廣義量詞系列：時間量化結構

John ∈ MUST-SING

2. 基本概念

A ∩ B ≠ Φ

some(A)(B)

2.2.2 數字的比較和運算

「三分結構」不僅可用來表達集合之間的關係，也可用來表達數字之間的比較和運算，這是因為根據集合論中有關「基數」的理論，「自然數」可以表達為某些特殊集合。舉例說，0、1、2便可分別表述為Φ、{Φ}、{Φ, {Φ}}，這樣自然數之間的「<」關係便可以表達為元素與集合之間的「∈」關係，例如1 < 2便可以表達為{Φ} ∈ {Φ, {Φ}}。另外，由於a ∈ A可以改寫為{a} ⊆ A，所以上式又可以改寫為{{Φ}} ⊆ {Φ, {Φ}}。現在如果我們用量詞"every"來代表集合之間的「⊆」關係，那麼我們便最終可以把1 < 2表達為以下的「三分結構式」：

every({{Φ}})({Φ, {Φ}})

S({Φ}) = {Φ} ∪ {{Φ}} = {Φ, {Φ}}

n + 0 = n
n + S(m) = S(n + m)

(<)(x)(y)
(=)(x + y)(z)

2.2.3 泛化量化結構

every({j})(SING)

every(U)(GOOD(e(X')))
(no ... except John)(U)(SING)

U ⊆ GOOD(e(X')) ⇔ U = GOOD(e(X'))
U ∩ SING = {j} ⇔ SING = {j}

There is no unicorn.
There are exactly 7 continents.
There are some students playing in the park.

no(UNICORN)(U) ⇔ UNICORN = Φ
(exactly 7)(CONTINENT)(U) ⇔ |CONTINENT| = 7
some(STUDENT ∩ {x: x is playing in the park})(U) ⇔ |STUDENT ∩ {x: x is playing in the park}| > 1

There is nobody in the room.

no(PERSON ∩ {x: x is in the room})(U) ⇔ PERSON ∩ {x: x is in the room} = Φ

2.2.4 結構化量詞的三分結構

(more ... than ...)(A, B)(C) ⇔ |A ∩ C| > |B ∩ C|
(the same ... as ...)(A)(B, C) ⇔ A ∩ B = A ∩ C

(>)(|A ∩ C|)(|B ∩ C|)
(=)(A ∩ B)(A ∩ C)

more(A ∩ C)(B ∩ C)
(the same)(A ∩ B)(A ∩ C)

more(A)(B) ⇔ |A| > |B|

2.2.5 迭代量詞的三分結構

|A ∩ {x: B ∩ {y: C ⊆ {z: D(x, y, z)}} = Φ}| / |A| > 0.5

most(A)({x: no(B)({y: every(C)({z: D(x, y, z)})})})

(more ... than ... some ... no)((A, B), (C, D))(E, F) ⇔
|A ∩ {x: C ∩ {z: E(x, z)} ≠ Φ}| > |B ∩ {y: D ∩ {w: F(y, w)} = Φ}|

more[A ∩ {x: some(C)({z: E(x, z)})}][B ∩ {y: no(D)({w: F(y, w)})}]

2.2.6 含複雜定語語句的三分結構

Every boy whom some girl loves is happy.

(BOY ∩ {x: GIRL ∩ {y: LOVE(y, x)} ≠ Φ}) ⊆ HAPPY(e(X'))

every[BOY ∩ {x: some(GIRL)({y: LOVE(y, x)})}][HAPPY(e(X'))]

At least 5 teachers' books are all boring.

At least 5 teachers each of whom has some book are such that for each of such teachers, all of his / her books are boring.

