Grice on numerical quantifiers: (∃x), "at least ONE" (WoW: 22)

Speranza

We are considering Grice's claim that "(∃x)" reads as "at least ONE", with emphasis on "ONE" qua allegedly numerical quantifier (WoW, 22).

In English, number-words (numerals) appear to be used as quantifiers, as in the following examples:

"Three students came to the party."

"There are four dogs in the yard."

On the other hand, numerals also figure in the following sorts of constructions.

"the three dogs"

"my four dogs"

"all five dogs"

"No two dogs are exactly alike."

"we three kings"

If numerals are quantifiers, then the first four phrases violate the prohibition against double-determiners.

In the last phrase, the appositive ‘three kings’ cannot be replaced by QPs such as ‘some kings’, ‘most kings’, or ‘all kings’. Given the latter data, we propose that numerals are fundamentally, not quantifiers, but are rather adjectives, as illustrated in the following tree.

"the three dogs"
DP
Det CNP
the
Adj CNP
three dogs

Here the numeral ‘three’ serves as a CNP-modifier, which is the principal role of adjectives.
Numerals can also be used as bare adjectives, although it is less common, as in the following poetic description of the arrival of one's first child.

"Now we are three."

S
S-Adv S
now
NP VP
we
Cop Adj
are three

Numerals are sometimes regarded as a species of number words – being logograms like ‘2’ rather than phonograms like‘two’.

Since this is not a semantically relevant distinction, we will simply use the terms interchangeably.

Also note this is in exact agreement with traditional lexicography.

There is an important difference, however, between numerical adjectives and most other

adjectives. In particular, most adjectives are conjunctive, but numerical adjectives are not. The latter concept may be defined as follows.
´
is conjunctive ü for any x1, …, xk
x
1 and … and xk are ´
if and only if
x
1 is ´, and … , and xk is ´

So, for example ‘happy’ is conjunctive, since for example
Jay and Kay are happy if and only if Jay is happy and Kay is happy.

On the other hand, ‘two’ is not conjunctive, since for example
Jay and Kay are two, but neither Jay nor Kay is two.

Notice carefully that the previous sentence introduces a novel use of ‘and’, which we refer to as
mereological ‘and’, which differs from the more common logical ‘and’. Recall that logical ‘and’ is
a two-place S-operator, which is illustrated in the following.

Jay and Kay are happy

.
(
1õ.)õ. 1õ.
are happy

1 .2õ. 1
Jay
1 and Kay1
H
[J] & H[K]
l
P1 { P(J) & P(K) } lx1 H[x]
.
are happy.
J
1 & K1
.
Jay1. .and. .Kay1.

We can apply a completely parallel treatment to

"Jay and Kay are two."

which accordingly analyzes this sentence as saying that Jay is two and Kay is two. This is an admissible reading of the sentence, to be sure, but it is not the most plausible reading.

Rather, the most plausible reading of this sentence treats  ‘and’, not as logical, but as mereological, the latter being categorially rendered as follows.

type(
and) = 2õ
.
and. = lxy{x+y}
Here,

+ is the mereological-sum operator, which is explained in the next section. In the meantime, the
following is an example grammatical analysis.

This is closely related to another notion – distributivity.

"Jay and Kay are two."

.
 õ
1 1õ.
[+1] are two
 
2õ 
Jay and Kay
2
[J+K]
J
+K lx{x1} lx1 2[x]
.
+1. .are two.
J
+ K
.
Jay. .and. .Kay.
Here,

2[.] is the two-predicate, in the metalanguage, which is explained in the next section.

We propose to expand the domain of entities to include the usual singular-objects (individuals)
as well as plural-objects (pluralities), the union of which is the class of count-objects.

On the one hand, singular-objects are the usual suspects of first-order logic, and include particular individual persons, places, and things.

On the other hand, plural-objects are entities that first-order logic studiously sidesteps, and include groups of individuals of various sorts and functions.

For example, whereas Nomar Garciapara is an individual, the Boston Red Sox infielders are a plurality.

Even the grammar of plurals sounds strange, as we strain to treat each plurality simultaneously as a "one" and as a "many".

Set theory presents a formal account of collections, which reduces every "many" to a "one", and in particular reduces all plural-talk about collections to singular-talk about sets, which are treated as singular-objects (individuals) completely on a par with "ordinary" singular-objects like numbers and
space-time points.

Although we don't officially propose to identify pluralities with (mathematical) sets, we do propose to model/explicate pluralities in terms of sets.

In this connection, we offer the following official definitions.

Where  Æ is a set of singular-entities (individuals), the associated set à of plural-entities
is defined as follows,

Ã
ü { X : X Ú Æ & X¹Æ & ;$y[X={y}] }

and the associated set
¶ of count-entities is defined as follows.
ü Æ È Ã

In other words, plural-entities correspond to non-empty non-singleton subsets of singular-entities.

We can now define  + as follows.

a
+b ü a+ È b+
where:
e+ ü {e} if eÎÆ
ü
e if eÎÃ

Count-objects, the counterparts of count noun phrases, are distinguished from mass-objects, the counterparts of mass noun phrases.

Some groups are mere collections; other groups take on a quasi-autonomous existence, and in particular can serve as agents of actions. For example, a jury can hand down a verdict.

To say that sets model pluralities is to say that many properties of pluralities correspond in a specified way to properties of sets.)

So, for example, granting that
J and K are individuals (i.e., elements of Æ),
J
+K = J+ È K+
=
{J} È {K}
=
{J, K}

This means, in particular, that the interpretation of the compound
‘Jay and Kay’ is (modeled by) the
set {Jay, Kay}, which consists of Jay and Kay and nothing else.

Granting that pluralities are (modeled by) sets, we can offer the following simple definitions of
the numerical adjectives.
1
[a] ü a+ has exactly one member
2
[a] ü a+ has exactly two members
3
[a] ü a+ has exactly three members
etc.

These in turn have strictly first-order rewrites, as follows.
ü
$x "y{yÎa+«y=x}
ü
$x1x2 { x1¹x2 & "y{yÎa+«. y=x1 Ú y=x2} }
ü
$x1x2x3 { x1¹x2 & x1¹x3 & x2¹x3 & "y{yÎa+ «. y=x1 Ú y=x2 Ú y=x3} }
etc.

We propose that plural-predication is a primitive notion on a par with singular-predication. So,
for example, our lexicon might include the following entries.
.
student. = lx0 S[x]
.
meeting. = lx0M[x]
Here, the variable ‘

x’ ranges over count-entities.

So, although we use singular-terms in our mathematical description, we understand them as modeling count-entities, including singular-entities and plural-entities.

So if x is a plural-entity, then S[x] means that x are students, and M[x] means that x are meeting.

We further propose that the lexicon will additionally include the following "logical" information.

(
L1) distributive[S]
(
L2) plural[M]
These in turn are expanded in accordance with the following definitions.
(d1) distributive[P]
ü "x { P[x] « "y{yáx ² P[y]} }
(d2) plural[P]
ü "x { P[x] ² Ã[x] }

Here, . is the mereological part-whole relation on the class ¶ of count-entities, which is officially
defined as follows.

a
.b ü a+Úb+
a
áb ü a.b & b./a
30

In other words, to say that P is (fully) distributive is to say that a count-entity
x is P if and only if every
count-entity that
x properly contains is P. And to say that P is (inherently) plural is to say that only plural-entities are P.

