The syntax of numbers and number words
1. The External Syntax of Numbers.
Post Determiners
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Numbers are post-determiners, whether cardinal, ordinal, or fractions.
The hundred old men
The three new cups
five pencils
The hundredth man
The fifth pencil
The two-thirds cups of flour.
It would be tempting to assign number words like "one", "five", "ten",
"hundred", "thousand", "million" and their ordinal and denominator forms to
that class. But number-words are better regarded as the building blocks of
post-determiners.
In an expression like "three hundred and fifty-three cups of coffee", "three
hundred and fifty-three" is the post-determiner.
Agreement for number words
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A. Cardinal
All cardinal numbers determine plural nouns except "one".
two men
*two man
five million three hundred and forty-three men
*five million three hundred and forty-three man
*one men
one man
But when one is part of a larger number the number determines a plural
noun:
five million three hundred and forty-one men
*five million three hundred and forty-one man
Similarly, when cardinal numbers appear with elided heads number for
subject/verb agreement parallels the restrictions noted above.
*Three men are found after the crash; two dies.
Three men are found after the crash; two die.
Three men are found after the crash; one dies.
*Three men are found after the crash; one die.
The code "agreement=singular;full_cardinal" is added to the entry for
"one" to indicate this special behavior.
B. Ordinal:
Ordinal numbers determine singular nouns:
the second man
*the second men
the three hundred and forty third man
*the three hundred and forty third men
except "first" which determines both singular and plural nouns:
the first man
the first men
but only when first is a full cardinal number.
the three hundred and forty first man
*the three hundred and forty first men
So the code "agreement=plural;full_ordinal" marks this exceptional behavior.
C. Fractions
Fractions all determine plural nouns, and in head elided constructions
agree as plurals.
Only three fifths plates of spinach were eaten.
*Only three fifths plate of spinach was eaten.
Three men are found, and three fifths die.
Three men are found, and three fifths dies.
So no lexical agreement= codes are needed to distinguish special behavior.
The only lexical record marked with "agreement=" codes is "one":
{base=one
cat=number_word
variant=first;ordinal
number_type=unit
agreement=singular;full_cardinal
agreement=plural;full_ordinal
}
2. The Internal Syntax of numbers and number words.
Numbers are of three types: Cardinal, Ordinal and Fractional. Each is
constructed out of number words. A number word alone (properly inflected)
can be a cardinal or ordinal number, but not a fraction. There may be many
ways to describe the gammar of these numbers, The grammar below is one
attempt. The grammar below descibes the american system and would need
modification to cover the British system.
A. Cardinal Numbers:
First we need lexical classes of number words. Membership in these
classes is indicated in the lexical entries for number words in a
number_type=slot.
"ten" and "hundred" are unique words, each in it's own class, I'll
just mention them by name.
units= {one,two,three,four,five,six,seven,eight,nine}
1. morphologically simple
2. occur in environment _ hundred
Units are marked number_type=unit in the lexicon.
teens= {eleven, twelve,thirteen,fourteen,fifteen,sixteen, seventeen,
eighteen, nineteen}
1. morphologically: unit+teen
2. occur in envrionment __ hundred
Teens are marked "number_type=teen" in the lexicon.
decades = {twenty, thirty,forty, fifty, sixty, seventy, eighty,
ninety}
1. morphologically unit+ty
2. cannot occur in __ hundred
3. occur in environment __ unit.
* ten five
* one five
twenty five
Decades are marked "number_type=decade" in the lexicon.
Magnitude words:
1. morphologically: latin number + illion
2. occur in a specified order with respect to
each other.
*five million ten billion.
*six trillion three thousand.
Magnitude words are have the lexical code "number_type=magnitude" with
an additional slot "power=". When power=N 1000^N is the value. Numbers
in the power= slot are also used to maintain the order of magnitude
words. Notice that the denotation of these words and their etymology
are off by one. Bi-illion is power 3. "hundred" is morphologically
different from the other magnitude words but semantically similar.
magnitude_word
hundred power=0
thousand power=1
million power=2
billion power=3
trillion power=4
quadrillion power=5
quintillion power=6
sextillion power=7
septillion power=8
octillion power=9
nonillion power=10
decillion power=11
undecillion power=12
duodecillion power=13
tredecillion power=14
quattuordecillion power=15
quindecillion power=16
sexdecillion power=17
septendecillion power=18
octodecillion power=19
novemdecillion power=20
vigintillion power=21
centillion power=101
The Grammar:
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Some terminology:
Basic cardinal numbers represent values less than 100.