(at least 5)(TEACHER ∩ {x: some(BOOK)(OWNx)})({x: every(BOOK ∩ OWNx)(BORING(e(X')))})

Every car of every member is red.

every(MEMBER ∩ {x: some(CAR)(OWNx)})({x: every(CAR ∩ OWNx)(RED(e(X')))})

Every car which is owned by every member is red.

every(CAR ∩ {y: every(MEMBER)({x: OWN(x, y)})})(RED(e(X')))

3. 時間論域

3.1 時刻、時段和時段集合

「時態邏輯」和形式語義學的早期理論「蒙太格語法」均以「時刻」(Moment)作為「時間論域」的基本元素，這種處理方法的優點是簡單直觀。由於具有一維性的時間與「實數」(Real Number)存在對應關係，我們可以把「時刻」比擬為「實數軸」(Real Number Line)上的點。可是以「時刻」為基礎的理論有時難以表達複雜的時間量。以語句

[2001.1.1, 2001.1.31]d (註2)

{2001.1.1, 2001.1.2, ... 2001.1.31}

John sang yesterday.
yesterday(X') = {i: i ∈ 語境X'中的說話時間的前一天}
= {2000.12.31}
John went on a trip between December 28th and 29th.
He had a happy time on those two days.
(on those 2 days)(X') = {2000.12.28, 2000.12.29}

Time(SING(j)) (註3)     (1)

3.2 事件和事件集合

e1: PROPOSITION(e1) = SING(j) ∧ TIME(e1) = [10:00, 10:11]min (註4)

Time(SING(j)) = [10:00, 10:10]min ∪ [10:30, 10:37]min ∪ [11:00, 11:10]min     (2)

{e ⊆ T: PROPOSITION(e) = SING(j)} = {e1, e2, e3}     (3)

e4
HUG(j, m)
[10:05, 10:11]min
e5
HUG(b, t)
[10:05, 10:11]min
e6
HUG(a, c)
[10:05, 10:11]min

3.3 時間論域上的關係和運算

i < i'
i先於i'
i > i'
i後於i'
i = i'
i與i'同時

Initial([10:00, 10:10]min) = 10:00
Final([10:00, 10:10]min) = 10:10

I < I'
I先於I'，即Final(I) < Initial(I')
I > I'
I後於I'，即Initial(I) > Final(I')
I ⊆ I'
I被包含於I'，即
Initial(I) ≥ Initial(I') ∧ Final(I) ≤ Final(I')

|I| = (Final(I) − Initial(I)) + 1

|I| = (Final(I) − Initial(I)) + 1 = (10:10 − 10:00) + 1 = 11 min

 |Time(SING(j))| = |[10:00, 10:10]min| + |[10:30, 10:37]min| + |[11:00, 11:10]min| = 11 min + 8 min + 11 min = 30 min

|{e ⊆ T: PROPOSITION(e) = SING(j)}| = |{e1, e2, e3}| = 3

TIME(e7) < TIME(e8)
Initial(TIME(e7)) = 2001.1.1
|(TIME(e7)| = 2 d

The committee holds a meeting twice a year.
The committee holds a meeting every half-year.

Period(E) = mine, e' ∈ E ∧ e ≠ e' (|Initial(TIME(e)) − Initial(TIME(e'))|)

Frequency(E) = 1 / Period(E)

Period(MEETING) = mine, e' ∈ MEETING ∧ e ≠ e' (|Initial(TIME(e)) − Initial(TIME(e'))|) = 181 d ≈ 6 mon = 1/2 yr
Frequency(MEETING) = 1 / Period(MEETING) = 1/181 d−1 ≈ 1/6 mon−1 = 2 yr−1

4. 時間量化結構

always
every
often
many
sometimes
some
mostly
most
seldom
few
never
no
at least twice
(at least 2)

4.2 含有一般量詞的結構

John wore uniform all day yesterday.     (4)

every(yesterday(X'))(Time(WEAR-UNIFORM(j)))

John wore T-shirt yesterday.     (5)

every(Time(WEAR-T-SHIRT(j))(yesterday(X'))     (6)

some(Time(WEAR-T-SHIRT(j))(yesterday(X'))     (7)

John wore the T-shirt.
John wore a T-shirt.