Notice that no (inherently) plural predicate is (fully) distributive, although many plural predicates are
plural-distributive, which may be defined as follows.

(d3) plural-distributive[P]
ü "x { P[x] «. Ã[x] & "y{Ã[y] & yáx .² P[y]} }

For example, it is fairly plausible to propose that  ‘meeting’ is plural-distributive.

For example, if Jay,
Kay, and Elle are meeting, then Jay and Kay are meeting, but we would say that Jay (or Kay, or Elle) is
meeting.

The following is an example that uses
‘and’ both logically and mereologically.
Jay and Kay, and Ray and Fay, are married
.
(
õ.)õ. õ1 1õ.
[+1] are married
 .
2õ. 
and
 
2õ   2õ 
Jay and Kay Ray and Fay
M
[J+K] & M[R+F]
l
P { P(J+K) & P(R+F) } lx{x1} lx1 M[x]
.
+1. .are married.
J
+K & R+F
.
and.
J
+ K R + F
.
Jay. .and. .Kay. .Ray. .and. .Fay.

First-order logic studiously avoids plural quantifiers, paraphrasing what it can, and ignoring the
rest, as illustrated in the following.

all dogs (are)
à every dog (is)
some dogs (are)
à some dog (is)
no dogs (are)
à no dog (is)
most dogs (are)
à Æ

Notwithstanding the clarity and precision afforded by singular-only speech, the overall
translation method is not entirely successful, as the following examples illustrate.

(1)

some students are waiting in the lounge

(2)

some students are gathering in the lounge

(3)

some theatre critics only listen to each other

For example, given the plural-inflection, (1) conveys the information that the students involved
are a plurality, which is grammatically analyzed as follows.

Some students are waiting in the lounge
.
(
1õ.)õ. 1õ.
are waiting…
Ïõ
[(1õ.)õ.] Ï
some
1 students
Ï ÏõÏ
student [+plural]
$
x { S[x] & Ã[x] & W[x] }
l
Q1 $x { S[x] & Ã[x] & Q(x) } lx1 W[x]
.
are waiting….
l
P0lQ1$x{…} lx0 { S[x] & Ã[x] }
.
some1 . .students.
l
x0 S[x] lP0 lx0 { P(x) & Ã[x] }
.
students. .+plural.

This reads the sentence as saying that there is a plural-set of students whose members are waiting in the lounge.

We are now in position to grammatically analyze numerical quantifiers, as in the following example.

Three dogs are barking.

In particular, we postulate that this sentence contains a covert existential quantifier, as in the following.

[some] three dogs are barking
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
[ some
1 ]
ÏõÏ Ï
three dogs
$
x { D[x] & 3[x] & B[x] }
l
Q1 $x { D[x] & 3[x] & Q(x) } lx1 B[x]
.
are barking.
l
P0lQ1$x{…} lx0 { D[x] & 3[x] }
.
some1 .
l
P0 lx0 { P(x) & 3[x] } lx0{D[x] & Ã[x]}
.
three. .dogs.

Note that the Ã-predicate becomes redundant after we add the 3-predicate, so it is dropped.

This analysis reads the original sentence as saying that the members of at least one 3-membered set of dogs are barking.

Notice that, granting distributivity, this is tantamount to saying that  at least three dogs are
barking.

In order to convey that exactly three dogs are barking, we need additional semantic
information, not postulated in the above analysis.

Compare this sentence to the following.

The three dogs are barking.

1 1õ.
are barking
Ïõ
1 Ï
the
1
ÏõÏ Ï
three dogs
B
[ ·x{D[x] & 3[x]} ]
·
x { D[x] & 3[x] }1 lx1 B[x]
.
are barking.
l
P0·xP(x) lx0 { D[x] & 3[x] }
.
the1 .
l
P0 lx0 { P(x) & 3[x] } lx0{D[x] & Ã[x]}
.
three. .dogs.
the dog is barking
.

1 1õ.
is barking
Ïõ
1 Ï
the
1 dog
ÏõÏ Ï
one dog
B
[ ·x{D[x] & 1[x]} ]
·
x { D[x] & 1[x] }1 lx1 B[x]
.
is barking.
l
P0·xP(x)1 lx0 { D[x] & 1[x] }
.
the. .dog.
l
P0 lx0 { P(x) & 1[x] } lx0 D[x]
.
one. .dog.

This first sentence is read as saying, in effect, that there is exactly one 3-membered set of dogs (in the
relevant domain), and its members are barking.

The second sentence is read as saying that there is exactly one 1-membered set of dogs (in the relevant domain), and its unique member is barking, which is to say there is exactly one dog, and it is barking.

Note that we employ the covert morpheme ‘one’ (as Grice does, WoW: 22).

We can alternatively employ singular-number inflection

[+singular].

Traditionally, definite-determiner phrases are logically mapped to definite descriptions involving
the iota-operator as we have done in previous examples (and also cited by Grice, p. 22).

The following example, however,
demonstrates the shortcomings of this approach.
the dogs are barking
.

1 1õ.
are barking
Ïõ
1 Ï
the
1 dogs
B
[ ·xD+[x] ]
·
x D+[x]1 lx1 B[x]
.
are barking.
l
P0 ·xP(x)1 lx0 D+[x]
.
the1. .dogs.

Here, we introduce the following abbreviation.

P
+[a] ü P[a] & Ã[a]
30

According to this analysis, the sentence says, in effect, that there is exactly one plural-set of dogs, and its members are barking.

The problem is immediate and fatal.

Suppose there are in fact three dogs barking; then how many plural-sets of dogs are there whose members are barking?

Well, since ‘barking’ is distributive, there are four such sets!

But the above analysis claims that there is exactly one such set.

What is needed is an alternative account of ‘the’.

The best-known alternative is due to Godehard Link, who proposes the following.

.
the. = lP0 mxP(x)
Here, ‘
m’ (mu) is short for ‘maximum’, which is defined relative to the mereological part-whole relation
.
among count-objects.8 In particular,
mnF
ü ·n{ F & "u{F[u/n] ² n.u} } [n free for u in F]

In other words,

mnF is the unique count-object that is F and furthermore contains every count-object
that is
F.

Let's see how this works in our earlier case.

Given our notation, we don't have to change much – just replace ‘

·’ with ‘m’.

the dogs are barking
.

1 1õ.
are barking
Ïõ
1 Ï
the
1 dogs
B
[ mxD+[x] ]
m
x D+[x]1 lx1 B[x]
.
are barking.
l
P0 mxP(x)1 lx0 D+[x]
.
the1. .dogs.

According to the analysis, the sentence says that there is a maximal plural-set of dogs (in the relevant
domain), and its members are barking.

Link's account of  ‘the’ works on problematic examples, but does it work on examples for
which we already have a satisfactory solution?

If not, then we must postulate two different meanings of ‘the’, which is, as anybody who loves Grice knows, theoretically inelegant.

Fortunately, our earlier examples work just as well with the new account of ‘the’, as seen in the following examples.

