# 廣義量詞系列：相關詞與度量結構

## 2. 相關詞

### 2.2 世界語的相關詞系統

人物／個體事物性質領屬時間空間原因方式數量

chiu (註5)
chio
chia
chies
chiam
chie
chial
chiel
chiom

iu
io
ia
ies
iam
ie
ial
iel
iom

neniu
nenio
nenia
nenies
neniam
nenie
nenial
neniel
neniom

tiu
tio
tia
ties
tiam
tie
tial
tiel
tiom

kiu
kio
kia
kies
kiam
kie
kial
kiel
kiom

kiu
kio
kia
kies
kiam
kie
kial
kiel
kiom

 Johano chiel gajnis la titolon "bravulo". (1) 約翰 以各種方法 得到 定冠詞 稱號 英雄 =「約翰用盡各種方法以得到『英雄』的稱號。」

### 2.3 數量論域

The number of male students is larger than the number of female students.

(>)(m)(f)

 Li faris iom da eraroj. (註7) 他 做 一些 介詞 錯誤 =「他犯了一些錯誤。」

John may borrow any number of books from the library.

every(QUAN)({n ∈ QUAN: n = the number of books that John may borrow from the library})

### 2.4 性質／謂詞論域

John met all kinds of people.     (2)

{P ∈ PRO: P(PEOPLE) ≠ Φ}

every({P ∈ PRO: P(PEOPLE) ≠ Φ})({P ∈ PRO: P(PEOPLE ∩ {x ∈ U: MEET(j, x)}) ≠ Φ})

(=)({P ∈ PRO: P(PEOPLE) ≠ Φ})({P ∈ PRO: P(PEOPLE ∩ {x ∈ U: MEET(j, x)}) ≠ Φ})

How is John?
What did John do to Mary?

A = {P ∈ PRED: P(j)}
A = {P ∈ PRED: P(j, m)}

### 2.5 方式論域

He examined the tissue microscopically.

John smokes heavily.
John beat Bill violently.
John gave the letter to Mary quickly.

every({j})(HEAVILY(SMOKE))
every({(j, b)})(VIOLENTLY(BEAT))
every({(j, l, m})(QUICKLY(GIVE))

John beat Bill with the stick.

every({s})({x ∈ U: WITH(x)(BEAT)(j, b)})

every(MANN)({M ∈ MANN: M(GAINI)(j, t)})

(=)(MANN)({M ∈ MANN: M(GAINI)(j, t)})     (3)

「方式量化結構」也可表現為「方式介詞 + 量化名詞組」的形式，例如以下語句：

John beat Bill with at least two sticks.     (4)

(at least 2)(STICK)({x ∈ U: WITH(x)(BEAT)(j, b)})     (5)

### 2.6 原因／因素論域

Laziness is one of the causes for John's failing the exam.
John failed the exam because of laziness.

 Nenial Johano faris tion. 無緣無故 約翰 做 此事 =「約翰無緣無故做此事。」

every({clazy})(Cause(FAIL(j, e)))
(=)({clazy})(Cause(FAIL(j, e)))
no(C)(Cause(FARI-TION(j)))

p □→ q ∧ ~p □→ ~q

John failed the exam because he was lazy.     (6)
John did not fail the exam although he was lazy.     (7)

vlazy = −kv~fail-exam ∧ k ≥ 1

vlazy = −kv~fail-exam ∧ k > 1
vlazy = −v~fail-exam

F = {v: −kv~fail-exam ∧ k ≥ 1}

C = {v: −kv~fail-exam ∧ k > 1}

### 2.7 焦點結構

John beat Bill [with the stick]F.

only(A)(B) ⇔ A ⊇ B

John only [borrowed]F the book (he didn't buy it).     (8)

only({BORROW})({P ∈ PRED: P(j, b)}) ⇔ {BORROW} ⊇ {P ∈ PRED: P(j, b)}     (9)

John didn't only [borrow]F the book (he also bought it).

every({BORROW})({P ∈ PRED: P(j, b)}) ⇔ {BORROW} ⊆ {P ∈ PRED: P(j, b)}     (10)

{BORROW} = {P ∈ PRED: P(j, b)}

## 3. 度量與相關概念

### 3.1 絕對度量

「度量」(Measure)是自然語言表達「量」的重要方式。在日常語言中，某些性質，如「長」、「重」、「熱」等物理量，都有明確和公認的計量標準，可以用某個實數代表該性質所達到的量。由此可見，「度量」實際上是前述「數量論域」的一種運用。不過，創立「向量空間語義學」的學者Winter等人利用向量的概念詮釋表達「度量」的語句，從而大大豐富了「度量」的語義學內容；而且筆者認為，我們可以把「度量」概念推廣引伸至「程度」、「比較結構」、「序數詞」、「不可數名詞」、「部分-整體關係」等多種語言結構，因此筆者特闢本節專門討論「度量」的語義問題。