4.3 泛化量化結構

The universe is always expanding.
God never dies.

every(T)(Time(EXPAND(u)))
no(T)(Time(DIE(g)))

John wins every time.

every(T)({e: PROPOSITION(e) = WIN(j)})

「第二類泛化量化結構」則具有Q(A)(T)的形式，這種結構可用來表達英語中涉及時間或「名詞性現在分詞」(即傳統語法所稱的「動名詞」Gerund)的「存在句」。試看以下例句：

There were many huggings in the party.

many(HUGGING ∩ {e: TIME(e) ⊆ PARTY})(T)

4.4 絕對時間量的表達法

John has slept for less than 6 hours.
John has won the game for at least 3 times.     (8)
John brushes his teeth (exactly) twice a day.
John visits Mary fortnightly.

(<)(|Time(SLEEP(j))|)(6 h)
(≥)(|{e: PROPOSITION(e) = WIN-GAME(j)}|)(3)
(=)(Frequency({e: PROPOSITION(e) = BRUSH-TEETH(j)}))(2 d−1)
(=)(Period({e: PROPOSITION(e) = VISIT(j, m)}))(14 d)

(at least 3)({e: PROPOSITION(e) = WIN-GAME(j)})(T)

John won the game for at least 3 times last year.     (9)

(at least 3)({e: PROPOSITION(e) = WIN-GAME(j)})({e: TIME(e) ⊆ (last year)(X')})

4.5 含有結構化量詞或疑問量詞的結構

「時間量化結構」亦可以包含複雜的量詞，包括「結構化量詞」和「疑問量詞」。先看以下語句：

John spent more time playing than studying yesterday.
John arrived home at the same time as Mary did yesterday.

more(yesterday(X') ∩ Time(PLAY(j)))(yesterday(X') ∩ Time(STUDY(j)))
(the same)(yesterday(X') ∩ Time(ARRIVE-HOME(j)))(yesterday(X') ∩ Time(ARRIVE-HOME(m)))

《廣義量詞系列：基本單式量詞》中介紹的大多數「結構化量詞」都可應用於「時間量化結構」中，這裡不作深入討論。

When did John sing yesterday?
How long did John sing yesterday?
How many times did John win the game last year?

whatd(yesterday(X'))(Time(SING(j)))(A)
A = yesterday(X') ∩ Time(SING(j))
(how many)(yesterday(X'))(Time(SING(j)))(A)
A = |yesterday(X') ∩ Time(SING(j))|
(how many)({e: TIME(e) ⊆ (last year)(X')})
({e: PROPOSITION(e) = WIN-GAME(j)})(A)
A = |{e: TIME(e) ⊆ (last year)(X')} ∩
{e: PROPOSITION(e) = WIN-GAME(j)}|

How often did John visit Mary?

A = Frequency({e: PROPOSITION(e) = VISIT(j, m)})

John在哪一次探望Mary時認識Bill？     (10)

A = VISITING ∩ {e': KNOWING ∩ DURING−1e' ≠ Φ}

4.6 含有迭代量詞的結構

John goes to church every Sunday.     (11)

DURING = {(e, e'): TIME(e) ⊆ TIME(e')}
DURING−1e' = {e: DURING(e, e')}

every(SUNDAY)({e': GOING-TO-CHURCH ∩ DURING−1e' ≠ Φ})

every(SUNDAY)({e': some(GOING-TO-CHURCH)(DURING−1e')})

John prays before having meals.     (12)

SHORTLY-BEFORE = {(e, e'): TIME(e) < TIME(e') ∧ Initial(TIME(e')) − Final(TIME(e)) < s}
SHORTLY-BEFORE−1e' = {e: SHORTLY-BEFORE(e, e')}

every(HAVING-MEAL)({e': PRAYING ∩ SHORTLY-BEFORE−1e' ≠ Φ})

every(HAVING-MEAL)({e': some(PRAYING)(SHORTLY-BEFORE−1e')})

5. 時間論域上的其他量化現象

5.1 時與體

[F(P(GO(j)))]i ⇔ ∃i'(i < i' ∧ ∃ i''(i'' < i' ∧ [GO(j)]i''))

[Prog(SING(j))]I ⇔ ∃I'(I ⊂ I' ∧ [SING(j)]I')

[Short(LOOK(j, m))]I ⇔ ∃I'(I' ⊂ I ∧ ~(I' ⊂initial I) ∧ ~(I' ⊂final I) ∧ [LOOK(j, m)]I')