7
reference+++
8

We note that the Link account of ‘the’ also applies to mass nouns like ‘water’ as in ‘the water in the bathtub’, although it requires postulating an expanded domain of entities that includes mass-entities, which include all manner of amorphous "quantities of matter".

For example, these entities figure in explaining what I mean when I say that the gold in
this ring was once scattered across the Galaxy [as currently suggested by cosmologists].

the three dogs are barking the dog is barking
B
[ mx{D[x] & 3[x]} ]
m
x { D[x] & 3[x] }1 lx1 B[x]
.
are barking.
l
P0 mxP(x)1 lx0 { D[x] & 3[x] }
.
the1.
l
P0 lx0 { P(x) & 3[x] } lx0 D+[x]
.
three. .dogs.
B
[ mx{D[x] & 1[x]} ]
m
x { D[x] & 1[x] }1 lx1 B[x]
.
is barking.
l
P0 mxP(x)1 lx0 { D[x] & 1[x] }
.
the1. .dog.
l
P0 lx0 { P(x) & 1[x] } lx0 D[x]
.
one. .dog.

The first sentence is read as saying that there is a unique maximal 3-membered set of dogs, and its
members are barking, and the second sentence is read as saying that there is a unique maximal 1-
membered set of dogs, and its members are barking.

Note that, as a matter of set theory, there is a maximal 3-membered set of dogs precisely if there is exactly one 3-membered set of dogs, and similarly, there is a maximal 1-membered set of dogs precisely if there is exactly one dog.

Accordingly, we can replace ‘m’ by ‘·’ in the above examples.

According to Link's account,
.
the. = lP0 mxP(x)
where
mxP(x) is the maximal set of count-entities that are P.

Consider the following analysis in accordance with this account.

the people I met-with this week

Ïõ Ï
the
Ï ÏõÏ
people [that] I met
with this week
m
x { P[x] & M[I,x] }
l
P0 mxP(x) lx0 { P[x] & M[I,x] }
.
the.
l
x0P[x] lP0 lx0 { P(x) & M[I,x] }
.
people. .that I met
with this week
.
Here, we take ‘meet with’ as a phrasal verb. Now, suppose that the following summarizes the meetwith events in which I was involved in this week.

I met with Jay on Monday at 10:00 a.m.;

I met with Kay and Elle on Tuesday at 11:00 a.m.;

In particular, on no occasion did I meet with Jay

+Kay+Elle, so there is no maximal set of individuals that I met with this week.

Yet it seems reasonable to claim that the people I met with this week include Jay, Kay, and Elle (and no one else).

To account for this reading, we propose the following cumulative reading of ‘the’.
.
the+. = lP0 ÅxP(x)
where:
Å
nF ü lub {n : F} least upper bound
ub
(S) ü {x : "y(yÎS² y.x} upper bound
l
(S) ü ·x{xÎS & "y{yÎS² x.y}} least

For example, the following is the set of all count-entities that I met with.

{Jay, Kay
+Elle}

So, the least upper bound of this set in the class of count-entities is:

Jay
+Kay+Elle [= {Jay, Kay, Elle}]

We next consider the use of universal quantifiers in plural domains, as illustrated in the
following examples.

(1)

All dogs are barking

(2)

All the dogs are barking

(3)

All three dogs are barking

Item (1) is fairly straightforward, being analyzed as follows.

All dogs are pets
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
all
1 dogs
"
x { D+[x] ²B[x] }
l
Q1 "x { D+[x] ² Q(x) } lx1 B[x]
.
are barking.
l
P0lQ1"x{…} lx0 D+[x]
.
all1. .dogs.

This reads the sentence as saying that every plural-set of dogs has the following property – its members are barking.

Granting that barking is distributive, this amounts to saying that every dog is barking.

Item (2) is not so straightforward, since it appears to have a double-determiner, involving in
particular a type-mismatch between ‘all’ and ‘the’.

In order to resolve this problem, we propose an optionally-pronounced partitive ‘of’ interposed between ‘all’ and ‘the’, as in the following grammatical analysis.

all [of] the dogs are
barking
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
all
1
õÏ 
[of]
Ïõ Ï
the dogs
"
x { D[x] ²B[x] }
l
Q1 "x { D[x] ² Q(x) } lx1 B[x]
.
are barking.
l
P0lQ1"x{…} lx0 D[x]
.
all1.
l
y lx0 {x.y} mx D+[x]
.
of.
l
P0 mxP(x) lx0 D+[x]
.
the. .dogs.

Thus, according to this analysis, the sentence says that every dog-entity "is" barking, which granting
distributivity is the same as every individual dog (in the relevant domain) is barking.

Note the partitive use of ‘of’, which is categorially rendered as follows.

type(
of) = õÏ
.
of. = ly lx0 {x.y}

Thus, ‘of’ converts a proper-noun phrase into a common-noun phrase, pretty much reversing the effect of
‘the’. 9

Item (3) is even less straightforward.

First, a naïve analysis goes as follows.

all three dogs are barking
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
all
1
ÏõÏ Ï
three dogs
"
x { D[x] & 3[x] .²B[x] }
l
Q1 "x { D[x] & 3[x] .² Q(x) } lx1 B[x]
.
are barking.
l
P0lQ1"x{…} lx0 { D[x] & 3[x] }
.
all1 .
l
P0 lx0 { P(x) & 3[x] } lx0 D+[x]
.
three. .dogs.

This reads the sentence as saying that the members of every 3-membered set of dogs are barking.

This is not a very plausible reading, unless we bring a lot of IMPLICATURE to save it (as Stephen Yablo, a former student of Grice's once said, "Implicatures happen").

A more plausible reading posits unpronounced material as in the following.

Not perfectly, however, since .dogs. includes only pluralities, whereas .of the dogs. includes individuals as well as pluralities.

all [of the] three dogs are barking
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
all
1
õÏ 
[of]
Ïõ Ï
[the]
ÏõÏ Ï
three dogs
"
x { x.mx{D[x]&3[x]} ² B[x] }
l
Q1"x{x.mx{D[x]&3[x]}²Q(x)} lx1 B[x]
.
are barking.
l
P0lQ1"x{…} lx0{x.mx{D[x]&3[x]}}
.
all1.
l
ylx0{x.y} mx{D[x]&3[x]}
.
of.
l
P0 mxP(x) lx0{D[x]&3[x]}
.
the.
l
P0lx0{P(x)&3[x]} lx0D+[x]
.
three. .dogs.

This reads the sentence as saying, in effect, that there are exactly three dogs, and they are
all barking.

For the sake of comparison, we conclude this section with counterpart examples that employ the
singular-quantifier

‘every’.

Every dog is barking.

.
(
1õ.)õ. 1õ.
is barking
Ïõ
[(1õ.)õ.] Ï
every
1 dog
ÏõÏ 
one dog
"
x { D[x] & 1[x] .²B[x] }
l
Q1"x{D[x] & 1[x] .² Q(x)} lx1 B[x]
.
is barking.
l
P0lQ1"x{…} lx0 { D[x] & 1[x] }
.
every1. .dog.
l
P0 lx0 { P(x) & 1[x] } lx0 D[x]
.
one. .dog.

