjunction Reduction.
(33)a.
b.

Few rules are both correct and easy to read.
Few rules are both correct, and few rules are easy to read.

The derivation of the surface form (33a), involving tough-movement, is not allowed, for we can
not regard them as a mutual variants. He attributes this fact to different truth value assigned toeach sentence. (33a) is true and (33b) is false, given the situation that “many rules are correct
but few of the correct ones are easy to read” ; and thus it is impossible for (33a) to entail (33b).
This fails in exactly what our test for upward entailment predicts.
Furthermore, Huddleston and Pullum (2002) provides us with a pair of data which indicatesthe downward entailment in more intuitive sense ; it is true that the set of “ignorer of big signs”
is a subset of “ignorer of signs”, and so “from the fact that (34a) entails (34b),” they say, “weknow that few good drivers is a downward entailing quantified NP”.
(34)a.
b.

Few good drivers ignore signs.
Few good drivers ignore big signs.

Our tests, however, find “few” downward entailing on their own ground.
(34)c.

Few drivers run slowly or ignore signs.
 Few drivers run slowly and few drivers ignore signs.

d.

Few boys smoke or few boys drink.
 Few boys smoke and drink.

This all means that the internal negation of “few”, as opposed to “affirmative few”, as we haveshown that a negation, internal or external, will render the monotonicity reversed. It has alreadyturned out that “many” is monotonic increasing and its internal negation monotonic decreasing.
Plus “few” shows “contrary” relation with “few-not”. This makes it possible to tentatively draw
the following square of opposition with “few” sitting on the left “upper deck”.
few (mon↓)few-not (mon↑)many-not (mon↓)many (mon↑)(Figure 14)

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Here our concern is with “contradictoriness”. As Declerck (1993) shows, “many” and “few”
are used in intensively contrastive manner.
Many are called but few are chosen.

(35)

Truly this, as such, does not necessarily reveal contradiction between “many” and “few”, but it
alludes to “contradictoriness” between “many” and “few”. Now observe the following sentences.
(36)a.

Not many women work for nonprofit organizations.

b.

It is not the case that many women work for nonprofit organizations.

c.

Not all people were created equal.

d.

It is not the case that all people were created equal.

It is very natural that we take (36a) as external negation (36b) on a par with (36c) vs. (36d) ;and indeed (36a) has the same interpretation as (37).
Few women work for nonprofit organizations.

(37)

Now observe the following ;(38)a.

Few merchants are generous.

b.

*Not few merchants are generous.

c.

Not a few merchants are generous.

Here we just have to take into account English idiosyncrasy. (38b) becomes well-formed if“few” is replaced with “a few” in (38c) ; and thus (38c) results in an external negative variant of(38a).3) Our observation is that the former contradicts the latter, and this consolidates our pictureof contradiction with “many” and “few”.
(39)a.

few＝manyb.

few＝many

Although we don’t ponder over contradictoriness-bearing negative variants of these quantifiers
here, yet what follows will hold (via substitution).
(40)a.

few＝many

b.

many＝few

Notice that monotonicity also consolidates contradictoriness found in Figure 14. Contradictoriness is guaranteed to hold as a result of externally negating a given proposition and negations, as
we saw, reverse monotonic directionality in an upside down fashion. This is exactly reflected inwhat the way in which relevant quantifiers are positioned in Figure 14 represents.
Then we have to ask ourselves why “many” (or “many-not” for that matter) are forced to siton the “lower deck”. To be definite on this question, “many” and “many-not” must stand in “subcontrary” relation ; that is, a sentence with “many” and the one with “many-not” can be both trueat the same time. Suppose quantifiers on the upper deck are contraries and they are both false.
Plus the categoricals, diagonally positioned across the square, are “contradictory” against each

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other. Then it must be necessary that both “many” and “many-not” are true, and obviously
enough they turn out to be “subcontraries”. This shows that it is considered correct for us tostack up both “few” and a negative variant of few on the upper deck, just one storey up from“many” and “many-not”.
One more additional argument for appropriateness of Figure 14 comes from entailing relations.
In order for Figure 14 to be correct, (41a-b) and (42a-b) must hold true at the same time.
(41)a.

few  manyb.

few many(42)a.

many
/ fewb.

many 
/ few

Suppose again that “contrariness” holds, i.e. both “few” and “few-not” are false : and then thismeans “many” and “many-not” are true because of contradiction between “few” and “many-not”,
and similarly between “few-not” and “many”. The truth of “many-not”, for example, cannot entail
the falsity of “few”, as (42a) indicates.
Now we will set about replacing “few” with “most-not”, and “few-not” with “most” respectively on Figure 14 to reduce it to Figure 13. Interestingly, Figure 13 consolidates itself on an independent ground. As things turns out, “most” is more closely related to “all”, and “many” toexistential quantifiers. Now observe that there are some differences in syntactic behavior among
these quantifiers.
(43)a.