V = {r • u: r ∈ [0.7, 3]}

John is short.     (11)
John is 1.3 m tall.     (12)
How tall is John?     (13)
How short is John?     (14)

TALL(e(X')) = {r • u: r ∈ R ∧ r ≥ s}
SHORT(e(X')) = {r • u: r ∈ R ∧ r < s}

Dim[HEIGHT](j)SHORT(e(X'))

TALL(e(X'))n = {r • u: r ∈ R}

 Dim[HEIGHT](j) ∈ TALL(e(X'))n ∩ {r • u: r = 1.3} = {1.3 • u}

A = {r • u: r ∈ R}
A = {r • u: r ∈ R ∧ r < s}

### 3.2 相對度量

「相對度量」(Relative Measure)表達「度量」之間的比較，典型的例句包括以下的「比較句」：

John is taller than Mary.     (15)
Mary is shorter than John.     (16)
Mary is less tall than John.     (17)
John is 0.1 m taller than Mary.     (18)

「向量空間語義學」使用「定位向量」(Located Vector)的概念來表達「相對度量」，「定位向量」就是不一定以「原點」為「起點」的向量。為了方便以下討論，我們假設所有「定位向量」都是垂直向上或向下的，並把以點p為「起點」的「單位定位向量」記作up。為讓讀者易於明白，以下首先解釋如何用「定位向量」表達語句(15)。從直觀上說，該句的意思就是存在一個代表John高度的「度量向量」Dim[HEIGHT](j)，其長度大於代表Mary高度的「度量向量」Dim[HEIGHT](m)。我們亦可以從另一個角度理解該句，Dim[HEIGHT](j)等於Dim[HEIGHT](m)與另一個「定位向量」w的「向量和」，這個wDim[HEIGHT](m)的「終點」為「起點」，因此亦可寫作x • uE-point(Dim[HEIGHT](m)) (其中x為實數)，其方向指向上(即x > 0)(請參看下圖)(註15)：

Sign(x) = Polarity(A)     (19)

 Dim[HEIGHT](j) ∈ {v ∈ V: v = Dim[HEIGHT](m) + x • uE-point(Dim[HEIGHT](m)) ∧ Sign(x) = Polarity(TALL)} = {v ∈ V: v = Dim[HEIGHT](m) + x • uE-point(Dim[HEIGHT](m)) ∧ x > 0}     (20)

Sign(x) = −Polarity(A)     (21)

Dim[HEIGHT](j)∈ {Dim[HEIGHT](m) + 0.1 • uE-point(Dim[HEIGHT](m))}

### 3.3 模糊集合與程度修飾語

John is tall.

μ[TALL(e(X'))](|Dim[HEIGHT](j)|)

「程度修飾語」可分為幾類，本文只擬討論其中兩類。第一類「程度修飾語」包括"completely" 、 "partially"、"half"、"to a certain extent"、"to a certain extent not"、"not at all"等，這類修飾語所表達的意義相當於某種比例(例如"completely"便相當於100%)，故可稱為「比例修飾語」(Proportional Modifier)。請注意「比例修飾語」所表達的意義是不模糊的，所以我們可以直接使用「度量函項」來表達含有這類修飾語的語句。此外，由於「比例修飾語」具有副詞的性質，我們應根據它們對「度量形容詞」的作用來確定它們的語義。下表列出各個「比例修飾語」的真值條件(在下表中，A代表「度量形容詞」，N代表與A相應的「度量名詞」，u代表個體)：

COMPLETELY(A)(e(X'))(u)
|Dim[N](u)| = 1
PARTIALLY(A)(e(X'))(u)
0 < |Dim[N](u)| < 1
HALF(A)(e(X'))(u)
|Dim[N](u)| = 0.5
TO-A-CERTAIN-EXTENT(A)(e(X'))(u)
|Dim[N](u)| > 0
TO-A-CERTAIN-EXTENT-NOT(A)(e(X'))(u)
|Dim[N](u)| < 1
NOT-AT-ALL(A)(e(X'))(u)
|Dim[N](u)| = 0

John is partially reliable.

PARTIALLY(RELIABLE)(e(X'))(j)

0 < |Dim[RELIABILITY](j)| < 1

μ[TRUTH](VERY(A)(e(X'))(u)) = F1(|Dim[N](u)|)

 0, if 0 ≤ x ≤ 0.8 F1(x) = (x − 0.8) / 0.1, if 0.8 ≤ x ≤ 0.9 1, if 0.9 ≤ x ≤ 1

John is very reliable.

μ[TRUTH](VERY(RELIABLE)(e(X'))(j)) = F1(|Dim[RELIABILITY](j)|)

### 3.4 度量與程度概念的推廣

John runs quickly.

μ[QUICKLY(RUN)](|Dim[SPEED](RUN(j))|)

μ[TRUTH](VERY(M)(P)(u)) = F1(|Dim[N](P(u))|)

John is dressed very neatly.