Every one of the dogs is barking
.
(
1õ.)õ. 1õ.
is barking
Ïõ
[(1õ.)õ.] Ï
every
1
ÏõÏ 
one
Ïõ Ï
of the dogs
"
x { D[x] & 1[x] .²B[x] }
l
Q1"x{D[x] & 1[x] .² Q(x)} lx1 B[x]
.
is barking.
l
P0lQ1"x{…} lx0 { D[x] & 1[x] }
.
every1.
l
P0 lx0 { P(x) & 1[x] } lx0 D[x]
.
one.
l
y lx0 {x.y} mx D+[x]
.
of. .the dogs.

Earlier, we proposed that the fundamental meaning of numerical QPs involves the phrase that Grice explicitly uses on p. 22 of WoW: "at least" ("at least one" for the meaning of "(∃x)"), rather
than "exactly".

In order to convey "exactly", the sentence either needs appropriate contextual factors, or it needs explicit modifiers such as ‘exactly’, ‘precisely’, and ‘only’.

The later are examples of exclusive adverbs, which also include the following, among others:

"just two" "just one"

"merely" -- "merely one", "merely two"

"simply"

"solely",

"alone",

"uniquely",

"exclusively",

"specifically",

"particularly",

"barely",

"scarcely"

The general idea is that an exclusive adverb focuses attention on a phrase, and conveys that other
possibilities are excluded.

What makes exclusive adverbs so interesting and perplexing is that the very same surface form
can receive many different interpretations.

Our favorite example comes from a popular song from the 1950's, whose title and key lyric is:

"I only have eyes for you."

We think we all sort of understand the sentiment of this song.

But imagine a considerably more gruesome scenario in which the village butcher, who saves body parts for the infamous surgeon Dr. Frankenstein, one day declares:

"Sorry, Dr. Frankenstein, but today I only have eyes for you."

It is also not hard to imagine a less flattering reading of the song in which the speaker (let's say, a boy) tells his girlfriend that only he has eyes for her.

So, depending upon how it is intonated, the sentence can be paraphrased in the following three different manners.

(1) I have eyes for you, but for no one else.

(2) I have eyes for you, but I have nothing else for you.

(3) I have eyes for you, but no one else does.

Given its focus-sensitivity, and given the variety of phrase types that ‘only’ can modify, the
semantics of ‘only’ is subtle and difficult.

We propose that ‘only’ is a multi-categorial adverb with the following multi-type.

type(
only) = (Kõ.)õ(Kõ.) [one for each type K]
Here,

K is the type of the focused phrase, and Kõ. is the type of the matrix that contains the focused
phrase.

For example, in

"I only have eyes for you"

the focused phrase is ‘you’, which has type 2, and the matrix is ‘I have-eyes-for…’, which has
type
2õ..

The semantics of  ‘only’ is a bit complicated.

Our first approximation goes as follows.

(We can also concoct a reading in which ‘have’ is focused, and one in which ‘for’ is focused, but these are grammatically far-fetched).

only. = lF ln { F(n) & ;$n¢ { F(n¢) & n¢¹n }
º lF ln "n¢
{ F(n¢) «n¢=n }
where
F Î .Kõ..
n
,n¢ Î .K.

For example, in our current example,
K=2, and
.
only. = lP2 ly2 { P(y) & ;$z{P(z) & z¹y} }
º l
P2 ly2 "z { P(z) «z=y }

Thus, we have the following grammatical analysis, in which we treat ‘have eyes for’ as an
idiomatic unit (lexical item).

"I only have-eyes-for you"
.

1 1õ.
I
1

2õ(1õ.) U2
you
2
(
2õ.)õ(2õ.) 2õ(1õ.)
only have-eyes-for
"
z { E[I,z] «z=U }
I
1 lx1 "z { E[x,z] «z=U }
.
I1.
l
y2 lx1 "z { E[x,z] «z=y } U2
.
you2.
l
P2 ly2"z{ P(z) «z=y } ly2 lx1 E[x,y]
.
only. .have-eyes-for.

The top node in the semantic tree says that the speaker of the sentence (I) has eyes for the audience of
the sentence (U), but for no one else.

The key composition is underwritten by the following derivation.

(1) (N
2²S)²(N2²S) 1 Pr lP2 ly2 "z { P(z) «z=y }
(2) N
2²(N1²S) 2 Pr ly2 lx1 E[x,y]
(3) N
2 3 As y2
(4) N
1 4 As x1
(5) N
2²S 24 2,4,MP2 ly2 E[x,y]
(6) N
2²S 124 1,5,²O ly2 "z { E[x,z] «z=y }
(7) S 1234 3,6,
²O "z { E[x,z] «z=y }
(8) N
1²S 123 4-7,²I lx1 "z { E[x,z] «z=y }
(9) N
2²(N1²S) 12 3-8,²I ly2 lx1 "z { E[x,z] «z=y }

Compare the above to the following alternative reading in which ‘I’ is focused.

In this case,
K
=1, and
.
only. = lP1 lx1 "z { P(z) «z=x }
30
I
only have-eyes-for you
.

1 1õ.
I
1

2õ(1õ.) U2
you
2
(
1õ.)õ(1õ.) 2õ(1õ.)
only have-eyes-for
"
z { E[z,U] «z=I }
I
1 lx1 "z { E[z,U] «z=x }
.
I1.
l
y2 lx1 "z { E[z,y] «z=x } U2
.
you2.
l
P1lx1"z{ P(z) «z=x } ly2 lx1 E[x,y]
.
only. .have-eyes-for.

The top node in the semantic tree says that the speaker of the sentence (I) has eyes for the audience of
the sentence (U), but no one else does.

The key composition is underwritten by the following derivation.
(1) (N
1²S)²(N1²S) 1 Pr lP1 lx1 "z { P(z) «z=x }
(2) N
2²(N1²S) 2 Pr ly2 lx1 E[x,y]
(3) N
2 3 As y2
(4) N
1 4 As x1
(5) (N
1²S)²S 14 1,4,MP2 lP1 "z { P(z) «z=x }
(6) N
2²S 124 2,5,TR ly2 "z { E[z,y] «z=x }
(7) S 1234 3,6,
²O "z { E[z,y] «z=x }
(8) N
1²S 123 4-8,²I lx1 "z "z { E[z,y] «z=x }
(9) N
2²(N1²S) 12 3-9,²I ly2 lx1 "z { E[z,y] «z=x }

Our account of  ‘only’ works great for the two previous examples, but consider the following
example,

"Jay only has eyes for Kay and Elle"

where we presume that ‘Kay and Elle’ is the focused phrase.

First, we must face the issue of whether ‘and’ is logical-conjunction or mereological conjunction.

If ‘and’ is mereological, then ‘Kay and Elle’ is a proper-noun phrase (), and Kayand-
Elle is a plural entity, in which case we obtain the following analysis.

Jay only has eyes for Kay and Elle
.

1 1õ.
Jay
1

2õ(1õ.) 2
Kay and
2 Elle
(
2õ.)õ(2õ.) 2õ(1õ.)
only has-eyes-for
"
z { E[J,z] «z=K+L }
J
1 lx1 "z { E[x,z] «z=K+L }
.
Jay1.
l
y2 lx1 "z { E[x,z] «z=y } (K+L)2
.
Kay and2 Elle.
l
P2ly2"z{ P(z) «z=y } ly2 lx1 E[x,y]
.
only. .has-eyes-for.