There are many students who like baseball.

b.

There are a few skyscrapers in Kyoto.

c.

There are some Japanese who play for MLB.

(44)a.

*There are all boys playing out in the field.

b.

*There are most students who distrust LDP.

(45)a.

b.

{ few }

There aremore than half percent of all students who distrust LDP.

{ over}*There are almost all (of the) students who distrust LDP.

The quantifiers in (43a-c) co-occur with there-construction and they seem to belong to the samegroup in the sense that they share existential implication ; however, (44) and (45) show that“all” does not turn up in there-construction, and “most” resists there-insertion. “Most” also shareswith “all” a property of not co-occuring with existential construction. “Most” seems to form itself
in the same group with “all”, and is plausibly allowed to sit up on the upper deck, one storey downfrom “all”. As 







(1985) points out, “most” in “most As are B” can be interpreted as“almost all”, and importantly the proposition with this interpretation is true even when the set

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designated by the subject term is empty. This is exactly a distinguishing characteristic, that, modern symbolic logicians claim, a standard quantifier “all” has ; that is, it does not carry existential
implication with it, and this hinders us from there-insertion with “all”. Similarly, this is responsible for non-occurrence of “most” in there-construction.
It might be reasonable to presume that “most” in the sense of “few-not” has the “almost all”
interpretation. Consider the following :(46)a.
b.

Most players on the team are non-Cubans.
Most players on the team are Cubans.

To make a defensible contradictory statement (46a) from (46b), it is necessary that PCshould be expressly greater than P−C. Put it another way, under the condition that when
PCgets closer to , (46a) should be a defensible contradiction of (46a) ; otherwise, underthe “more than half” interpretation, there is very little difference in cardinality difference between
PC
P−Cand PCP−C, and (46a) is just not good enough to label it a defensible contradiction.
Here we can transform Figure 13 into the following rearranged Figure 15, with affirmatives positioned on the left column.
most (mon↑)most-not (mon↓)many (mon↑)many-not (mon↓)(Figure 15)“Most” and its denial, i.e. external negation of “most”, stand in contradictory relation. S−PSPis true with “most Ss are P”. As far as “(most Ss are P)” is concerned, S−P
S−P
SP. This is bluntly expressed by (47), and is easily ilSPholds, which means lustrated by a pair of sentences in (48).
(47)

Many Ss are not P.

(48)a.

It is not the case that most children like hot chocolate.

b.

Many children don’t like hot chocolate.

Given that most＝many, “most” and “many-not” are contradictory against each other.
Notice also that monotonicity directionality is diagonally opposite, and this gives us further supporting evidence that “most” and “many-not” are contradictories.
“Most” and “most-not”, independently of how large or small the set designated by the subject

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桃山学院大学総合研究所紀要 第30巻第２号

term is, cannot be both true at the same time.
(49)a.
b.
(50)a.
b.

Most of his classmates are happy with their results of the final.
Most of his classmates are not happy with their results of the final.
Most of Americans are satisfied with the result of the last presidential election.
Most of Americans are not satisfied with the result of the last presidential election.

Each pair of (49) and (50) cannot be true at the same time, respectively.
“Most”, on the square of Figure 15, entails “many”. As is shown in (43a), “many” appears in
there-construction, and sure it does have existential implication. The same is true of “some”. It
generally is interpreted as “there is at least one x such that ...” : and thus “one or more”. Thatmeans that “some” enjoys liberality of interpretation. So does “many”, and thus it has the interpretation : “there are many or more x’s such that ...”, which allows “many” to be entailed by“most”.
There is another way of verifying that “most”, on the square of Figure 15, entails “many”.
Assume that “most” does not entail “many”. This means that the categorical with “most” is true ;whereas the one with “many” is false. Since “many” and “most-not” stand in contradictory relation as we have seen, “most-not” falls into truth if “many” is false. Then we come to the conclusion that both “most” and “most-not” are true. As we have seen, and by the nature of square of
opposition as well, “most” and “most-not” are in “contrary” relation by which it is meant that bothof them cannot be simultaneously true : and therefore, by reductio ad absurdum, the initial assumption is false that “most” does not entail “many”. “Most” does indeed entail “many”.
§5.