5.2 動詞的情狀類型與語義分解

「情狀類型」(Situation Type)是對動詞的一種分類，這種分類主要根據動詞所表達的動作行為是否具有動態性、持續性和明確的終結點而作出，最初由Vendler提出，後經Smith等人改良。英語的動詞可分為五種「情狀類型」－「狀態動詞」(Stative Verb)、「完結動詞」(Accomplishment Verb)、「活動動詞」(Activity Verb)、「實現動詞」(Achievement Verb)和「單動作動詞」(Semelfactive Verb)，各類動詞的例子分別為"know"、"run a mile"、"run"、"die"和"knock"。

John has been running for an hour.
John ran a mile in one hour.

[p]I ⇒ ∀I'(I' ⊆ I ⇒ [p]I')

「語義分解」(Semantic Decomposition)是指利用少數幾個代表基本語義的詞項來界定其他詞匯的語義，這種方法曾經成為「生成語義學」(Generative Semantics)的重要分析方法。舉例說，該學派便曾以下列方式界定"die"和"kill"的語義：

die = BECOME(~ALIVE)
kill = CAUSE(BECOME(~ALIVE))

 [BECOME(p)]I ⇔ (1) ∃J (Initial(I) ⊆ J ∧ [~p]J) ∧ (2) ∃K (Final(I) ⊆ K ∧ [p]K) ∧ (3) ~∃I' (I' ≠ Φ ∧ I' ⊂ I ∧ I'滿足(1)和(2)中I的條件)

5.3 事件推理、內崁分句與題元角色

「事件語義學」區別於形式語義學其他分支理論的最大特點是，以「事件」作為基本論元。在這種理論下，表達普通事件的語句成為「存在量化句」。舉例說，語句"John gave the book to Mary yesterday."便可表達為(在下式中，b和y分別代表當前語境所指的"the book"和"yesterday")：

∃e (PREDICATE(e) = GIVE ∧ AGENT(e) = j ∧ PATIENT(e) = b ∧ RECIPIENT(e) = m ∧ TIME(e) = y)

p ∧ q ⇒ p

John gave the book to Mary yesterday. ⇒ The book was given to Mary.

John did something that caused Bill to work.

∃e (AGENT(e) = j ∧ ∃e' (PROPOSITION(e') = WORK(b) ∧ CAUSE(e, e')))

「感知陳述」的例句如"John knows that Bill left."，這一句同樣包含兩個事件："John knows something."和"Bill left."，而且第二個事件充當第一個事件的其中一個題元角色。基於上述分析，我們可以把上句表達為：

∃e ∃e' (PROPOSITION(e) = KNOW(j, e') ∧ PROPOSITION(e') = LEAVE(b))

「事件語義學」的另一個特點是，把「題元角色」視為原始概念作為「事件」定義的一部分。可是，有關「題元角色」的定義、種類和數目等等，這些都是極富爭議性的問題，學界尚未有定論。那麼在定義「事件」時，我們如何確定某一類「事件」應包含哪些「題元角色」？例如，我們如何確定VISIT類「事件」包含「施事」和「受事」這兩個角色，而KNOW類「事件」則包含「感知者」和「主題」這兩個角色？對此問題的其中一個簡單答案是，這些「題元角色」是從眾多同類動詞的共同語義中歸納出來的。我們可以把這種歸納結果表達為「意義公設」。舉例說，對於VISIT和KNOW這兩種「事件」，我們有以下「意義公設」：

∀e (PREDICATE(e) = VISIT ⇒ ∃x (AGENT(e) = x) ∧ ∃y (PATIENT(e) = y))
∀e (PREDICATE(e) = KNOW ⇒ ∃x (EXPERIENCER(e) = x) ∧ ∃y (THEME(e) = y))

{i ∈ T: SING(i)[j(i)]}

e1: SING(e1) ∧ AGENT(e1) = j ∧ TIME(e1) = [10:00, 10:11]min

Fc = {y: F(c, y)}

F−1d = {x: F(x, d)}