According to this analysis, the sentence says that Jay has eyes for an entity if and only if that entity is
(identical to) the plurality Kay +Elle.

Therefore, since Kay ¹ Kay+Elle, Jay does not have eyes for Kay, and similarly he does not have eyes for Elle.

Jay only has eyes for Kay-and-Elle (as a unit so to speak).

Perhaps the envisaged circumstances are odd (or even kinky!), but it is nevertheless an admissible reading of the sentence.

On the other hand, if  ‘and’ is logical-conjunction, then ‘Kay and Elle’ is a quantifier phrase [
(õ.)õ.], and Kay-and-Elle is a QP-object, in which case we categorially render ‘only’ as
follows.

type(
only) = [(2õ.)õ.]õ[(2õ.)õ.]
.
only. = lÄ2 lP2 "Q2 { Ä(Q) « Q=P }
º l
Ä2 lP2 { Ä(P) & ;$Q { Ä(Q) & Q¹P } }
where
Ä2 Î .(2õ.)õ..
P
2,Q2 Î .2õ..

Unfortunately, this does not yield appropriate truth-conditions for the above sentence, since it implies
that the above sentence says that Jay has eyes for neither Kay nor Elle (exercise).

In light of the difficulties faced by our original account of .only., we now consider the
following  second approximation account of .only..
.
only. = lF ln { F(n) & ;$n¢ { F(n¢) & n¢^n } }
where
F Î .Kõ..
n
,n¢ Î .K.

Note that this account is obtained from the first approximation by replacing ‘

By ‘^’, where ‘^’ refers to the disjointness relation, the exact definition of which varies from sort/type to sort/type.

For example, when applied exclusively to singular-entities, disjointness (^) coincides with nonidentity, and the revised account subsumes our earlier account.

This is to be expected, since the earlier account makes correct predictions when applied to singular-entities.

On the other hand, the revised account does not coincide with the original account when applied to plural-entities.

For example, when applied to our standing example, we obtain the following analysis.

"Jay only has eyes for Kay and Elle."

E
[J,K+L] & ;$z{E[J,z] & z^K+L}
J
1 lx1 { E[x,K+L] & ;$z{E[x,z] & z^K+L} }
.
Jay1.
l
y2 lx1 { E[x,y] & ;$z{E[x,z] & z^y} } (K+L)2
.
Kay and2 Elle.
l
P2 ly2 { P(z) & ;$z{P(z) & z^y} } ly2 lx1 E[x,y]
.
only. .has-eyes-for.

This analysis reads the sentence as saying that Jay has eyes for the plural-entity Kay-and-Elle, but for no entity disjoint from Kay-and-Elle.

So it does not say that Jay does not have eyes for Kay, or for Elle.

Unfortunately, it does not say that Jay  does have eyes for Kay or Elle, unless we further hypothesize that the E-relation is distributive.

According to the most plausible reading , the sentence under scrutiny says that Jay has eyes for Kay,
and Jay has eyes for Elle, but Jay has eyes for no one else.

This suggests that the appropriate reading of ‘and’ is the logical reading, in which case the focus of
‘only’ is a QP, but the alternatives considered are not all QP-objects, but only those QP-objects that
are "like" Kay, Elle, and Kay-and-Elle.

Putting all this together, we obtain the following categorial rendering of
‘only’.
.
only. = lt2 lÄ2 {x(Ä)} { t(Ä) & ;$Ã{x(Ã) & t(Ã) & Ã^Ä} }
where
t Î .[(õ.)õ.]õ..
Ã
,Ä Î .(õ.)õ..
x
(Ã) ü $P [ $xP(x) & Ã = lQ"x{P(x)²Q(x)} ]
Ã
^Ä ü ;$P{ Ã(P) & Ä(P) }

Note the introduction of the additional restriction on the domain of QP-objects, which in particular
restricts the domain to universal-QP-objects.

The following is the associated analysis, where

E(a) ülnE[a,n] [a free for n].

Jay only has eyes for Kay and Elle
.

1 1õ.
Jay
1
[(
2õ.)õ.]õ(1õ.) (2õ.)õ.
Kay and
2 Elle
{ [(
2õ.)õ.]õ. } õ { [(2õ.)õ.]õ.} 2õ(1õ.)
only has-eyes-for
E
[J,K] & E[J,L] & ;$Ã{ x(Ã) & Ã(lxE[J,x]) & Ã^lP{P(K) & P(L)} }
J
1 lx1 { E[x,K] & E[x,L] & ;$Ã{ x(Ã) & Ã(lyE[x,y]) & Ã^lP{P(K) & P(L)} } }
.
Jay1.
l
Ä2 {x(Ä)} { Ä(lyE[x,y]) & ;$Ã{x(Ã) & Ã(lyE[x,y]) & Ã^Ä} } lP2 { P(K) & P(L) }
.
Kay and2 Elle.
l
t2 lÄ2 {x(Ä)} { t(Ä) & ;$Ã{x(Ã) & t(Ã) & Ã^Ä} } ly2 lx1 E[x,y]
.
only. .has-eyes-for.

First, note that, in the second composition, the input is of the appropriate sort for the functor.

Next,
since the top node quantifies over second-order predicates, it is not obvious what it says.

As it turns out, it is equivalent to the following, which is exactly what we want.
"
x { E[J,x] «. x=K Ú x=L }

The equivalence is demonstrated in the following type-theory derivations, where ‘
E’ is short for
‘
lxE[J,x]’.
(1)
"x { E(x) «. x=K Ú x=L } Pr
(2)
­: E(K) & E(L) & ;$Ã{ x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } 3,4,SL
(3)
E(K) & E(L) 1,IL
(4)
­: ;$Ã{ x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } ID
(5)
$Ã{ x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } As
(6)
x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } $O
(7)
­: þ 16,19,SL
(8)
$P { $xP(x) & Ã = lQ"x{P(x)²Q(x)} } 6a,Def x
(9)
$xP(x) $&O
(10) P(
a) $O
(11)
à = lQ "x { P(x) ² Q(x) } 10, $&O
(12)
lQ "x { P(x) ² Q(x) } ^ lP{P(K) & P(L)} } 6c,11,IL
(13)
lP{P(K) & P(L)} = lQ"x { x=K Ú x=L .² Q(x) } lC
(14)
lQ "x { P(x) ² Q(x) } ^ lQ "x { x=K Ú x=L .² Q(x) } 12,13,IL
(15)
;$x { P(x) &. x=K Ú x=L } 14,Def ^
(16)
a¹K & a¹L 10,15,QL
(17)
"x { P(x) ² E(x) } 6b,8b,IL,lC
(18)
E(a) 10,17,QL
(19)
a=K Ú a=L 1,18,QL
30
(1)
E(K) & E(L) & ;$Ã{ x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } Pr
(2)
­: "x { E(x) «. x=K Ú x=L } 1a,1b,3,QL
(3)
­: "x { E(x) ². x=K Ú x=L } UCD
(4)
E(a) As
(5)
­: a=K Ú a=L DD
(6) [
lP{P(a)}](E) 4,lC
(7) P(
a) «"x{x=a² P(x)} IL
(8)
lP{P(a)} = lP"x{ [lx{x=a}](x) ² P(x) } 7,lC
(9)
x( lP{P(a)} ) 8,QL,Def x
(10)
lP{P(K) & P(L)} ^/ lP{P(a)} 1c,6,9,QL
(11)
lP{P(K) & P(L)} = lQ"x { x=K Ú x=L .² Q(x) } lC
(12)
lP{P(a)} = lQ"x { x=a² Q(x) } lC
(13)
lQ"x { x=K Ú x=L .² Q(x) } ^/ lQ"x { x=a² Q(x) } 10-12,IL
(14)
$x { x=K Ú x=L & x=a } 13,Def ^
(15)
a=K Ú a=L 14,IL

What happens when the focus is a CNP?