Conclusion. To integrate or incorporate “most” and “many” into the traditional square of

opposition, we have made good use of one of the developments of formal semantics, i.e. monotonicity in the study of generalized quantifiers ranging from “standard” ones like “all” and “some”
to “non-standard” ones like “most” and “many”. We examined methods of testing monotonic directionality, and validated them.
We have taken notice of Keynes’s observation, i.e. an idea of “few” as an equivalent counterpart of “most-not”, by way of setting about our task of organizing the traditional square of opposition with “most” and “many” included in there. It turned out to be good and easy access to ourreasoning. Not fully complete though it may be, our conclusion is Figure 16 with directionality ofmonotonicity onto each quantifier. Any quantifier on either column is in opposition to the corresponding counterpart on the other end of line, which is overtly reflected monotonic directionality
on each quantifier : and it depends on kind of the line what opposing relation they are in. A successive sequence of quantifiers, in top-to-bottom fashion on either column, is in subalternating relation partly because each of them enjoys the same directionality of monotonicity.
The present discussion should be considered to involve an integral characterization of a very

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no(mon↓)(mon↑)all(mon↑)mostmost-not(mon↓)(mon↑)many

many-not(mon↓)some-not(mon↓)(mon↑)some(Figure 16)(where “” stands for “contraries”, ““contradictories”, and “” for “subcontraries”, “” for” for “subalternation (or entailment)”)

small set of quantifiers in a natural language in terms of opposing relations and entailment, andit remains to be seen if a further research can see a possibility of counting out more quantifiers
including numerals, proportional quantifiers, and etc..4)

NOTES
1) This research was supported by Momoyama Gakuin University under the research project 98KYO119.
2) The symbol “〜” used in (22) corresponds to our notation “” in what it stands for.
3) An additional argument is made for claiming that an external negation of “many” is an equivalent counterpart of “few”. Jackendoff (1972) provides a following pair of sentences.
( i ) a.

Many of the arrows didn’t hit the target, but many of them hit it.

b.

*Not many of the arrows hit the target, but many of them hit it.

In his argument of negation, (i a) is taken as an example of VP-negation, which corresponds to our internal negation. The status of (i a) indicates that, in the case of internal negation, one conjunct is not regarded
as denial of the other conjunct, i.e. the first and second conjuncts are true at the same time. Unacceptabilityof (i b) indicates that its first and second conjuncts cannot be true simultaneously in a given situation.
Jackendoff regards this negation as sentential negation as opposed to VP-negation. We take the former tobe external negation. Given our square of opposition, “few” is found to be sitting on the deck, one story
up from and diagonally right across “many”, and it follows from this that many＝few.
4) We could suggest that there be a possibility of ( i ) where “almost all” and “most” are treated separately.








(1985) points out that ‘most As are B’ under the “more than half ” interpretation is false when
the subject term is empty. Existential constructions claim that there are As designated by the subject term,
and it is thus false when the set is ; and so the “most” must share the same property as existential constructions do. This predicts that there must be some “most” that appear in there-construction, but he does
not give us any examples. We can provide the following examples indeed :( i ) a.
b.

Most butterflies that there are deep in the mountains are huge.
Most of the students that there were in the classroom are ones she had met before.

For as yet unknown reason, they are all relativized “most” NP. If they turns out to have “more than half ”
interpretation one way or another, there will possibly be some new proposal in which we can divide

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桃山学院大学総合研究所紀要 第30巻第２号“MOST” into two “most’s”, say, “most” and “almost all”, and pair up the latter with “few”, resulting in (ii).

all

no

almost most

few

most

most-notmany

many-notsome

some-not(ii)

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Keynes, J. N.: 1906, Formal Logic (4th ed.), MaCMillan.
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Publishers.
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Tidman P. and H. Kahane : 1999, Logic and Philosophy (8th ed.), Wadworth Publishing Company.
van Eijck, J. : 1985, ‘Generalized Quantifiers and Traditional Logic’ in van Benthem, J. and A. ter Meulen(eds.), Generalized Quantifiers in Natural Language, Foris, Dordrecht, 119.

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, D. : ‘Logical Constants in Quantifier Languages’, Linguistics and Philosophy 8, 387413.
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346.
【邦語文献】
ガストン・カゼノヴァ： ブール代数』 村田 全・小室博昭 共訳（1968)．白水社．

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The Square of Opposition with “most” and “many”

Shin-ichi SHIMIZU

Our concern is with constructing a traditional square of opposition into which “most” and“many” are integrated. Our basic position is in favor of traditional formal logic that originated with
Aristotle, but the present discussion have taken liberties with recent developments of formal semantics to such an extent that they make contribution to more understanding of what the squareof opposition looks like. The concept of directionality of monotonicity and especially our tests
contribute to our conclusion. Still our discussion finds crucial insight in Keynes, one of formal logicians of a century ago. We propose as a conclusion a traditional square of opposition in whicha near universal and near particular are neatly incorporated into a traditional proto-type square insuch a way that they are entirely wrapped up.

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