μ[TRUTH](VERY(NEATLY)(DRESSED)(j)) = F1(|Dim[NEATNESS](DRESSED(j))|)

### 3.5 比較結構

John is the tallest student in his class.

every(CLASS–{j})({x ∈ U: |Dim[HEIGHT](j)| > |Dim[HEIGHT](x)|})

John is taller than at least one of his classmates. (註18)

some(CLASS–{j})({x ∈ U: |Dim[HEIGHT](j)| > |Dim[HEIGHT](x)|})

most(CLASS–{j})({x ∈ U: |Dim[HEIGHT](j)| > |Dim[HEIGHT](x)|})

John is taller than most of his classmates.

### 3.6 序數詞

「數詞」(Numeral)一般可分為「基數詞」(Cardinal Numeral)和「序數詞」(Ordinal Numeral)這兩種，前者表達數目多少，例如英語的"one"、"two"、"hundred"、"thousand"等；後者表達次序，例如英語的"first" 、 "second"、"hundredth"、"thousandth"等。「基數詞」是「數量論域」的元素，在前面已討論了很多，本節集中討論「序數詞」。

John is the second tallest student in his class.

(all except 1)(CLASS–{j})({x ∈ U: |Dim[HEIGHT](j)| > |Dim[HEIGHT](x)|})

John is the nth tallest student in his class.

(all except n − 1)(CLASS–{j})({x ∈ U: |Dim[HEIGHT](j)| > |Dim[HEIGHT](x)|})

「序數詞」並不局限於表達性質之間的比較，它們更常用於表達空間和時間位置，例見以下語句：

John is the second person in the queue.
Mary was the third student to arrive.

(all except 1)(QUEUE–{j})({x ∈ U: (in front of)(Loc(j), Loc(x))})
(all except 2)(STUDENT–{m})({x ∈ U: Time(ARRIVE(m)) < Time(ARRIVE(x))})

John is the first person in the queue.
Mary is the last student to arrive.

every(QUEUE–{j})({x ∈ U: (in front of)(Loc(j), Loc(x))})
no(STUDENT–{m})({x ∈ U: Time(ARRIVE(m)) < Time(ARRIVE(x))})

### 3.7 不可數名詞

Cresswell的"The Semantics of Degree"一文是研究與「度量」有關的語義問題的早期文獻，上面3.1小節介紹的把「度量」處理成向量的方法可以說是對Cresswell文中某些概念的改良和發展。Cresswell的文章除了討論「度量」的表達形式外，還把這種表達方式推廣至「可數名詞」(Count Noun)和「不可數名詞」(Mass Noun)，以表達這兩種名詞的量。由此可見，「度量」與「可數名詞」和「不可數名詞」的量其實有某種相通之處。以下沿襲Cresswell一文的精神，把前面有關「度量」的某些概念推廣應用於這兩種名詞。

「可數名詞」的量可分為「絕對量」和「相對量」兩種，「絕對量」就是代表該名詞的集合的基數。利用前面3.1小節介紹的概念，我們可以用向量Dim[MODULUS](A)代表集合A的基數，即

|Dim[MODULUS](A)| = |A|

|Dim[PROPORTIONB](A)| = |A| / |B|

All that John drank was water.     (22)

{x ∈ U: DRINK(j, x)} ⊆ WATER

{x ⊆ U: DRINK(j, x)}

∪{x ⊆ U: DRINK(j, x)}

Bunt亦把上式寫為

[x ⊆ U: DRINK(j, x)]

[x ⊆ U: DRINK(j, x)] ⊆ WATER     (23)

There is 500 ml of water.

|Dim[QUANTITY](WATER)| = 500

|Dim[PROPORTIONB](A)| = |Dim[QUANTITY](A)| / |Dim[QUANTITY](B)|     (24)

John drank at least half of the water.

|Dim[PROPORTIONX ∩ WATER]([x ⊆ X ∩ WATER: DRINK(j, x)])| > 0.5

### 3.8 部分-整體關係

John's whole body is soaked through.     (25)

The whole set of furniture is new.
The whole family supports him.

JOHN'S-BODY ⊆ [x ⊆ JOHN'S-BODY: SOAKED(x)]

JOHN'S-BODY = [x ⊆ JOHN'S-BODY: SOAKED(x)]

|Dim[PROPORTIONJOHN'S-BODY]([x ⊆ JOHN'S-BODY: SOAKED(x)])| = 1

## 4. 混合量化結構

John must return at least two books tomorrow.

WD** = World(Time(|BOOK ∩ {x ∈ U: RETURN(j, x)}| ≥ 2) ⊆ tomorrow(X'))

「混合量化結構」的「轄域」可能會隨著所用詞匯的位置分佈、所用的語言結構、句子焦點的所在位置等因素而變化，由於這將涉及很複雜的問題，討論只能到此為止。