Consider the following example.

Only poisonous snakes are dangerous

Depending on the focus of ‘only’, this is ambiguous among the following.


Snakes are dangerous, but non-poisonous snakes are not dangerous
poisonous

Snakes are dangerous, but other poisonous things are not dangerous
poisonous.

Snakes are dangerous, but other things are not dangerous

We first consider the last one, since its overall structure is the simplest.

The first thing to notice is that the paraphrase is not well-formed by simple categorial formation rules.

poisonous snakes are dangerous…
?
CNP VP
are dangerous
Adj CNP
poisonous snakes

We need an NP to serve as the subject of the VP, but what we have is a CNP.

There seems to be a missing determiner – but which one?

When we say that poisonous snakes are dangerous, do we mean "all" poisonous snakes, "most" poisonous snakes, "some" poisonous snakes, or *what*?

There does not seem to be a generally agreed upon answer to this question. Grice says that "it depends on context", which hardly illuminates (His analysis of "Englishmen are brave" -- in Jill's reasoning, "Jack is an Englishman; he is, therefore, brave", "Aspects of Reasoning" -- Grice further explores the basis for such a generalisation: empirical, tautological, deductive?)

However, we propose that, at least in the presence of ‘only’, the missing determiner is ‘some’.

So, for example, the sentence

Only men play NFL football

says that some men play NFL football, but no non-men play NFL football.

It most certainly does not say that all, or most, or generally, men play NFL football.

How does this fit into our account of  ‘only’?

The focus phrase is ‘men’ which is a CNP, so the applicable sub-category of ‘only’ is:

(Ïõ.)õ(Ïõ.)

and the associated interpretation of  ‘only’ is:

l
Ä0 lQ0 { Ä(Q) & ;$P{Ä(P) & P^Q} }
where
P
^Q ü ;$x { P(x) & Q(x) }

With this account of .only. in hand, we offer the following analysis.

Note the covert determiner ‘some’.

Also note that, for the sake of brevity, we occasionally write ‘M’ in place of ‘lxM[x]’.

only men play NFL football
.
(
1õ.)õ. 1õ.
play NFL
Ïõ
[(1õ.)õ.] (Ïõ.)õ.
[some
1]
(
Ïõ.)õ(Ïõ.) Ï
only men
$
x{ M[x] & F[x] } & ;$P{ $x{P(x) & F[x]} & P^M}
l
Q1{$x{M[x]&Q(x)} & ;$P{$x{P(x)&Q(x)} & P^M}} lx1 F[x]
.
play NFL.
l
P0lQ1{…} lÄ0 { Ä(M) & ;$P{Ä(P) & P^M} }
.
some1.
l
Ä0 lQ0 { Ä(Q) & ;$P{Ä(P) & P^Q} } lx0 M[x]
.
only. .men.

The key composition is underwritten by the following derivation.

(1) (
¤²S)²S 1 Pr lÄ0 { Ä(M) & ;$P{Ä(P) & P^M} }
(2)
¤²[(N1²S)²S] 2 Pr lP0 lQ1 $x{ P(x) & Q(x) }
(3) N
1²S 3 As Q1
(4)
¤²S 23 2,3,MP2 lP0 $x{ P(x) & Q(x) }
(5) S 123 1,4,
²O $x{M[x]&Q(x)} & ;$P{ $x{P(x)&Q(x)} & P^M}
(6) (N
1²S)²S 12 3-5,²I lQ1 $x{M[x]&Q(x)} & ;$P{ $x{P(x)&Q(x)} & P^M}

Now, let us examine the top node more carefully. It clearly contains the information that some men play NFL football.

The question is whether it also says (or merely "implicates", to use Grice's insidious parlance) that no one else does.

In other words, is the top node equivalent to the following.

Note carefully, however, that there is also a weak sense of ‘only’.

$
x{ M[x] & F[x] } & ;$x { ;M[a] & F[a] }

By way of answering this question, we offer the following type-theory derivations.
(1)
;$P{ $x{P(x) & F[x]} & P^M } Pr
(2)
­: ;$x { ;M[a] & F[a] } ID
(3)
$x { ;M[a] & F[a] } As
(4)
;M[a] & F[a] 2,$O
(5)
­: þ 4a,12,SL
(6) [
lx(x=a)](a) IL,lC
(7)
$x{ [lx(x=a)](x) & F[x] } 4b,6,QL
(8)
lx(x=a) ^/ M 1b,7,QL
(9)
$x{ [lx(x=a)](x) & M[x] } 8,Def ^
(10) [
lx(x=a)](b) & M[b] 9,$O
(11)
b=a 9a,lC
(12)
M[a] 9b,10,IL
(1)
;$x { ;M[x] & F[x] } Pr
(2)
­: ;$P{ $x{P(x) & F[x]} & P^M } ID
(3)
$P{ $x{P(x) & F[x]} & P^M } As
(4)
­: þ 6b,8,SL
(5)
$x{P(x) & F[x]} & P^M 3,$O
(6) P(
a) & F[a] 4a,$O
(7)
;$x { P(x) & M[x] } 3b,Def ^
(8)
;M[a] 6a,7,QL
(8)
;F[a] 1,7,QL

We next consider an example in which the focus is an adjective, as in the following example.

Only poisonous snakes are dangerous.

In this case the focus-type is

ÏõÏ, and the matrix-type is (ÏõÏ)õ., and the associated interpretation of
‘only’
is as follows.
.
only. = l» lh { »(h) & ;$h¢{»(h¢) & h¢^h} }
where
» Î .(ÏõÏ)õ..
h Î
ÏõÏ
h¢^h
ü ; $P0 $x0 { h¢(P0)(x0) & h(P0)(x0) }
Hardegree,
Numerical Quantifiers page 23 of 30 30

Note carefully that the construction only makes sense for subsective adjectives,12 so in particular, the h-variables range over subsective adjectives.

Compare, if you have the time, the following two pieces of nonsense involving nonsubsective
adjectives:

Only alleged snakes are dangerous.

Only former snakes are dangerous

The following is the associated grammatical analysis.