A cardinal number of rank N (Card_Number(N)) means a cardinal number
representing a value between 1,000 to the Nth power and 1,000 to the
N+1th power. The cardinal numbers of rank 0 represent numbers in the
hundreds. The cardinal numbers of rank 1 are in the thousands and
those of rank 2 are in the millions, 3 in the billions etc. Cardinal
numbers of rank 21 are in the vigintillions.
Card_Number( can be read "consists of".
Rules:
________
Rule 1: Basic Cardinal Numbers
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Basic_Card_Number ==> {unit, "ten", teen, decade(+("-")+unit)}
A basic cardinal number may be a unit alone, a teen alone, the number
"ten" or a decade optionally followed by a unit. A dash "-" may occur
between the decade and the unit.
e.g.
five unit
ten "ten"
twelve teen
forty decade
twenty five decade+unit
forty-six decade+"-"+unit
Only decades may be followed by a unit:
*ten four
*fifteen six
Rule 2: Cardinal Numbers of Rank 0
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Card_Number(0) ==> unit + hundred + ((and) + Basic_Card_Number)
A cardinal number of rank 0 consists of a unit or decade+unit followed
by "hundred" optionally followed by a basic cardinal number. "and" may
optionally occur before the basic cardinal number
e.g.
five hundred unit+hundred
six hundred and seventy unit+hundred+basic cardinal number
Since "hundred" is Magnitude(0) this rule can almost be assimilated to
the rule schema below.
Rule 3: Cardinal Numbers of Rank 1
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Card_Number(1)==>{teen,decade+unit}+hundred + ((and) + Basic_Card_Number)
A cardinal number of rank 1 may consist of the word "hundred" preceded
by a teen or decade and unit, optionally followed by a basic cardinal
number. The basic cardinal number may be preceded by "and".
e.g.
twelve hundred teen+hundred
twenty five hundred and ten decade+unit+hundred+basic cardinal number
The decade must be followed by a unit: *twenty hundred,
"ten" cannot precede hundred either: *ten hundred.
Since the following rule schema also covers cardinal numbers of rank 1,
this rule creates a systematic homonymy among cardinal numbers of
rank 1.
e.g.
fifteen hundred == one thousand five hundred
twenty five hundred and six == two thousand five hundred and six
Rule schema for Cardinal Numbers of Rank > 0;
___________________________________________
Card_Number(N) ==> Card_Number(<1) + Magnitude(N)-word +
({(and) + Card_Number( 0.
A cardinal number of rank N (N>0) consists of a magnitude word of
power N preceded by a cardinal number of rank <1 and optionally
followed by a cardinal number of rank decade+"="+unit
Basic_Card_Number
______|______
| | |
decade - unit
| | |
twenty - three
Card_Number(0) ==> unit + hundred + and + Basic_Card_Number
Card_Number(0)
________|_______________
| | | |
unit hundred and Basic_Card_Number
| | | |
| | | unit
| | | |
one hundred and two
Card_Number(1) ==>
Cardinal_Number(0) + Magnitude(1)-word + Card_Number(0)
Card_Number(1)
________________|_______________
| | |
Card_Number(0) Magnitude(1)-word Card_Number(0)
________|_________ | ________|__________________
| | | | | | | | | |
unit hundred and unit | unit hundred and decade unit
| | | | | | | | | |
one hundred and two thousand four hundred and fifty six
Card_Number(1) ==>
teen+hundred+basic cardinal number
Cardinal_Number(1)
__________|_________________
| | | |
teen "hundred" "and" Basic_Card_Number
| | | |
| | | unit
| | | |
fifteen hundred and six
Card_Number(1) ==>
Cardinal_Number(0) + magnitude(1)-word + and + Basic_Card_Number
Card_Number(1)
_________________|_______________________
| | | |
Card_Number(0) Magnitude(1)-word and Basic_Card_Number
| | | |
basic_number | | |
| | | |
unit | | unit
| | | |
one thousand and six
B. Ordinal Numbers:
Ordinal numbers are identical to cardinal numbers except that the
right most number word is in Ordinal form:
Cardinal Ordinal
one first
two second
twenty-three twenty-third
one thousand and six one thousand and sixth
C. Fractions
A fraction consists of numerator followed by a denominator. The
numberator is just a cardinal number. The denominator is identical to
a cardinal number except that its right most number word in in
denominator form. In most cases the denominator form is the same as
the ordinal form. The exceptions are "one" which has no denominator
form, and "two" which has a special full_denominator form when it
exhausts the denominator.
The rule is:
Fraction ==> cardinal(alpha number) + denominator(alpha number)
(alpha number) indicates that the when the numerator is a number
other than "one" the denominator must be in plural denominator form.
one-third
two-thirds
one five hundred and thirty-second
two five hundred and thiry-seconds
five thousand three millionths