Only poisonous snakes are dangerous
.
(1õ.)õ. 1õ.
are dangerous
Ïõ
[(1õ.)õ.] (Ïõ.)õ.
[some
1]
[(
ÏõÏ)õ.]õ. Ï
snakes
[(
ÏõÏ)õ.]õ[(ÏõÏ)õ.] ÏõÏ
only poisonous
$
x{ P(S)(x) & D[x] } & ;$h{ $x{h(S)(x) & D[x]} & h^P }
l
Q1{ $x{P(S)(x) & Q(x)} & ;$h{ $x{h(S)(x) & Q(x)} & h^P} } lx1 D[x]
.
are dangerous.
l
P0 lQ1 $x { P(x) & Q(x) } lÄ0 { Ä0( P(S) ) & ;$h{ Ä0( h(S) ) & h^P} }
.
some1.
l
» { »(P) & ;$h{»(h) & h^P} } lx0 S[x]
.
snakes.
l
» lh { »(h) & ;$h¢{»(h¢) & h¢^h} } P
.
only. .poisonous.

The key compositions are underwritten by the following derivations.

Basically, h is subsective if h(S) Ú S, for every set S of entities.

For example, ‘poisonous’ is subsective since every poisonous N is an N.)

Note that these two alleged pieces of nonsense make sense if ‘alleged snakes’ and ‘former snakes’ are focused, although they describe rather odd worlds, even for Grice.

(1) [(
¤²¤)²S]²S 1 Pr l» { »(P) & ;$h{»(h) & h^P} }
(2)
¤ 2 Pr lx0 S[x]
(3)
¤²S 3 As Ä0
(4)
¤²¤ 4 As h
(5)
¤ 24 2,4,²O h(S)
(6) S 234 3,5,
²O Ä0( h(S) )
(7) (
¤²¤)²S 23 4-6,²I lh Ä0( h(S) )
(8) S 123 1,7,
²O Ä0( P(S) ) & ;$h{ Ä0( h(S) ) & h^P}
(9) (
¤²S)²S 12 3-8,²I lÄ0 { Ä0( P(S) ) & ;$h{ Ä0( h(S) ) & h^P} }
(1) (
¤²S)²S 1 Pr lÄ0 { Ä0( P(S) ) & ;$h{ Ä0( h(S) ) & h^P} }
(2)
¤²[(N1²S)²S] 2 Pr lP0 lQ1 $x { P(x) & Q(x) }
(3) N
1²S 3 As Q1
(4)
¤²S 23 2,3,MP2 lP0 $x { P(x) & Q(x) }
(5) S 123 1,4,
²O $x{P(S)(x) & Q(x)} & ;$h{$x{h(S)(x) & Q(x)} & h^P}
(6) (N
1²S)²S 12 3-5,²I lQ1 $x{P(S)(x) & Q(x)} & ;$h{$x{h(S)(x) & Q(x)} & h^P}

Let us now examine the top node which is
$
x{ P(S)(x) & D[x] } & ;$h{ $x{h(S)(x) & D[x]} & h^P }

This clearly says that some poisonous snakes are dangerous. The question is whether it denies that any non-poisonous snakes are dangerous, which is to say whether it is equivalent to the following.
$
x{ P(S)(x) & D[x] } & ;$x { S[x] & ;P(S)(x) & D[x] }

This is settled in the following type-theory derivations.

(1)
;$h { $x{h(S)(x) & D[x]} & h^P } Pr
(2)
­: ;$x { S[x] & ;P(S)(x) & D[x] } ID
(3)
$x { S[x] & ;P(S)(x) & D[x] } As
(4)
­: þ 13b,13c,SL
(5)
S[a] & ;P(S)(a) & D[a] 3,$O
(6) [let]
h¢ = lQ0 lx0 { Q(x) & ;P(Q0)(x0) } lC
(7)
h¢(S) = lx0 { S[x] & ;P(S)(x0) } 6,IL
(8)
h¢(S)(a) 5a,b,7,IL,lC
(9)
$x{h¢(S)(x) & D[x] 5c,8,QL
(10)
h ^/ P 1,9,QL
(11)
$Q0 $x0 { h¢(Q0)(x0) & P(Q0)(x0) } 10,Def ^
(12)
h¢(Q0)(b0) & P(Q0)(b0) 11,$O
(13)
Q(b) & ;P(Q0)(b0) & P(Q0)(b0) 6,12,IL,lC
(Note 14

Note carefully the difference between a non-poisonous snake and a non(poisonous snake). Here, it is critical to the semantics that the  h-variables range over subsective adjectives.

(1)
;$x { S[x] & ;P(S)(x) & D[x] } Pr
(2)
­: ;$h { $x{h(S)(x) & D[x]} & h^P } ID
(3)
$h{ $x{h(S)(x) & D[x]} & h^P } As
(4)
­: þ 5b,9,SL
(5)
$x{h(S)(x) & D[x]} & h^P 3,$O
(6)
h(S)(a) & D[a] 5a,$O
(7)
;P(S)(a) 6b,5a,Def ^
(8)
S[a] 6a,h is subsective
(9)
;D[a] 1,7,8,QL

One of the more perplexing combinations involves ‘the’ and ‘only’.

First, note that the following are not equivalent.

"The only people I respect are kind."

"Only the people I respect are kind."

So, clearly ‘the only’ =/= ‘only the’, so we have to be careful in constructing our categorial
analysis.

We propose that, when it appears in the phrase ‘the only’, the role of ‘only’ is largely
emphatic, similar to how ‘unique’ operates inside ‘the unique’.

In other words, ‘the’ does the real work, and ‘only’ simply provides EXPLICATURAL emphasis (as we may call it, to use a noun that Grice avoided like the plague, 'explicature').

Let us apply this proposal to a few examples.

First consider the following:

The only people I respect are Jay and Kay.

which is equivalent to

"Jay and Kay are the only people I respect"

which suggests that ‘be’ is transitive.

This is further clarified in the following.
.

1 1õ.
Ïõ
1 Ï 1õ(1õ.) 1
the
1 are Jay and1 Kay
Æ
Ï
only people I respect
30
m
x R[x] = J+K
m
x R[x]1 lx1 { x = J+K }
l
P0 mx P(x)1 lx0R[x] ly1 lx1 { x=y } (J+K)1
.
the1. .are. .Jay and1 Kay.
Æ l
x0R[x]
.
only. .people I respect.

According to this analysis, the sentence says that the maximal count entity containing all the people I
respect is identical to the count-entity Jay+Kay.

By standard mereological reasoning, granting that "respect" is distributive, this is equivalent to saying that I respect an individual if and only if that individual is Jay or Kay.

Now, back to our original example.

"The only people I respect are kind"

Note that this is not equivalent to

"Kind are the only people I respect"

since the latter is, well, ill-formed (even if Speranza can utter it -- and worse, get understood!).

This suggests that ‘be’ is a copula.

This is further clarified in the following.
.

1 1õ.
are kind
Ïõ
1 Ï
the
1
Æ
Ï
only people I respect
K
[ mx R(x) ]
m
x R[x]1 lx1K[x]
.
are kind.
l
P0 mx P(x)1 lx0R[x]
.
the1.
Æ l
x0R[x]
.
only. .people I respect.

According to this analysis, the sentence says that the maximal count-entity containing all the people I
respect is counted among the entities that are kind.

This entails that everyone Speranza respect is kind, provided we read the predicate
‘is kind’ as distributive.

This is strongly encouraged by the presence of the word ‘only’ in Speranza's original phrase.

We next consider how  ‘only’ interacts with ‘if’ in phrases such as:

"I will get into Cal Tech only if I ace all my exams"

"A number is even only if it is divisible by two."

We propose that, in ‘only if’ clauses, the focus of ‘only’ is the antecedent (i.e., the complement
of ‘if’).

Thus, applying our general analysis of ‘only’, we have the following.

type(
only) = (.õ.)õ(.õ.)
.
only. = lh lp { h(p) & ;$q{h(q) & q^p} }
where
h Î ..õ..
q
^p ü q&;p

In other words,

The following is an example analysis.

"I will get into Cal Tech only if I ace all my exams."

.
.õ. .
I ace all my exams
.
0 .0õ(.õ.)
I will get into Cal Tech
(
.õ.)õ(.õ.) .õ(.0õ.)
only if
{
A²C} & ;$r{ (r²C) & r^A}
l
p{ {p²C} & ;$r{ (r²C) & r^p} } A
.
I ace all my exams.
C
0 lq0 lp { {p²q} & ;$r{ (r²q) & r^p} }
.
I will get into Cal Tech.
lh l
p { h(p) & ;$q{h(q) & q^p} } lp lq0 {p²q}
.
only. .if.
(1) (S
²S)²(S²S) 1 Pr lh lp { h(p) & ;$r{h(r) & r^p} }
(2) S
²(S0²S) 2 Pr lp lq0 {p²q}
(3) S
0 3 As q0
(4) S
²S 23 2,3,MP2 lp {p²q}
(5) S
²S 123 1,4,²O lp { {p²q} & ;$r{ (r²q) & r^p} }
(6) S
0²(S²S) 12 3-5,²I lq0 lp { {p²q} & ;$r{ (r²q) & r^p} }
30

The following type-theory derivations demonstrate that the top node is equivalent to the following.
A
«C
(1) {
A²C} & ;$r{ (r²C) & r^A} Pr
(2)
­: A«C 1a,3,SL
(3)
­: C²A 6,SL
(4)
C²C SL
(5)
C ^/ A 1b,4,QL
(6)
;(C & ;A) 5, Def ^
(1)
A«C Pr
(2)
­: {A²C} & ;$r{ (r²C) & r^A} 1,SL,3,SL
(3)
­: ;$r{ (r²C) & r^A} ID
(4)
$r{ (r²C) & r^A} As
(5)
­: þ 6a,7,SL
(6) B
²C & B^A 4,$O
(7) B &
;A 6b,Def ^

Now, here is the problem.

It does not seem that ‘only if’ is equivalent to ‘if and only if’ (cfr. Pears in Berlin et al, essays on Austin), so we propose that ‘only’ has both a strong sense and a weak sense, the latter being the negative half of the former.
.
onlyw. = lF ln ;$n¢ { F(n¢) & n¢^n }
where
F Î .Kõ..
n
,n¢ Î .K.

This gives us the following truth conditions for
‘C only if A’.15
C only if A
º if C then A

The weak sense of  ‘only’ also applies to CNP applications, according to which
only A's are B's does not logically entail

some A's are B's
but only (the weak half):
no non-A's are B's

This is left as an exercise for the Grice Club.

Note carefully that this equivalence, which is counterintuitive, is largely a product of the oddity (so well explained by Grice) of our truth conditions for ‘if’, according to which ‘if’ is truth-functional.

We should not automatically expect this equivalence to obtain for other (more robust) versions of
‘if…then’.

Finally, we consider how  ‘exactly’ works in phrases such as

Exactly three dogs are barking.

We propose that, in this context, ‘exactly’ behaves semantically like ‘only’ where the focus is the
numerical adjective ‘three’.

In particular, for numerical adjectives, we propose the following typeanalysis of
‘only’.

type(
only) = [(ÏõÏ)õ.]õ[(ÏõÏ)õ.]
.
only. = l» lh { »(h) & ;$h¢ { »(h¢) & h¢^h } }
where
» Î .(ÏõÏ)õ..
h
,h¢ Î Á [the class of numeral objects, a subclass of .ÏõÏ.]
where
h¢^h ü h¢ > h
With this proposal in hand, we now offer the following analysis.
exactly three dogs are barking
S
l
(1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] (Ïõ.)õ.
[ some
1 ]
[
(ÏõÏ)õ.]õ. Ï
dogs
[
(ÏõÏ)õ.]õ[(ÏõÏ)õ.] ÏõÏ
exactly three
$
x { D[x] & 3[x] & B[x] } & ;$h { $x{ h(D)(x) & B[x] } & h>3 } }
l
Q1 $x{D[x] & 3[x] & Q(x)} & ;$h{ $x{h(D)(x) & Q(x)} & h>3 } } lx1 B[x]
.
are barking.
l
P0 lQ1 $x { P(x) & Q(x) } lÃ0 { Ã0(lx{D[x]&3[x]}) & ;$h{ Ã0(h(D)) & h¢>3 } }
.
some1 .
l
» { »(lP0lx{P(x) & 3[x]}) & ;$h¢ { »(h¢) & h¢>3 } } lx0 D[x] [üD]
.
dogs.
l
» lh { »(h) & ;$h¢ { »(h¢) & h¢>h } } lP0lx{P(x) & 3[x]} [ü3]
.
exactly. .three.

According to this analysis, the sentence says that there is a three-membered set of dogs whose members are barking and there is no  larger set of dogs whose members are barking. The following derivations underwrite the key compositions. Note the abbreviation:
3D[a] üD[a]&3[a]

(1) [(

¤²¤)²S]²S 1 Pr l» { »(lP0lx{P(x) & 3[x]}) & ;$h¢ { »(h¢) & h¢>3 } }

(2)

¤ 2 Pr lx0 D[x]

(3)

¤²S 3 As Ã0

(4)

¤²¤ 4 As h

(5)

¤ 24 2,4,²O h ( lx0 D[x] )

(6) S 234 3,5,

²O Ã0 ( h ( lx0 D[x] ) )

(7) (

¤²¤)²S 23 4-6,²I lh { Ã0 ( h ( lx0 D[x] ) ) }

(8) S 123 1,7,

²O Ã0(lx{3D[x]}) & ;$h¢{Ã0(h¢(lx0 D[x])) & h¢>3 }

(9) (

¤²S)²S 12 3-8,²I lÃ0 {Ã0(lx{3D[x]}) & ;$h{Ã0(h(lx0D[x])) & h¢>3}}

(1) (

¤²S)²S 1 Pr lÃ0 {Ã0(lx{3D[x]}) & ;$h{Ã0(h(lx0D[x])) & h>3}}

(2)

¤²[(N1²S)²S] 2 Pr lP0 lQ1 $x { P(x) & Q(x) }

(3) N

1²S 3 As Q1

(4)

¤²S 23 2,3,MP2 lP0 $x { P(x) & Q(x) }

(5) S 123 1,4,

²O $x{3D[x]&Q(x)} & ;$h{$x{h(D)(x) & Q(x)} & h>3 }}

(6) (N1²S)²S 12 3-5,²O lQ1 $x{3D[x]&Q(x)} & ;$h{$x{h(D)(x) & Q(x)} & h>3 }}

There are possibly other problems left out in this account, but as Grice says, "leave it to implicature".

--- By courtesy of G. Hardegree.